[Next][Prev] [Right] [Left] [Up] [Index] [Root]

Ideals

Ideals for function field orders O are O-modules I ⊆F for which there is a d ∈F such that dI ⊆O is a non-zero ideal of O, that is they are fractional ideals of O. Over the coefficient ring of O they are also free modules of rank n, where n equals the degree [F:k(x, α1, ..., αr)].

Subsections

Creation of Ideals

ideal< O | a1, a2, ... , am > : RngFunOrd, RngElt, ..., RngElt -> RngFunOrdIdl
Given an order O, as well as elements a1, a2, ..., am coercible into the field of fractions F of O, create the fractional ideal of O generated by these elements.

Note that, contrary to the general case for the constructors, the right hand side elements are not necessarily contained in the left hand side.

ideal< O | T, d > : RngFunOrd, AlgMatElt, RngElt -> RngFunOrdIdl
ideal< O | T, d > : RngFunOrd, ModDed, RngElt -> RngFunOrdIdl
The ideal of the order O of an algebraic function field whose basis is the matrix or dedekind module T over the coefficient ring of O divided by the element d of the denominator ring of O.
ideal< O | T, S > : RngFunOrd, AlgMatElt, [RngFunOrdIdl] -> RngFunOrdIdl
ideal< O | T, I1, ..., In > : RngFunOrd, AlgMatElt, RngFunOrdIdl, ..., RngFunOrdIdl -> RngFunOrdIdl
The ideal of the order O of an algebraic function field whose basis is the matrix T over the coefficient ring of O along with the coefficient ideals I1, ..., In or those in S.
x * O : RngElt, RngFunOrd -> RngFunOrdIdl
O * x : RngFunOrd, RngElt -> RngFunOrdIdl
Create the ideal x * O where x is coercible into the function field of the order O.
Ideal(P) : PlcFunElt -> RngFunOrdIdl
Create a prime ideal corresponding to the place P.
Ideals(D) : DivFunElt -> RngFunOrdIdl, RngFunOrdIdl
Create two ideals of the `finite' and `infinite' maximal order respectively corresponding to the divisor D.
O !! I : RngFunOrd, RngFunOrdIdl -> RngFunOrdIdl
Return the ideal I as an ideal of the order O.

Parent and Category

Parent(I) : RngFunOrdIdl -> PowIdl
Category(I) : RngFunOrdIdl -> Cat

Arithmetic Operators

I + J : RngFunOrdIdl, RngFunOrdIdl -> RngFunOrdIdl
I * J : RngFunOrdIdl, RngFunOrdIdl -> RngFunOrdIdl
I / J : RngFunOrdIdl, RngFunOrdIdl -> RngFunOrdIdl
I ^ k : RngFunOrdIdl, RngIntElt -> RngFunOrdIdl
The ideal J is required to be invertible. The ideal I is required to be invertible for negative k.

c * I : RngElt, RngFunOrdIdl -> RngFunOrdIdl
I * c : RngFunOrdIdl, RngElt -> RngFunOrdIdl
I / c : RngFunOrdIdl, RngElt -> RngFunOrdIdl

c / I : RngElt, RngFunOrdIdl -> RngFunOrdIdl
The principal ideal generated by the ring element c divided by the ideal I.
IdealQuotient(I, J) : RngFunOrdIdl, RngFunOrdIdl -> RngFunOrdIdl
ColonIdeal(I, J) : RngFunOrdIdl, RngFunOrdIdl -> RngFunOrdIdl
The colon ideal [I:J] of elements which multiply all elements of the ideal J into the ideal I.
ChineseRemainderTheorem(I1, I2, e1, e2) : RngFunOrdIdl, RngFunOrdIdl, RngFunOrdElt, RngFunOrdElt -> RngFunOrdElt
CRT(I1, I2, e1, e2) : RngFunOrdIdl, RngFunOrdIdl, RngFunOrdElt, RngFunOrdElt -> RngOrdElt
Returns an element e of the order O such that (e1 - e) is in the ideal I1 of O and (e2 - e) is in the ideal I2.

