sage:R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1]), R([0, 1, 0, 1]));
magma:R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1], R![0, 1, 0, 1]);
sage:X = HyperellipticCurve(R([0, 4, 1, 0, 2, 0, 1]))
magma:X,pi:= SimplifiedModel(C);
Conductor: | N | = | 1047 | = | 3⋅349 |
magma:Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | Δ | = | 3141 | = | 32⋅349 |
magma:Discriminant(C); Factorization(Integers()!$1);
|
I2 | = | 8 | = |
23 |
I4 | = | 604 | = |
22⋅151 |
I6 | = | 1017 | = |
32⋅113 |
I10 | = | −12564 | = |
−22⋅32⋅349 |
J2 | = | 4 | = |
22 |
J4 | = | −100 | = |
−22⋅52 |
J6 | = | −1 | = |
−1 |
J8 | = | −2501 | = |
−41⋅61 |
J10 | = | −3141 | = |
−32⋅349 |
g1 | = | −1024/3141 |
g2 | = | 6400/3141 |
g3 | = | 16/3141 |
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),(0:0:1),(−1:1:1)
All points:
(1:0:0),(1:−1:0),(0:0:1),(−1:1:1)
All points:
(1:−1:0),(1:1:0),(0:0:1),(−1:0:1)
magma:[C![-1,1,1],C![0,0,1],C![1,-1,0],C![1,0,0]]; // minimal model
magma:[C![-1,0,1],C![0,0,1],C![1,-1,0],C![1,1,0]]; // 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/10Z
magma:MordellWeilGroupGenus2(Jacobian(C));
Generator |
D0 |
|
|
|
|
|
Height |
Order |
(−1:1:1)−(1:−1:0) |
z(x+z) |
= |
0, |
y |
= |
z3 |
0 |
10 |
Generator |
D0 |
|
|
|
|
|
Height |
Order |
(−1:1:1)−(1:−1:0) |
z(x+z) |
= |
0, |
y |
= |
z3 |
0 |
10 |
Generator |
D0 |
|
|
|
|
|
Height |
Order |
(−1:0:1)−(1:−1:0) |
z(x+z) |
= |
0, |
y |
= |
x3+xz2+2z3 |
0 |
10 |
2-torsion field: 4.0.1396.1
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);