Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + 1)y = -4x^6 + 2x^5 - 14x^4 + 18x^3 - 20x^2 + 25x - 23$ | (homogenize, simplify) |
$y^2 + (x^3 + z^3)y = -4x^6 + 2x^5z - 14x^4z^2 + 18x^3z^3 - 20x^2z^4 + 25xz^5 - 23z^6$ | (dehomogenize, simplify) |
$y^2 = -15x^6 + 8x^5 - 56x^4 + 74x^3 - 80x^2 + 100x - 91$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-23, 25, -20, 18, -14, 2, -4]), R([1, 0, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-23, 25, -20, 18, -14, 2, -4], R![1, 0, 0, 1]);
sage: X = HyperellipticCurve(R([-91, 100, -80, 74, -56, 8, -15]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(548740\) | \(=\) | \( 2^{2} \cdot 5 \cdot 27437 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-548740\) | \(=\) | \( - 2^{2} \cdot 5 \cdot 27437 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(167212\) | \(=\) | \( 2^{2} \cdot 17 \cdot 2459 \) |
\( I_4 \) | \(=\) | \(824940097\) | \(=\) | \( 12973 \cdot 63589 \) |
\( I_6 \) | \(=\) | \(38082033914927\) | \(=\) | \( 9533 \cdot 3994758619 \) |
\( I_{10} \) | \(=\) | \(70238720\) | \(=\) | \( 2^{9} \cdot 5 \cdot 27437 \) |
\( J_2 \) | \(=\) | \(41803\) | \(=\) | \( 17 \cdot 2459 \) |
\( J_4 \) | \(=\) | \(38439613\) | \(=\) | \( 71 \cdot 109 \cdot 4967 \) |
\( J_6 \) | \(=\) | \(39312521235\) | \(=\) | \( 3^{3} \cdot 5 \cdot 13 \cdot 19 \cdot 1178963 \) |
\( J_8 \) | \(=\) | \(41444369399234\) | \(=\) | \( 2 \cdot 83^{3} \cdot 36241091 \) |
\( J_{10} \) | \(=\) | \(548740\) | \(=\) | \( 2^{2} \cdot 5 \cdot 27437 \) |
\( g_1 \) | \(=\) | \(127654829703532651729243/548740\) | ||
\( g_2 \) | \(=\) | \(2808027502126164181351/548740\) | ||
\( g_3 \) | \(=\) | \(13739653907355965823/109748\) |
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);
|
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: \(\Z/{2}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(3x^2 + 2xz + 7z^2\) | \(=\) | \(0,\) | \(18y\) | \(=\) | \(17xz^2 - 23z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(3x^2 + 2xz + 7z^2\) | \(=\) | \(0,\) | \(18y\) | \(=\) | \(17xz^2 - 23z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(3x^2 + 2xz + 7z^2\) | \(=\) | \(0,\) | \(18y\) | \(=\) | \(x^3 + 34xz^2 - 45z^3\) | \(0\) | \(2\) |
2-torsion field: 6.4.24089247008000.1
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(4\) |
Regulator: | \( 1 \) |
Real period: | \( 1.286350 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 2 \) |
Leading coefficient: | \( 2.572701 \) |
Analytic order of Ш: | \( 8 \) (rounded) |
Order of Ш: | twice a square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(2\) | \(2\) | \(1\) | \(1 + T^{2}\) | |
\(5\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 2 T + 5 T^{2} )\) | |
\(27437\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 186 T + 27437 T^{2} )\) |
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);