Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x + 1)y = x^6 + 5x^5 + 7x^4 - 10x^2 - 4x + 4$ | (homogenize, simplify) |
$y^2 + (x^3 + xz^2 + z^3)y = x^6 + 5x^5z + 7x^4z^2 - 10x^2z^4 - 4xz^5 + 4z^6$ | (dehomogenize, simplify) |
$y^2 = 5x^6 + 20x^5 + 30x^4 + 2x^3 - 39x^2 - 14x + 17$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([4, -4, -10, 0, 7, 5, 1]), R([1, 1, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![4, -4, -10, 0, 7, 5, 1], R![1, 1, 0, 1]);
sage: X = HyperellipticCurve(R([17, -14, -39, 2, 30, 20, 5]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(75625\) | \(=\) | \( 5^{4} \cdot 11^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(75625\) | \(=\) | \( 5^{4} \cdot 11^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(6428\) | \(=\) | \( 2^{2} \cdot 1607 \) |
\( I_4 \) | \(=\) | \(1809025\) | \(=\) | \( 5^{2} \cdot 269^{2} \) |
\( I_6 \) | \(=\) | \(4688688575\) | \(=\) | \( 5^{2} \cdot 23 \cdot 839 \cdot 9719 \) |
\( I_{10} \) | \(=\) | \(-9680000\) | \(=\) | \( - 2^{7} \cdot 5^{4} \cdot 11^{2} \) |
\( J_2 \) | \(=\) | \(1607\) | \(=\) | \( 1607 \) |
\( J_4 \) | \(=\) | \(32226\) | \(=\) | \( 2 \cdot 3 \cdot 41 \cdot 131 \) |
\( J_6 \) | \(=\) | \(-21867176\) | \(=\) | \( - 2^{3} \cdot 19 \cdot 293 \cdot 491 \) |
\( J_8 \) | \(=\) | \(-9044766727\) | \(=\) | \( - 33967 \cdot 266281 \) |
\( J_{10} \) | \(=\) | \(-75625\) | \(=\) | \( - 5^{4} \cdot 11^{2} \) |
\( g_1 \) | \(=\) | \(-10717151840024807/75625\) | ||
\( g_2 \) | \(=\) | \(-133737756368718/75625\) | ||
\( g_3 \) | \(=\) | \(56470866794024/75625\) |
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 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: \(\Z/{5}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 + xz - z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2\) | \(0\) | \(5\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 + xz - z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2\) | \(0\) | \(5\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 + xz - z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - xz^2 + z^3\) | \(0\) | \(5\) |
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 1 \) |
Real period: | \( 18.94815 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 5 \) |
Leading coefficient: | \( 3.031704 \) |
Analytic order of Ш: | \( 4 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(5\) | \(4\) | \(4\) | \(1\) | \(1\) | |
\(11\) | \(2\) | \(2\) | \(1\) | \(( 1 - T )^{2}\) |
Galois representations
For primes $\ell \ge 5$ the Galois representation data has not been computed for this curve since it is not generic.
For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not all of $\GSp(4,\F_\ell)$.
Prime \(\ell\) | mod-\(\ell\) image | Is torsion prime? |
---|---|---|
\(2\) | 2.12.2 | no |
\(3\) | 3.432.4 | no |
Sato-Tate group
\(\mathrm{ST}\) | \(\simeq\) | $\mathrm{SU}(2)\times\mathrm{SU}(2)$ |
\(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\times\mathrm{SU}(2)\) |
Decomposition of the Jacobian
Simple over \(\overline{\Q}\)
magma: HeuristicDecompositionFactors(C);
Endomorphisms of the Jacobian
Of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
\(\End (J_{})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{5}}{2}]\) |
\(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{5}) \) |
\(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
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);