Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x + 1)y = x^3 + x^2 + x$ | (homogenize, simplify) |
$y^2 + (x^3 + xz^2 + z^3)y = x^3z^3 + x^2z^4 + xz^5$ | (dehomogenize, simplify) |
$y^2 = x^6 + 2x^4 + 6x^3 + 5x^2 + 6x + 1$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, 1, 1]), R([1, 1, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, 1, 1], R![1, 1, 0, 1]);
sage: X = HyperellipticCurve(R([1, 6, 5, 6, 2, 0, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(31329\) | \(=\) | \( 3^{2} \cdot 59^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(31329\) | \(=\) | \( 3^{2} \cdot 59^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(92\) | \(=\) | \( 2^{2} \cdot 23 \) |
\( I_4 \) | \(=\) | \(625\) | \(=\) | \( 5^{4} \) |
\( I_6 \) | \(=\) | \(-128553\) | \(=\) | \( - 3 \cdot 73 \cdot 587 \) |
\( I_{10} \) | \(=\) | \(-4010112\) | \(=\) | \( - 2^{7} \cdot 3^{2} \cdot 59^{2} \) |
\( J_2 \) | \(=\) | \(23\) | \(=\) | \( 23 \) |
\( J_4 \) | \(=\) | \(-4\) | \(=\) | \( - 2^{2} \) |
\( J_6 \) | \(=\) | \(1980\) | \(=\) | \( 2^{2} \cdot 3^{2} \cdot 5 \cdot 11 \) |
\( J_8 \) | \(=\) | \(11381\) | \(=\) | \( 19 \cdot 599 \) |
\( J_{10} \) | \(=\) | \(-31329\) | \(=\) | \( - 3^{2} \cdot 59^{2} \) |
\( g_1 \) | \(=\) | \(-6436343/31329\) | ||
\( g_2 \) | \(=\) | \(48668/31329\) | ||
\( g_3 \) | \(=\) | \(-116380/3481\) |
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
Known points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (-3 : 7 : 2),\, (-3 : 24 : 2)\)
magma: [C![-3,7,2],C![-3,24,2],C![0,-1,1],C![0,0,1],C![1,-1,0],C![1,0,0]]; // minimal model
magma: [C![-3,-17,2],C![-3,17,2],C![0,-1,1],C![0,1,1],C![1,-1,0],C![1,1,0]]; // 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: \(\Z \oplus \Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.312443\) | \(\infty\) |
\((0 : -1 : 1) - (1 : 0 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0.231589\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.312443\) | \(\infty\) |
\((0 : -1 : 1) - (1 : 0 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0.231589\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + xz^2 + z^3\) | \(0.312443\) | \(\infty\) |
\((0 : -1 : 1) - (1 : 1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 + xz^2 - z^3\) | \(0.231589\) | \(\infty\) |
2-torsion field: 5.1.2005056.1
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(2\) |
Mordell-Weil rank: | \(2\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 0.065821 \) |
Real period: | \( 13.74196 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 0.904512 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(3\) | \(2\) | \(2\) | \(1\) | \(( 1 + T )^{2}\) | |
\(59\) | \(2\) | \(2\) | \(1\) | \(( 1 + T )^{2}\) |
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);