Minimal equation
Minimal equation
Simplified equation
| $y^2 + (x^3 + 1)y = -2x^6 - 2x^5 + 2x^3 - 2x - 2$ | (homogenize, simplify) |
| $y^2 + (x^3 + z^3)y = -2x^6 - 2x^5z + 2x^3z^3 - 2xz^5 - 2z^6$ | (dehomogenize, simplify) |
| $y^2 = -7x^6 - 8x^5 + 10x^3 - 8x - 7$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-2, -2, 0, 2, 0, -2, -2]), R([1, 0, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-2, -2, 0, 2, 0, -2, -2], R![1, 0, 0, 1]);
sage: X = HyperellipticCurve(R([-7, -8, 0, 10, 0, -8, -7]))
magma: X,pi:= SimplifiedModel(C);
Invariants
| Conductor: | \( N \) | \(=\) | \(2500\) | \(=\) | \( 2^{2} \cdot 5^{4} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(-400000\) | \(=\) | \( - 2^{7} \cdot 5^{5} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(860\) | \(=\) | \( 2^{2} \cdot 5 \cdot 43 \) |
| \( I_4 \) | \(=\) | \(36865\) | \(=\) | \( 5 \cdot 73 \cdot 101 \) |
| \( I_6 \) | \(=\) | \(8199455\) | \(=\) | \( 5 \cdot 11 \cdot 43 \cdot 3467 \) |
| \( I_{10} \) | \(=\) | \(16384\) | \(=\) | \( 2^{14} \) |
| \( J_2 \) | \(=\) | \(1075\) | \(=\) | \( 5^{2} \cdot 43 \) |
| \( J_4 \) | \(=\) | \(9750\) | \(=\) | \( 2 \cdot 3 \cdot 5^{3} \cdot 13 \) |
| \( J_6 \) | \(=\) | \(107500\) | \(=\) | \( 2^{2} \cdot 5^{4} \cdot 43 \) |
| \( J_8 \) | \(=\) | \(5125000\) | \(=\) | \( 2^{3} \cdot 5^{6} \cdot 41 \) |
| \( J_{10} \) | \(=\) | \(400000\) | \(=\) | \( 2^{7} \cdot 5^{5} \) |
| \( g_1 \) | \(=\) | \(459401384375/128\) | ||
| \( g_2 \) | \(=\) | \(1937983125/64\) | ||
| \( g_3 \) | \(=\) | \(9938375/32\) |
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
Automorphism group
| \(\mathrm{Aut}(X)\) | \(\simeq\) | $C_2^2$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
|
| \(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $C_2^2$ | 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 $\R$ and $\Q_{5}$.
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 | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - D_\infty\) | \(2x^2 + 3xz + 2z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2 - 2z^3\) | \(0\) | \(5\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - D_\infty\) | \(2x^2 + 3xz + 2z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2 - 2z^3\) | \(0\) | \(5\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - D_\infty\) | \(2x^2 + 3xz + 2z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(x^3 - 2xz^2 - 3z^3\) | \(0\) | \(5\) |
BSD invariants
| Hasse-Weil conjecture: | verified |
| Analytic rank: | \(0\) |
| Mordell-Weil rank: | \(0\) |
| 2-Selmer rank: | \(0\) |
| Regulator: | \( 1 \) |
| Real period: | \( 3.411821 \) |
| Tamagawa product: | \( 5 \) |
| Torsion order: | \( 5 \) |
| Leading coefficient: | \( 0.682364 \) |
| Analytic order of Ш: | \( 1 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(2\) | \(2\) | \(7\) | \(5\) | \(( 1 - T )( 1 + T )\) | |
| \(5\) | \(4\) | \(5\) | \(1\) | \(1\) |  |
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? |
|---|---|---|
| \(2\) | 2.60.2 | no |
| \(3\) | 3.2880.2 | no |
Sato-Tate group
| \(\mathrm{ST}\) | \(\simeq\) | $J(E_1)$ |
| \(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\) |
Decomposition of the Jacobian
Splits over \(\Q\)
Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
Elliptic curve isogeny class 50.b
Elliptic curve isogeny class 50.a
magma: HeuristicDecompositionFactors(C);
Endomorphisms of the Jacobian
Of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
| \(\End (J_{})\) | \(\simeq\) | an order of index \(2\) in \(\Z \times \Z\) |
| \(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q\) |
| \(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\sqrt{5}) \) with defining polynomial \(x^{2} - x - 1\)
Not of \(\GL_2\)-type over \(\overline{\Q}\)
Endomorphism ring over \(\overline{\Q}\):
| \(\End (J_{\overline{\Q}})\) | \(\simeq\) | an Eichler order of index \(15\) in a maximal order of \(\End (J_{\overline{\Q}}) \otimes \Q\) |
| \(\End (J_{\overline{\Q}}) \otimes \Q \) | \(\simeq\) | \(\mathrm{M}_2(\)\(\Q\)\()\) |
| \(\End (J_{\overline{\Q}}) \otimes \R\) | \(\simeq\) | \(\mathrm{M}_2 (\R)\) |
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);