y2+(x3+1)y=−x5+x3+2x2+x−1 |
(homogenize, simplify) |
y2+(x3+z3)y=−x5z+x3z3+2x2z4+xz5−z6 |
(dehomogenize, simplify) |
y2=x6−4x5+6x3+8x2+4x−3 |
(homogenize, minimize) |
sage:R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, 1, 2, 1, 0, -1]), R([1, 0, 0, 1]));
magma:R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, 1, 2, 1, 0, -1], R![1, 0, 0, 1]);
sage:X = HyperellipticCurve(R([-3, 4, 8, 6, 0, -4, 1]))
magma:X,pi:= SimplifiedModel(C);
Conductor: | N | = | 975 | = | 3⋅52⋅13 |
magma:Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | Δ | = | −63375 | = | −3⋅53⋅132 |
magma:Discriminant(C); Factorization(Integers()!$1);
|
I2 | = | 148 | = |
22⋅37 |
I4 | = | −48575 | = |
−52⋅29⋅67 |
I6 | = | −4076175 | = |
−3⋅52⋅17⋅23⋅139 |
I10 | = | −8112000 | = |
−27⋅3⋅53⋅132 |
J2 | = | 37 | = |
37 |
J4 | = | 2081 | = |
2081 |
J6 | = | 35929 | = |
19⋅31⋅61 |
J8 | = | −750297 | = |
−3⋅383⋅653 |
J10 | = | −63375 | = |
−3⋅53⋅132 |
g1 | = | −69343957/63375 |
g2 | = | −105408893/63375 |
g3 | = | −49186801/63375 |
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),(−1:0:1),(3:−14:1)
All points:
(1:0:0),(1:−1:0),(−1:0:1),(3:−14:1)
All points:
(1:−1:0),(1:1:0),(−1:0:1),(3:0:1)
magma:[C![-1,0,1],C![1,-1,0],C![1,0,0],C![3,-14,1]]; // minimal model
magma:[C![-1,0,1],C![1,-1,0],C![1,1,0],C![3,0,1]]; // simplified model
Number of rational Weierstrass points: 2
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/2Z⊕Z/6Z
magma:MordellWeilGroupGenus2(Jacobian(C));
Generator |
D0 |
|
|
|
|
|
Height |
Order |
D0−(1:−1:0)−(1:0:0) |
x2−3xz+z2 |
= |
0, |
y |
= |
−4xz2+z3 |
0 |
2 |
D0−(1:−1:0)−(1:0:0) |
2x2−2xz+z2 |
= |
0, |
2y |
= |
4xz2−z3 |
0 |
6 |
Generator |
D0 |
|
|
|
|
|
Height |
Order |
D0−(1:−1:0)−(1:0:0) |
x2−3xz+z2 |
= |
0, |
y |
= |
−4xz2+z3 |
0 |
2 |
D0−(1:−1:0)−(1:0:0) |
2x2−2xz+z2 |
= |
0, |
2y |
= |
4xz2−z3 |
0 |
6 |
Generator |
D0 |
|
|
|
|
|
Height |
Order |
D0−(1:−1:0)−(1:1:0) |
x2−3xz+z2 |
= |
0, |
y |
= |
x3−8xz2+3z3 |
0 |
2 |
D0−(1:−1:0)−(1:1:0) |
2x2−2xz+z2 |
= |
0, |
2y |
= |
x3+8xz2−z3 |
0 |
6 |
2-torsion field: Q(−3,5)
The mod-ℓ Galois representation
has maximal image GSp(4,Fℓ)
for all primes ℓ
except those listed.
Simple over Q
magma:HeuristicDecompositionFactors(C);
Not of GL2-type over Q
Endomorphism ring over Q:
End(J) | ≃ | Z |
End(J)⊗Q | ≃ | Q |
End(J)⊗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);