Roots of Ideals

IsPower(I, n) : RngFunOrdIdl, RngIntElt -> BoolElt, RngFunOrdIdl
Return whether the ideal I has an nth root and if so return an nth root.
Root(I, n) : RngFunOrdIdl, RngIntElt -> RngFunOrdIdl
Return the nth root of the ideal I.
IsSquare(I) : RngFunOrdIdl -> BoolElt, RngFunOrdIdl
Return whether the ideal I is a square and if so return a square root.
SquareRoot(I) : RngFunOrdIdl -> RngFunOrdIdl
Sqrt(I) : RngFunOrdIdl -> RngFunOrdIdl
Return a square root of the ideal I.

Example FldFunG_ideal-is-square (H42E30)

A simple creation of an ideal and the use of IsSquare is shown below.

> P<x> := PolynomialRing(GF(79));
> P<y> := PolynomialRing(P);
> Fa<a> := FunctionField(y^2 - x);
> P<y> := PolynomialRing(Fa);
> Fb<b> := FunctionField(y^2 - a);
> P<y> := PolynomialRing(Fb);
> Fc<c> := FunctionField(y^2 + a*b);
> I := a*b*c*MaximalOrderInfinite(Fc);
> IsSquare(I^2);
true Fractional ideal of Maximal Order of Fc over Maximal Order of Fb over 
Maximal Equation Order of Fa over Valuation ring of Univariate rational function
field over GF(79) with generator 1/x
Basis:
Pseudo-matrix over Maximal Order of Fb over Maximal Equation Order of Fa over 
Valuation ring of Univariate rational function field over GF(79) with generator 
1/x
Fractional ideal of Maximal Order of Fb over Maximal Equation Order of Fa over 
Valuation ring of Univariate rational function field over GF(79) with generator 
1/x
Generators:
1
((78*x^2 + 71*x + 78)/x^2*a + (23*x^2 + 15*x + 78)/x^2)*b + (67*x^2 + 60*x + 
    78)/x^2*a + (3*x^2 + 47*x + 78)/x * ( 1 0 ) 
Fractional ideal of Maximal Order of Fb over Maximal Equation Order of Fa over 
Valuation ring of Univariate rational function field over GF(79) with generator 
1/x
Generators:
1
((59*x^2 + 44*x + 78)/x^2*a + (4*x^2 + 48*x + 78)/x^2)*b + (29*x^2 + 46*x + 
    78)/x^2*a + (71*x + 78)/x * ( 0 1 ) 
> _, II := $1; 
> II eq I;
true
> MaximalOrderFinite(Fc)!!I;
Ideal of Maximal Order of Fc over Maximal Equation Order of Fb over Maximal 
Equation Order of Fa over Univariate Polynomial Ring in x over GF(79)
Generators:
a*b*c
a*b*c

Equality and Membership

I eq J : RngFunOrdIdl, RngFunOrdIdl -> BoolElt
I ne J : RngFunOrdIdl, RngFunOrdIdl -> BoolElt
I in S: RngFunOrdIdl, PowIdl -> BoolElt
I notin S: RngFunOrdIdl, PowIdl -> BoolElt

