Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x + 1)y = x^3 + x^2 - 2x$ | (homogenize, simplify) |
$y^2 + (x^3 + xz^2 + z^3)y = x^3z^3 + x^2z^4 - 2xz^5$ | (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, -2, 1, 1]), R([1, 1, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -2, 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: | \( N \) | \(=\) | \(8649\) | \(=\) | \( 3^{2} \cdot 31^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(77841\) | \(=\) | \( 3^{4} \cdot 31^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(92\) | \(=\) | \( 2^{2} \cdot 23 \) |
\( I_4 \) | \(=\) | \(17689\) | \(=\) | \( 7^{2} \cdot 19^{2} \) |
\( I_6 \) | \(=\) | \(603507\) | \(=\) | \( 3 \cdot 37 \cdot 5437 \) |
\( I_{10} \) | \(=\) | \(-9963648\) | \(=\) | \( - 2^{7} \cdot 3^{4} \cdot 31^{2} \) |
\( J_2 \) | \(=\) | \(23\) | \(=\) | \( 23 \) |
\( J_4 \) | \(=\) | \(-715\) | \(=\) | \( - 5 \cdot 11 \cdot 13 \) |
\( J_6 \) | \(=\) | \(-3645\) | \(=\) | \( - 3^{6} \cdot 5 \) |
\( J_8 \) | \(=\) | \(-148765\) | \(=\) | \( - 5 \cdot 29753 \) |
\( J_{10} \) | \(=\) | \(-77841\) | \(=\) | \( - 3^{4} \cdot 31^{2} \) |
\( g_1 \) | \(=\) | \(-6436343/77841\) | ||
\( g_2 \) | \(=\) | \(8699405/77841\) | ||
\( g_3 \) | \(=\) | \(23805/961\) |
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 | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((1 : 0 : 1)\) | \((-1 : -1 : 1)\) |
\((-2 : 0 : 1)\) | \((-1 : 2 : 1)\) | \((1 : -3 : 1)\) | \((1 : -5 : 2)\) | \((1 : -8 : 2)\) | \((4 : 8 : 3)\) |
\((-2 : 9 : 1)\) | \((4 : -135 : 3)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((1 : 0 : 1)\) | \((-1 : -1 : 1)\) |
\((-2 : 0 : 1)\) | \((-1 : 2 : 1)\) | \((1 : -3 : 1)\) | \((1 : -5 : 2)\) | \((1 : -8 : 2)\) | \((4 : 8 : 3)\) |
\((-2 : 9 : 1)\) | \((4 : -135 : 3)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((-1 : -3 : 1)\) | \((-1 : 3 : 1)\) |
\((1 : -3 : 1)\) | \((1 : 3 : 1)\) | \((1 : -3 : 2)\) | \((1 : 3 : 2)\) | \((-2 : -9 : 1)\) | \((-2 : 9 : 1)\) |
\((4 : -143 : 3)\) | \((4 : 143 : 3)\) |
magma: [C![-2,0,1],C![-2,9,1],C![-1,-1,1],C![-1,2,1],C![0,-1,1],C![0,0,1],C![1,-8,2],C![1,-5,2],C![1,-3,1],C![1,-1,0],C![1,0,0],C![1,0,1],C![4,-135,3],C![4,8,3]]; // minimal model
magma: [C![-2,-9,1],C![-2,9,1],C![-1,-3,1],C![-1,3,1],C![0,-1,1],C![0,1,1],C![1,-3,2],C![1,3,2],C![1,-3,1],C![1,-1,0],C![1,1,0],C![1,3,1],C![4,-143,3],C![4,143,3]]; // 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 | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : -1 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.066384\) | \(\infty\) |
\((-1 : 2 : 1) + (1 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2 + z^3\) | \(0.070872\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : -1 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.066384\) | \(\infty\) |
\((-1 : 2 : 1) + (1 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2 + z^3\) | \(0.070872\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : -3 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + xz^2 - z^3\) | \(0.066384\) | \(\infty\) |
\((-1 : 3 : 1) + (1 : 3 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - xz^2 + 3z^3\) | \(0.070872\) | \(\infty\) |
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(2\) |
Mordell-Weil rank: | \(2\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 0.004684 \) |
Real period: | \( 18.14230 \) |
Tamagawa product: | \( 4 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 0.339964 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(3\) | \(2\) | \(4\) | \(4\) | \(( 1 + T )^{2}\) | |
\(31\) | \(2\) | \(2\) | \(1\) | \(( 1 + T )^{2}\) |
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? |
---|---|---|
\(2\) | 2.120.2 | 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);