Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^2 + x + 1)y = 3x^6 + 26x^5 + 40x^4 + 25x^3 + 22x^2 + 7x - 8$ | (homogenize, simplify) |
$y^2 + (x^2z + xz^2 + z^3)y = 3x^6 + 26x^5z + 40x^4z^2 + 25x^3z^3 + 22x^2z^4 + 7xz^5 - 8z^6$ | (dehomogenize, simplify) |
$y^2 = 12x^6 + 104x^5 + 161x^4 + 102x^3 + 91x^2 + 30x - 31$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-8, 7, 22, 25, 40, 26, 3]), R([1, 1, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-8, 7, 22, 25, 40, 26, 3], R![1, 1, 1]);
sage: X = HyperellipticCurve(R([-31, 30, 91, 102, 161, 104, 12]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(2739\) | \(=\) | \( 3 \cdot 11 \cdot 83 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(-2739\) | \(=\) | \( - 3 \cdot 11 \cdot 83 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(21044\) | \(=\) | \( 2^{2} \cdot 5261 \) |
\( I_4 \) | \(=\) | \(-927505967\) | \(=\) | \( -927505967 \) |
\( I_6 \) | \(=\) | \(-9952421552727\) | \(=\) | \( - 3 \cdot 547 \cdot 6064851647 \) |
\( I_{10} \) | \(=\) | \(-350592\) | \(=\) | \( - 2^{7} \cdot 3 \cdot 11 \cdot 83 \) |
\( J_2 \) | \(=\) | \(5261\) | \(=\) | \( 5261 \) |
\( J_4 \) | \(=\) | \(39799337\) | \(=\) | \( 39799337 \) |
\( J_6 \) | \(=\) | \(82088193169\) | \(=\) | \( 82088193169 \) |
\( J_8 \) | \(=\) | \(-288030310344365\) | \(=\) | \( - 5 \cdot 69193 \cdot 832541761 \) |
\( J_{10} \) | \(=\) | \(-2739\) | \(=\) | \( - 3 \cdot 11 \cdot 83 \) |
\( g_1 \) | \(=\) | \(-4030338368178862301/2739\) | ||
\( g_2 \) | \(=\) | \(-5795364321847592797/2739\) | ||
\( g_3 \) | \(=\) | \(-2272046943202955449/2739\) |
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
Automorphism group
\(\mathrm{Aut}(X)\) | \(\simeq\) | $C_2$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
|
\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $C_2$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
|
Rational points
This curve has no rational points.
magma: []; // minimal model
magma: []; // 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);
Mordell-Weil group of the Jacobian
Group structure: \(\Z/{2}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(4x^2 + 32xz + 31z^2\) | \(=\) | \(0,\) | \(8y\) | \(=\) | \(28xz^2 + 27z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(4x^2 + 32xz + 31z^2\) | \(=\) | \(0,\) | \(8y\) | \(=\) | \(28xz^2 + 27z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(4x^2 + 32xz + 31z^2\) | \(=\) | \(0,\) | \(8y\) | \(=\) | \(x^2z + 57xz^2 + 55z^3\) | \(0\) | \(2\) |
2-torsion field: 6.2.3961119888.3
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 1 \) |
Real period: | \( 0.664267 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 2 \) |
Leading coefficient: | \( 0.664267 \) |
Analytic order of Ш: | \( 4 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(3\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - T + 3 T^{2} )\) | |
\(11\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 3 T + 11 T^{2} )\) | |
\(83\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 83 T^{2} )\) |
Galois representations
The mod-$\ell$ Galois representation has maximal image \(\GSp(4,\F_\ell)\) for all primes \( \ell \) except those listed.
Prime \(\ell\) | mod-\(\ell\) image | Is torsion prime? |
---|---|---|
\(2\) | 2.15.1 | yes |
\(3\) | 3.80.2 | no |
Sato-Tate group
\(\mathrm{ST}\) | \(\simeq\) | $\mathrm{USp}(4)$ |
\(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{USp}(4)\) |
Decomposition of the Jacobian
Simple over \(\overline{\Q}\)
magma: HeuristicDecompositionFactors(C);
Endomorphisms of the Jacobian
Not of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
\(\End (J_{})\) | \(\simeq\) | \(\Z\) |
\(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
\(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\R\) |
All \(\overline{\Q}\)-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);