Predicates on Ideals

IsZero(I) : RngFunOrdIdl -> BoolElt
Returns true if and only if the ideal I is the zero ideal of the order O.
IsOne(I) : RngFunOrdIdl -> BoolElt
Returns true if and only if the ideal I is the identity ideal of the order O, i.e. I = O.
IsIntegral(I) : RngFunOrdIdl -> BoolElt
Returns true if and only if the ideal I is integral (a true ideal of its order).
IsPrime(I) : RngFunOrdIdl -> BoolElt
Returns true if and only if the ideal I is prime.
IsPrincipal(I) : RngFunOrdIdl -> BoolElt, FldFunElt
Returns true and a generator if the fractional ideal I is principal, false otherwise. The function field has to be global.
Predicates on Prime Ideals
IsInert(P) : RngFunOrdIdl -> BoolElt
Return true if the inertia degree of the prime ideal P is the degree of its order.
IsInert(P, O) : RngFunOrdIdl, RngFunOrd -> BoolElt
IsInert(P, O) : RngElt, RngFunOrd -> BoolElt
Return true if there is an inert ideal in the order O above the prime ideal P.
IsRamified(P) : RngFunOrdIdl -> BoolElt
Return true if the ramification index of the prime ideal P is not 1.
IsRamified(P, O) : RngFunOrdIdl, RngFunOrd -> BoolElt
IsRamified(P, O) : RngElt, RngFunOrd -> BoolElt
Return true if there is a ramified ideal in the order O above the prime ideal P.
IsSplit(P) : RngFunOrdIdl -> BoolElt
Return true if the prime ideal P is not the only ideal lying above the prime ideal it lies above.
IsSplit(P, O) : RngFunOrdIdl, RngFunOrd -> BoolElt
IsSplit(P, O) : RngElt, RngFunOrd -> BoolElt
Return true if there are at least 2 distinct ideals which lie in the order O above the prime ideal P.
IsTamelyRamified(P) : RngFunOrdIdl -> BoolElt
Return whether the prime ideal P is not wildly ramified.
IsTamelyRamified(P, O) : RngFunOrdIdl, RngFunOrd -> BoolElt
IsTamelyRamified(P, O) : RngElt, RngFunOrd -> BoolElt
Return whether the prime ideal P is not wildly ramified in the order O.
IsTotallyRamified(P) : RngFunOrdIdl -> BoolElt
Return whether the ramification index of the prime ideal P is the same as the degree of its order over its coefficient order.
IsTotallyRamified(P, O) : RngFunOrdIdl, RngFunOrd -> BoolElt
IsTotallyRamified(P, O) : RngElt, RngFunOrd -> BoolElt
Return whether there are any totally ramified ideals in the order O lying above the prime ideal P.
IsTotallySplit(P) : RngFunOrdIdl -> BoolElt
Return whether there are as many ideals as the degree of the order of the prime ideal P lying over the prime P lies over.
IsTotallySplit(P, O) : RngFunOrdIdl, RngFunOrd -> BoolElt
IsTotallySplit(P, O) : RngElt, RngFunOrd -> BoolElt
Return whether there are as many ideals of the order O which lie above the prime ideal P as the degree of O.
IsUnramified(P) : RngFunOrdIdl -> BoolElt
Return whether the ramification index of the prime ideal P is 1.
IsUnramified(P, O) : RngFunOrdIdl, RngFunOrd -> BoolElt
IsUnramified(P, O) : RngElt, RngFunOrd -> BoolElt
Return whether all the ideals of the order O which lie above the prime ideal P are unramified.
IsWildlyRamified(P) : RngFunOrdIdl -> BoolElt
Return whether the ramfication index of the prime ideal P is a multiple of the characteristic of the residue field of P.
IsWildlyRamified(P, O) : RngFunOrdIdl, RngFunOrd -> BoolElt
IsWildlyRamified(P, O) : RngElt, RngFunOrd -> BoolElt
Return whether any of the ideals of the order O which lie above the prime ideal P are wildly ramified.

Further Ideal Operations

I meet J : RngFunOrdIdl, RngFunOrdIdl -> RngFunOrdIdl
The intersection of the ideals I and J.
Gcd(I, J) : RngFunOrdIdl, RngFunOrdIdl -> RngFunOrdIdl
Given invertible ideals of an order O, returns the greatest common divisor of the ideals I and J.
Lcm(I, J) : RngFunOrdIdl, RngFunOrdIdl -> RngFunOrdIdl
Given invertible ideals of an order O, returns the least common multiple of the ideals I and J.
Factorization(I) : RngFunOrdIdl -> [ <RngFunOrdIdl, RngIntElt> ]
Factorisation(I) : RngFunOrdIdl -> [<RngFunOrdIdl, RngIntElt>]
Factorization of the ideal I (as sequence of prime ideal, exponent pairs). The order must be maximal.
Decomposition(O, p) : RngFunOrd, RngElt -> [ RngFunOrdIdl ]
Decomposition(O, p) : RngFunOrd, RngFunOrdIdl -> [ RngFunOrdIdl ]
A sequence containing all prime ideals of the order O lying above the prime element or ideal p of the coefficient ring of O.
Decomposition(O) : RngFunOrd -> [ RngFunOrdIdl ]
A sequence containing all prime ideals of the `infinite' maximal order O.
DecompositionType(O, p) : RngFunOrd, RngElt -> [ <RngIntElt, RngIntElt> ]
DecompositionType(O, p) : RngFunOrd, RngFunOrdIdl -> [ <RngIntElt, RngIntElt> ]
Sequence of tuples of residue degrees and ramification indices of the prime ideals of the order O lying over p, a prime polynomial or ideal or element of valuation ring of valuation 1.
DecompositionType(O) : RngFunOrd -> [ <RngIntElt, RngIntElt> ]
Sequence of tuples of residue degrees and ramification indices of the prime ideals of the `infinite' maximal order O.
MultiplicatorRing(I) : RngFunOrdIdl -> RngFunOrd
Returns the multiplicator ring of the ideal I of the order O, that is, the subring of elements of the field of fractions of O that multiply I into itself.
pMaximalOrder(O, p) : RngFunOrd, RngFunOrdIdl -> RngFunOrd
The p-maximal over order of the order O where p is a prime ideal of the coefficient ring of O. See also the description in Section Orders and Ideals.
pRadical(O, p) : RngFunOrd, RngFunOrdIdl -> RngFunOrdIdl
Returns the p-radical of an order O for a prime ideal p of the coefficient ring of O, defined as the ideal consisting of elements of O for which some power lies in the ideal pO.

