The Jacobian of this curve is isogenous to that of the modular curve X0(29) (which has discriminant 295).
y2+(x3+x2+x)y=x4+x3+3x2+x+2 |
(homogenize, simplify) |
y2+(x3+x2z+xz2)y=x4z2+x3z3+3x2z4+xz5+2z6 |
(dehomogenize, simplify) |
y2=x6+2x5+7x4+6x3+13x2+4x+8 |
(homogenize, minimize) |
sage:R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([2, 1, 3, 1, 1]), R([0, 1, 1, 1]));
magma:R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![2, 1, 3, 1, 1], R![0, 1, 1, 1]);
sage:X = HyperellipticCurve(R([8, 4, 13, 6, 7, 2, 1]))
magma:X,pi:= SimplifiedModel(C);
Conductor: | N | = | 841 | = | 292 |
magma:Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | Δ | = | −841 | = | −292 |
magma:Discriminant(C); Factorization(Integers()!$1);
|
I2 | = | 1420 | = |
22⋅5⋅71 |
I4 | = | 4201 | = |
4201 |
I6 | = | 1973899 | = |
61⋅32359 |
I10 | = | 107648 | = |
27⋅292 |
J2 | = | 355 | = |
5⋅71 |
J4 | = | 5076 | = |
22⋅33⋅47 |
J6 | = | 93408 | = |
25⋅3⋅7⋅139 |
J8 | = | 1848516 | = |
22⋅3⋅154043 |
J10 | = | 841 | = |
292 |
g1 | = | 5638216721875/841 |
g2 | = | 227094529500/841 |
g3 | = | 11771743200/841 |
sage:C.igusa_clebsch_invariants(); [factor(a) for a in _]
magma:IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
Aut(X) | ≃ |
C2 |
magma:AutomorphismGroup(C); IdentifyGroup($1);
|
Aut(XQ) | ≃ |
C2 |
magma:AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
|
All points:
(1:0:0),(1:−1:0)
All points:
(1:0:0),(1:−1:0)
All points:
(1:−1:0),(1:1:0)
magma:[C![1,-1,0],C![1,0,0]]; // minimal model
magma:[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/7Z
magma:MordellWeilGroupGenus2(Jacobian(C));
Generator |
D0 |
|
|
|
|
|
Height |
Order |
D0−(1:−1:0)−(1:0:0) |
x2+z2 |
= |
0, |
y |
= |
z3 |
0 |
7 |
Generator |
D0 |
|
|
|
|
|
Height |
Order |
D0−(1:−1:0)−(1:0:0) |
x2+z2 |
= |
0, |
y |
= |
z3 |
0 |
7 |
Generator |
D0 |
|
|
|
|
|
Height |
Order |
D0−(1:−1:0)−(1:1:0) |
x2+z2 |
= |
0, |
y |
= |
x3+x2z+xz2+2z3 |
0 |
7 |
2-torsion field: 6.0.53824.1
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).
Simple over Q
magma:HeuristicDecompositionFactors(C);
Of GL2-type over Q
Endomorphism ring over Q:
End(J) | ≃ | Z[2] |
End(J)⊗Q | ≃ | Q(2) |
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);