Minimal equation
Minimal equation
Simplified equation
$y^2 + xy = -8x^6 - 28x^5 - 65x^4 - 88x^3 - 88x^2 - 51x - 20$ | (homogenize, simplify) |
$y^2 + xz^2y = -8x^6 - 28x^5z - 65x^4z^2 - 88x^3z^3 - 88x^2z^4 - 51xz^5 - 20z^6$ | (dehomogenize, simplify) |
$y^2 = -32x^6 - 112x^5 - 260x^4 - 352x^3 - 351x^2 - 204x - 80$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-20, -51, -88, -88, -65, -28, -8]), R([0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-20, -51, -88, -88, -65, -28, -8], R![0, 1]);
sage: X = HyperellipticCurve(R([-80, -204, -351, -352, -260, -112, -32]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(2872\) | \(=\) | \( 2^{3} \cdot 359 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-367616\) | \(=\) | \( - 2^{10} \cdot 359 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(52152\) | \(=\) | \( 2^{3} \cdot 3 \cdot 41 \cdot 53 \) |
\( I_4 \) | \(=\) | \(30585\) | \(=\) | \( 3 \cdot 5 \cdot 2039 \) |
\( I_6 \) | \(=\) | \(530058255\) | \(=\) | \( 3 \cdot 5 \cdot 2999 \cdot 11783 \) |
\( I_{10} \) | \(=\) | \(45952\) | \(=\) | \( 2^{7} \cdot 359 \) |
\( J_2 \) | \(=\) | \(52152\) | \(=\) | \( 2^{3} \cdot 3 \cdot 41 \cdot 53 \) |
\( J_4 \) | \(=\) | \(113305906\) | \(=\) | \( 2 \cdot 7 \cdot 1291 \cdot 6269 \) |
\( J_6 \) | \(=\) | \(328168275184\) | \(=\) | \( 2^{4} \cdot 317 \cdot 1249 \cdot 51803 \) |
\( J_8 \) | \(=\) | \(1069100888228783\) | \(=\) | \( 543019 \cdot 1968809357 \) |
\( J_{10} \) | \(=\) | \(367616\) | \(=\) | \( 2^{10} \cdot 359 \) |
\( g_1 \) | \(=\) | \(376751407549293075168/359\) | ||
\( g_2 \) | \(=\) | \(15695150888732498127/359\) | ||
\( g_3 \) | \(=\) | \(871642853702611839/359\) |
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);
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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 $\R$.
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: \(\Z \oplus \Z/{4}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 + xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.379949\) | \(\infty\) |
\(D_0 - D_\infty\) | \(6x^2 + 7xz + 8z^2\) | \(=\) | \(0,\) | \(18y\) | \(=\) | \(-7xz^2 + 4z^3\) | \(0\) | \(4\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 + xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.379949\) | \(\infty\) |
\(D_0 - D_\infty\) | \(6x^2 + 7xz + 8z^2\) | \(=\) | \(0,\) | \(18y\) | \(=\) | \(-7xz^2 + 4z^3\) | \(0\) | \(4\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 + xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2\) | \(0.379949\) | \(\infty\) |
\(D_0 - D_\infty\) | \(6x^2 + 7xz + 8z^2\) | \(=\) | \(0,\) | \(18y\) | \(=\) | \(-13xz^2 + 8z^3\) | \(0\) | \(4\) |
2-torsion field: 6.4.8248384.1
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(1\) |
Mordell-Weil rank: | \(1\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 0.379949 \) |
Real period: | \( 4.123655 \) |
Tamagawa product: | \( 2 \) |
Torsion order: | \( 4 \) |
Leading coefficient: | \( 0.391694 \) |
Analytic order of Ш: | \( 2 \) (rounded) |
Order of Ш: | twice a square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(3\) | \(10\) | \(2\) | \(1 + T + 2 T^{2}\) | |
\(359\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 359 T^{2} )\) |
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);