It is possible to call this function even if p is not prime. In this case the p-trace-radical will be computed, i.e. { x∈F | Tr(xO)⊆C} for F the field of fractions of O and C the order of p (if p is an ideal) or the parent of p otherwise. If p is square free and all divisors are larger than the field degree, this is the intersection of the radicals for all l dividing p.

Valuation(a, P) : RngElt, RngFunOrdIdl -> RngIntElt
Valuation(I, P) : RngFunOrdIdl, RngFunOrdIdl -> RngIntElt
The valuation of a or the ideal I at the prime ideal P. The element a must be coercible into the field of fractions of P's order.
Order(I) : RngFunOrdIdl -> RngFunOrd
The order of the ideal I.
Denominator(I) : RngFunOrdIdl -> RngElt
The "smallest" element d of the coefficient ring of the order O of the ideal I such that dI ⊆O.
Minimum(I) : RngFunOrdIdl -> Any
A generator m of the ideal R ∩dI where R is the coefficient ring of the ideal's order and d is the denominator of the ideal I (d is the second return value).
I meet R : RngFunOrdIdl, Rng -> Any
The intersection of the ideal I with a coefficient ring R of its order.
IntegralSplit(I) : RngFunOrdIdl -> RngFunOrdIdl, RngElt
The integral ideal dI and d, where d is the denominator of the ideal I.
Norm(I) : RngFunOrdIdl -> Any
The norm of the ideal I, as element of the coefficient field of the algebraic function field to which I belongs.
TwoElement(I) : RngFunOrdIdl -> RngElt, RngElt
Given an ideal I with function field F as the function field of its order O, returns two elements a, b ∈F such that I = aO + bO.
Generators(I) : RngFunOrdIdl -> [ RngFunOrdElt ]
Given a (fractional) ideal I of the order O, return a sequence of elements of the function field F that generate I as an ideal.
Basis(I) : RngFunOrdIdl -> [FldFunElt]
Basis(I, R) : RngFunOrdIdl, Rng -> [ RngElt ]
A basis of the ideal I as a free module over the coefficient ring of its order, coerced into the ring R if given.
BasisMatrix(I) : RngFunOrdIdl -> AlgMatElt
Let (b1, ..., bn) be the basis of the ideal I and let (ω1, ..., ωn) be the basis of the order O. A matrix B with coefficients in the rational function field is returned such that (b1, ..., bn) = (ω1, ..., ωn) Bt.
TransformationMatrix(I) : RngFunOrdIdl -> AlgMatElt, RngElt
Let (b1, ..., bn) be the basis of the ideal I and let (ω1, ..., ωn) be the basis of the order O. A matrix T with coefficients in the coefficient ring of O and a denominator d are returned such that (b1, ..., bn) = (ω1, ..., ωn) Tt / d.
CoefficientIdeals(I) : RngOrdFracIdl -> [RngOrdFracIdl]
The coefficient ideals of the ideal I in a relative extension. These are the ideals {Ai} of the coefficient ring of the order of I such that for every element e ∈I, e = ∑i ai * bi where {bi} is the basis returned for I and each ai ∈Ai.
Different(I) : RngFunOrdIdl -> RngFunOrdIdl
The different of the (possibly fractional) ideal I of an order of an algebraic function field.
Codifferent(I) : RngFunOrdIdl -> RngFunOrdIdl
The codifferent of the ideal I. This will be the inverse of the different of I if I is an ideal of a maximal order.
Divisor(I) : RngFunOrdIdl -> DivFunElt
The divisor corresponding to the ideal factorization of the ideal I.
Divisor(I, J) : RngFunOrdIdl, RngFunOrdIdl -> DivFunElt
The divisor corresponding to the ideal factorization of the ideals I and J belonging to the `finite' and `infinite' maximal order.

