Minimal equation
Minimal equation
Simplified equation
| $y^2 + (x^2 + x)y = -x^6 + 3x^5 - 6x^4 + 6x^3 - 6x^2 + 3x - 1$ | (homogenize, simplify) |
| $y^2 + (x^2z + xz^2)y = -x^6 + 3x^5z - 6x^4z^2 + 6x^3z^3 - 6x^2z^4 + 3xz^5 - z^6$ | (dehomogenize, simplify) |
| $y^2 = -4x^6 + 12x^5 - 23x^4 + 26x^3 - 23x^2 + 12x - 4$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, 3, -6, 6, -6, 3, -1]), R([0, 1, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, 3, -6, 6, -6, 3, -1], R![0, 1, 1]);
sage: X = HyperellipticCurve(R([-4, 12, -23, 26, -23, 12, -4]))
magma: X,pi:= SimplifiedModel(C);
Invariants
| Conductor: | \( N \) | \(=\) | \(676\) | \(=\) | \( 2^{2} \cdot 13^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(-17576\) | \(=\) | \( - 2^{3} \cdot 13^{3} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(1244\) | \(=\) | \( 2^{2} \cdot 311 \) |
| \( I_4 \) | \(=\) | \(1249\) | \(=\) | \( 1249 \) |
| \( I_6 \) | \(=\) | \(129167\) | \(=\) | \( 37 \cdot 3491 \) |
| \( I_{10} \) | \(=\) | \(2249728\) | \(=\) | \( 2^{10} \cdot 13^{3} \) |
| \( J_2 \) | \(=\) | \(311\) | \(=\) | \( 311 \) |
| \( J_4 \) | \(=\) | \(3978\) | \(=\) | \( 2 \cdot 3^{2} \cdot 13 \cdot 17 \) |
| \( J_6 \) | \(=\) | \(72332\) | \(=\) | \( 2^{2} \cdot 13^{2} \cdot 107 \) |
| \( J_8 \) | \(=\) | \(1667692\) | \(=\) | \( 2^{2} \cdot 13^{2} \cdot 2467 \) |
| \( J_{10} \) | \(=\) | \(17576\) | \(=\) | \( 2^{3} \cdot 13^{3} \) |
| \( g_1 \) | \(=\) | \(2909390022551/17576\) | ||
| \( g_2 \) | \(=\) | \(4602275343/676\) | ||
| \( g_3 \) | \(=\) | \(10349147/26\) |
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
Automorphism group
| \(\mathrm{Aut}(X)\) | \(\simeq\) | $D_6$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
|
| \(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $D_6$ | 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 $\R$, $\Q_{2}$, and $\Q_{11}$.
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: \(\Z/{3}\Z \oplus \Z/{3}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - D_\infty\) | \(2x^2 - 2xz + z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2\) | \(0\) | \(3\) |
| \(D_0 - D_\infty\) | \(x^2 - xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2\) | \(0\) | \(3\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - D_\infty\) | \(2x^2 - 2xz + z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2\) | \(0\) | \(3\) |
| \(D_0 - D_\infty\) | \(x^2 - xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2\) | \(0\) | \(3\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - D_\infty\) | \(2x^2 - 2xz + z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(x^2z - xz^2\) | \(0\) | \(3\) |
| \(D_0 - D_\infty\) | \(x^2 - xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^2z - xz^2\) | \(0\) | \(3\) |
BSD invariants
| Hasse-Weil conjecture: | verified |
| Analytic rank: | \(0\) |
| Mordell-Weil rank: | \(0\) |
| 2-Selmer rank: | \(0\) |
| Regulator: | \( 1 \) |
| Real period: | \( 7.177120 \) |
| Tamagawa product: | \( 3 \) |
| Torsion order: | \( 9 \) |
| Leading coefficient: | \( 0.265819 \) |
| Analytic order of Ш: | \( 1 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(2\) | \(2\) | \(3\) | \(1\) | \(( 1 + T )^{2}\) | |
| \(13\) | \(2\) | \(3\) | \(3\) | \(( 1 - T )^{2}\) |  |
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? |
|---|---|---|
| \(2\) | 2.120.4 | no |
| \(3\) | 3.17280.1 | yes |
Sato-Tate group
| \(\mathrm{ST}\) | \(\simeq\) | $E_1$ |
| \(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\) |
Decomposition of the Jacobian
Splits over \(\Q\)
Decomposes up to isogeny as the square of the elliptic curve isogeny class:
Elliptic curve isogeny class 26.a
magma: HeuristicDecompositionFactors(C);
Endomorphisms of the Jacobian
Not of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
| \(\End (J_{})\) | \(\simeq\) | an Eichler order of index \(3\) in a maximal order of \(\End (J_{}) \otimes \Q\) |
| \(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\mathrm{M}_2(\)\(\Q\)\()\) |
| \(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\mathrm{M}_2 (\R)\) |
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);