Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x)y = -x^6 - 7x^5 - 15x^4 + 12x^3 + 16x^2 + x - 9$ | (homogenize, simplify) |
$y^2 + (x^3 + xz^2)y = -x^6 - 7x^5z - 15x^4z^2 + 12x^3z^3 + 16x^2z^4 + xz^5 - 9z^6$ | (dehomogenize, simplify) |
$y^2 = -3x^6 - 28x^5 - 58x^4 + 48x^3 + 65x^2 + 4x - 36$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-9, 1, 16, 12, -15, -7, -1]), R([0, 1, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-9, 1, 16, 12, -15, -7, -1], R![0, 1, 0, 1]);
sage: X = HyperellipticCurve(R([-36, 4, 65, 48, -58, -28, -3]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(417152\) | \(=\) | \( 2^{7} \cdot 3259 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(-417152\) | \(=\) | \( - 2^{7} \cdot 3259 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(5468\) | \(=\) | \( 2^{2} \cdot 1367 \) |
\( I_4 \) | \(=\) | \(10622470\) | \(=\) | \( 2 \cdot 5 \cdot 47 \cdot 97 \cdot 233 \) |
\( I_6 \) | \(=\) | \(7817476501\) | \(=\) | \( 7817476501 \) |
\( I_{10} \) | \(=\) | \(-52144\) | \(=\) | \( - 2^{4} \cdot 3259 \) |
\( J_2 \) | \(=\) | \(5468\) | \(=\) | \( 2^{2} \cdot 1367 \) |
\( J_4 \) | \(=\) | \(-5835854\) | \(=\) | \( - 2 \cdot 2917927 \) |
\( J_6 \) | \(=\) | \(4185810564\) | \(=\) | \( 2^{2} \cdot 3 \cdot 13 \cdot 439 \cdot 61121 \) |
\( J_8 \) | \(=\) | \(-2792294936341\) | \(=\) | \( - 23 \cdot 121404127667 \) |
\( J_{10} \) | \(=\) | \(-417152\) | \(=\) | \( - 2^{7} \cdot 3259 \) |
\( g_1 \) | \(=\) | \(-38188496456892856/3259\) | ||
\( g_2 \) | \(=\) | \(7453838285890001/3259\) | ||
\( g_3 \) | \(=\) | \(-1955494539257649/6518\) |
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);
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\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $C_2$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
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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 except over $\Q_{2}$.
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 | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 + 4xz - 4z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-21xz^2 + 16z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 + 4xz - 4z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-21xz^2 + 16z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 + 4xz - 4z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(x^3 - 41xz^2 + 32z^3\) | \(0\) | \(2\) |
2-torsion field: 6.2.21751973888.4
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 1 \) |
Real period: | \( 0.691503 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 2 \) |
Leading coefficient: | \( 2.766013 \) |
Analytic order of Ш: | \( 16 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(7\) | \(7\) | \(1\) | \(1 - T\) | |
\(3259\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 84 T + 3259 T^{2} )\) |
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 |
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);