Example FldFunG_ideals (H42E31)

> PR<x> := FunctionField(Rationals());
> P<y> := PolynomialRing(PR);
> FR1<a> := FunctionField(y^3 - x + 1/x^3);
> P<y> := PolynomialRing(FR1);
> FR2<c> := FunctionField(y^2 - a/x^3*y + 1);
> I := ideal<MaximalOrderFinite(FR2) | 
> [ x^9 + 1639*x^8 + 863249*x^7 + 148609981*x^6 + 404988066*x^5 + 567876948*x^4 + 
> 468363837*x^3 + 242625823*x^2 + 68744019*x + 8052237, x^9 + 1639*x^8 + 
> 863249*x^7 + 148609981*x^6 + 404988066*x^5 + 567876948*x^4 + 468363837*x^3 +
> 242625823*x^2 + 68744019*x + 8052237, (x^15 + 1639*x^14 + 863249*x^13 + 
> 148609981*x^12 + 404988066*x^11 + 567876948*x^10 + 468363837*x^9 + 
> 242625823*x^8 + 68744019*x^7 + 8052237*x^6)*c, (x^15 + 1639*x^14 + 
> 863249*x^13 + 148609981*x^12 + 404988066*x^11 + 567876948*x^10 + 
> 468363837*x^9 + 242625823*x^8 + 68744019*x^7 + 8052237*x^6)*c ]>;
> I;
Ideal of Maximal Order of FR2 over Maximal Order of FR1 over Univariate 
Polynomial Ring in x over Rational Field
Basis:
Pseudo-matrix over Maximal Order of FR1 over Univariate Polynomial Ring in x 
over Rational Field
Ideal of Maximal Order of FR1 over Univariate Polynomial Ring in x over Rational
Field
Generator:
x^9 + 1639*x^8 + 863249*x^7 + 148609981*x^6 + 404988066*x^5 + 567876948*x^4 + 
    468363837*x^3 + 242625823*x^2 + 68744019*x + 8052237 * ( 1 0 ) 
Fractional ideal of Maximal Order of FR1 over Univariate Polynomial Ring in x 
over Rational Field
Generator:
(x^9 + 1639*x^8 + 863249*x^7 + 148609981*x^6 + 404988066*x^5 + 567876948*x^4 + 
    468363837*x^3 + 242625823*x^2 + 68744019*x + 8052237)/x^2 * ( 0 1 ) 
> J := ideal<MaximalOrderFinite(FR2) | 
> [ x^3 + 278*x^2 + 164*x + 742, x^3 + 278*x^2 + 164*x + 742, (x^9 + 278*x^8 + 
> 164*x^7 + 742*x^6)*c, (x^9 + 278*x^8 + 164*x^7 + 742*x^6)*c ]>;
> J;
Ideal of Maximal Order of FR2 over Maximal Order of FR1 over Univariate 
Polynomial Ring in x over Rational Field
Basis:
Pseudo-matrix over Maximal Order of FR1 over Univariate Polynomial Ring in x 
over Rational Field
Ideal of Maximal Order of FR1 over Univariate Polynomial Ring in x over Rational
Field
Generator:
x^3 + 278*x^2 + 164*x + 742 * ( 1 0 ) 
Fractional ideal of Maximal Order of FR1 over Univariate Polynomial Ring in x 
over Rational Field
Generator:
(x^3 + 278*x^2 + 164*x + 742)/x^2 * ( 0 1 ) 
> Generators(J);
[
    x^3 + 278*x^2 + 164*x + 742,
    x^3 + 278*x^2 + 164*x + 742,
    (x^7 + 278*x^6 + 164*x^5 + 742*x^4)*c,
    (x^7 + 278*x^6 + 164*x^5 + 742*x^4)*c
]

