y2+(x+1)y=−x5−x4 |
(homogenize, simplify) |
y2+(xz2+z3)y=−x5z−x4z2 |
(dehomogenize, simplify) |
y2=−4x5−4x4+x2+2x+1 |
(homogenize, minimize) |
sage:R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, 0, 0, -1, -1]), R([1, 1]));
magma:R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, 0, 0, -1, -1], R![1, 1]);
sage:X = HyperellipticCurve(R([1, 2, 1, 0, -4, -4]))
magma:X,pi:= SimplifiedModel(C);
Conductor: | N | = | 1051 | = | 1051 |
magma:Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | Δ | = | −1051 | = | −1051 |
magma:Discriminant(C); Factorization(Integers()!$1);
|
I2 | = | 64 | = |
26 |
I4 | = | −200 | = |
−23⋅52 |
I6 | = | 185 | = |
5⋅37 |
I10 | = | 4204 | = |
22⋅1051 |
J2 | = | 32 | = |
25 |
J4 | = | 76 | = |
22⋅19 |
J6 | = | −241 | = |
−241 |
J8 | = | −3372 | = |
−22⋅3⋅281 |
J10 | = | 1051 | = |
1051 |
g1 | = | 33554432/1051 |
g2 | = | 2490368/1051 |
g3 | = | −246784/1051 |
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),(0:0:1),(−1:0:1),(0:−1:1)
All points:
(1:0:0),(0:0:1),(−1:0:1),(0:−1:1)
All points:
(1:0:0),(−1:0:1),(0:−1:1),(0:1:1)
magma:[C![-1,0,1],C![0,-1,1],C![0,0,1],C![1,0,0]]; // minimal model
magma:[C![-1,0,1],C![0,-1,1],C![0,1,1],C![1,0,0]]; // 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/8Z
magma:MordellWeilGroupGenus2(Jacobian(C));
Generator |
D0 |
|
|
|
|
|
Height |
Order |
(0:1:1)−(1:0:0) |
x |
= |
0, |
y |
= |
xz2+z3 |
0 |
8 |
2-torsion field: 4.2.4204.1
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);