Minimal equation
Minimal equation
Simplified equation
| $y^2 + (x^3 + x)y = -1$ | (homogenize, simplify) |
| $y^2 + (x^3 + xz^2)y = -z^6$ | (dehomogenize, simplify) |
| $y^2 = x^6 + 2x^4 + x^2 - 4$ | (minimize, homogenize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1]), R([0, 1, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1], R![0, 1, 0, 1]);
sage: X = HyperellipticCurve(R([-4, 0, 1, 0, 2, 0, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
| Conductor: | \( N \) | \(=\) | \(784\) | \(=\) | \( 2^{4} \cdot 7^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(12544\) | \(=\) | \( 2^{8} \cdot 7^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(116\) | \(=\) | \( 2^{2} \cdot 29 \) |
| \( I_4 \) | \(=\) | \(445\) | \(=\) | \( 5 \cdot 89 \) |
| \( I_6 \) | \(=\) | \(16259\) | \(=\) | \( 71 \cdot 229 \) |
| \( I_{10} \) | \(=\) | \(1568\) | \(=\) | \( 2^{5} \cdot 7^{2} \) |
| \( J_2 \) | \(=\) | \(116\) | \(=\) | \( 2^{2} \cdot 29 \) |
| \( J_4 \) | \(=\) | \(264\) | \(=\) | \( 2^{3} \cdot 3 \cdot 11 \) |
| \( J_6 \) | \(=\) | \(-1280\) | \(=\) | \( - 2^{8} \cdot 5 \) |
| \( J_8 \) | \(=\) | \(-54544\) | \(=\) | \( - 2^{4} \cdot 7 \cdot 487 \) |
| \( J_{10} \) | \(=\) | \(12544\) | \(=\) | \( 2^{8} \cdot 7^{2} \) |
| \( g_1 \) | \(=\) | \(82044596/49\) | ||
| \( g_2 \) | \(=\) | \(1609674/49\) | ||
| \( g_3 \) | \(=\) | \(-67280/49\) |
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
All points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (-1 : 1 : 1),\, (1 : -1 : 1)\)
magma: [C![-1,1,1],C![1,-1,0],C![1,-1,1],C![1,0,0]];
Number of rational Weierstrass points: \(2\)
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: \(\Z/{2}\Z \times \Z/{6}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2 + xz + 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0\) | \(2\) |
| \(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2 + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2\) | \(0\) | \(6\) |
2-torsion field: \(\Q(\sqrt{-7}) \)
BSD invariants
| Hasse-Weil conjecture: | verified |
| Analytic rank: | \(0\) |
| Mordell-Weil rank: | \(0\) |
| 2-Selmer rank: | \(2\) |
| Regulator: | \( 1 \) |
| Real period: | \( 11.27010 \) |
| Tamagawa product: | \( 4 \) |
| Torsion order: | \( 12 \) |
| Leading coefficient: | \( 0.313058 \) |
| Analytic order of Ш: | \( 1 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(2\) | \(4\) | \(8\) | \(4\) | \(1 + T\) | |
| \(7\) | \(2\) | \(2\) | \(1\) | \(( 1 - T )^{2}\) |  |
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
Splits over \(\Q\)
Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
Elliptic curve 56.b2
Elliptic curve 14.a5
Endomorphisms of the Jacobian
Of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
| \(\End (J_{})\) | \(\simeq\) | an order of index \(2\) in \(\Z \times \Z\) |
| \(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q\) |
| \(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).