> TwoElement(J);
x^3 + 278*x^2 + 164*x + 742
((3/2*x^10 + 419*x^9 + 802*x^8 + 1441*x^7 + 1484*x^6)*a^2 + (3/2*x^8 + 417*x^7 +
    246*x^6 + 1113*x^5)*a + (3/2*x^8 + 837/2*x^7 + 663*x^6 + 1359*x^5 + 
    1113*x^4))*c + (x^6 + 277*x^5 - 114*x^4 + 578*x^3 - 742*x^2)*a^2 + (3*x^5 + 
    834*x^4 + 492*x^3 + 2226*x^2)*a - x^3 - 278*x^2 - 164*x - 742
> Minimum(I);
Ideal of Maximal Order of FR1 over Univariate Polynomial Ring in x over Rational
Field
Generator:
x^9 + 1639*x^8 + 863249*x^7 + 148609981*x^6 + 404988066*x^5 + 567876948*x^4 + 
    468363837*x^3 + 242625823*x^2 + 68744019*x + 8052237 1
> Basis(J);
[ 1, x^6*c ]
> Basis(I);
[ 1, x^6*c ]
> I eq J;
false
> II, d := IntegralSplit(I);
> II;
Ideal of Maximal Order of FR2 over Maximal Order of FR1 over Univariate 
Polynomial Ring in x over Rational Field
Basis:
Pseudo-matrix over Maximal Order of FR1 over Univariate Polynomial Ring in x 
over Rational Field
Ideal of Maximal Order of FR1 over Univariate Polynomial Ring in x over Rational
Field
Generator:
x^9 + 1639*x^8 + 863249*x^7 + 148609981*x^6 + 404988066*x^5 + 567876948*x^4 + 
    468363837*x^3 + 242625823*x^2 + 68744019*x + 8052237 * ( 1 0 ) 
Fractional ideal of Maximal Order of FR1 over Univariate Polynomial Ring in x 
over Rational Field
Generator:
(x^9 + 1639*x^8 + 863249*x^7 + 148609981*x^6 + 404988066*x^5 + 567876948*x^4 + 
    468363837*x^3 + 242625823*x^2 + 68744019*x + 8052237)/x^2 * ( 0 1 ) 
> d;
1
> IsIntegral(I);
true
> GCD(I, J)*LCM(I, J) eq I*J;
true

Functions on Prime Ideals
RamificationIndex(I) : RngFunOrdIdl -> RngIntElt
RamificationDegree(I) : RngFunOrdIdl -> RngIntElt
The ramification index of the prime ideal I over the corresponding prime of its coefficient ring.
Degree(I) : RngFunOrdIdl -> RngIntElt
InertiaDegree(I) : RngFunOrdIdl -> RngIntElt
ResidueClassDegree(I) : RngFunOrdIdl -> RngIntElt
The residue class degree (inertia degree) of the prime ideal I over the corresponding prime of its coefficient ring.
ResidueClassField(I) : RngFunOrdIdl -> Rng, Map
The residue class field of the prime ideal I and the residue class mapping.
Place(I) : RngFunOrdIdl -> PlcFunElt
The place corresponding to the prime ideal I, where I is defined over the `finite' or `infinite' maximal order.
SafeUniformizer(P) : RngFunOrdIdl -> RngFunOrdElt
For an ideal I of a maximal order in a function field, this returns an element which has valuation 1 at the given prime and which has valuation 0 at all other primes lying over the same prime of the underlying rational function field. See also LocalUniformizer (for places) below.

Example FldFunG_order-ideals (H42E32)

> R<x> := FunctionField(GF(3));
> P<y> := PolynomialRing(R);
> f := y^4 + x*y + x^4 + x + 1;
> F<a> := FunctionField(f);
> O := MaximalOrderFinite(F);
> x*O;
Ideal of O
Generator:
x
> L := Factorization(x*O);
> L;
[ <Ideal of O
Generators:
x
a^2 + a + 2, 1>, <Ideal of O
Generators:
x
a^2 + 2*a + 2, 1> ]
> P1 := L[1][1];
> P2 := L[2][1];
> BasisMatrix(P1);
[x 0 0 0]
[0 x 0 0]
[2 1 1 0]
[1 1 0 1]
> P1 meet P2 eq x*O;
true
> IsPrime(P1);
true
> Place(P1);
(x, a^2 + a + 2)

 [Next][Prev] [Right] [Left] [Up] [Index] [Root]

Version: V2.19 of Wed Apr 24 15:09:57 EST 2013