y2+(x3+1)y=x5+x4+5x2+12x+8 |
(homogenize, simplify) |
y2+(x3+z3)y=x5z+x4z2+5x2z4+12xz5+8z6 |
(dehomogenize, simplify) |
y2=x6+4x5+4x4+2x3+20x2+48x+33 |
(homogenize, minimize) |
sage:R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([8, 12, 5, 0, 1, 1]), R([1, 0, 0, 1]));
magma:R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![8, 12, 5, 0, 1, 1], R![1, 0, 0, 1]);
sage:X = HyperellipticCurve(R([33, 48, 20, 2, 4, 4, 1]))
magma:X,pi:= SimplifiedModel(C);
Conductor: | N | = | 588 | = | 22⋅3⋅72 |
magma:Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | Δ | = | −18816 | = | −27⋅3⋅72 |
magma:Discriminant(C); Factorization(Integers()!$1);
|
I2 | = | 748 | = |
22⋅11⋅17 |
I4 | = | 11545 | = |
5⋅2309 |
I6 | = | 2902787 | = |
2902787 |
I10 | = | 2408448 | = |
214⋅3⋅72 |
J2 | = | 187 | = |
11⋅17 |
J4 | = | 976 | = |
24⋅61 |
J6 | = | −192 | = |
−26⋅3 |
J8 | = | −247120 | = |
−24⋅5⋅3089 |
J10 | = | 18816 | = |
27⋅3⋅72 |
g1 | = | 228669389707/18816 |
g2 | = | 398891383/1176 |
g3 | = | −34969/98 |
sage:C.igusa_clebsch_invariants(); [factor(a) for a in _]
magma:IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
Aut(X) | ≃ |
C22 |
magma:AutomorphismGroup(C); IdentifyGroup($1);
|
Aut(XQ) | ≃ |
C22 |
magma:AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
|
All points:
(1:0:0),(1:−1:0),(−1:−1:1),(−1:1:1),(−2:3:1),(−2:4:1)
All points:
(1:0:0),(1:−1:0),(−1:−1:1),(−1:1:1),(−2:3:1),(−2:4:1)
All points:
(1:−1:0),(1:1:0),(−2:−1:1),(−2:1:1),(−1:−2:1),(−1:2:1)
magma:[C![-2,3,1],C![-2,4,1],C![-1,-1,1],C![-1,1,1],C![1,-1,0],C![1,0,0]]; // minimal model
magma:[C![-2,-1,1],C![-2,1,1],C![-1,-2,1],C![-1,2,1],C![1,-1,0],C![1,1,0]]; // simplified model
Number of rational Weierstrass points: 0
magma:#Roots(HyperellipticPolynomials(SimplifiedModel(C)));
This curve is locally solvable everywhere.
magma:f,h:=HyperellipticPolynomials(C); g:=4*f+h^2; HasPointsEverywhereLocally(g,2) and (#Roots(ChangeRing(g,RealField())) gt 0 or LeadingCoefficient(g) gt 0);
Group structure: Z/24Z
magma:MordellWeilGroupGenus2(Jacobian(C));
Generator |
D0 |
|
|
|
|
|
Height |
Order |
(−2:4:1)+(−1:−1:1)−(1:−1:0)−(1:0:0) |
(x+z)(x+2z) |
= |
0, |
y |
= |
−5xz2−6z3 |
0 |
24 |
Generator |
D0 |
|
|
|
|
|
Height |
Order |
(−2:4:1)+(−1:−1:1)−(1:−1:0)−(1:0:0) |
(x+z)(x+2z) |
= |
0, |
y |
= |
−5xz2−6z3 |
0 |
24 |
Generator |
D0 |
|
|
|
|
|
Height |
Order |
(−2:1:1)+(−1:−2:1)−(1:−1:0)−(1:1:0) |
(x+z)(x+2z) |
= |
0, |
y |
= |
x3−10xz2−11z3 |
0 |
24 |
2-torsion field: 8.0.12446784.1
For primes ℓ≥5 the Galois representation data has not been computed for this curve since it is not generic.
For primes ℓ≤3, the image of the mod-ℓ Galois representation is listed in the table below, whenever it is not all of GSp(4,Fℓ).
Splits over Q
Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
Elliptic curve isogeny class 14.a
Elliptic curve isogeny class 42.a
magma:HeuristicDecompositionFactors(C);
Of GL2-type over Q
Endomorphism ring over Q:
End(J) | ≃ | an order of index 2 in Z×Z |
End(J)⊗Q | ≃ | Q × Q |
End(J)⊗R | ≃ | R×R |
All Q-endomorphisms of the Jacobian are defined over Q.
magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
magma:HeuristicIsGL2(C); HeuristicEndomorphismDescription(C); HeuristicEndomorphismFieldOfDefinition(C);
magma:HeuristicIsGL2(C : Geometric := true); HeuristicEndomorphismDescription(C : Geometric := true); HeuristicEndomorphismLatticeDescription(C);