Minimal equation
Minimal equation
Simplified equation
| $y^2 + (x^2 + x + 1)y = -x^6 + 3x^5 - 3x^4 - x$ | (homogenize, simplify) |
| $y^2 + (x^2z + xz^2 + z^3)y = -x^6 + 3x^5z - 3x^4z^2 - xz^5$ | (dehomogenize, simplify) |
| $y^2 = -4x^6 + 12x^5 - 11x^4 + 2x^3 + 3x^2 - 2x + 1$ | (minimize, homogenize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, 0, 0, -3, 3, -1]), R([1, 1, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, 0, 0, -3, 3, -1], R![1, 1, 1]);
sage: X = HyperellipticCurve(R([1, -2, 3, 2, -11, 12, -4]))
magma: X,pi:= SimplifiedModel(C);
Invariants
| Conductor: | \( N \) | \(=\) | \(980\) | \(=\) | \( 2^{2} \cdot 5 \cdot 7^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(7840\) | \(=\) | \( 2^{5} \cdot 5 \cdot 7^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(276\) | \(=\) | \( 2^{2} \cdot 3 \cdot 23 \) |
| \( I_4 \) | \(=\) | \(3945\) | \(=\) | \( 3 \cdot 5 \cdot 263 \) |
| \( I_6 \) | \(=\) | \(280149\) | \(=\) | \( 3 \cdot 93383 \) |
| \( I_{10} \) | \(=\) | \(1003520\) | \(=\) | \( 2^{12} \cdot 5 \cdot 7^{2} \) |
| \( J_2 \) | \(=\) | \(69\) | \(=\) | \( 3 \cdot 23 \) |
| \( J_4 \) | \(=\) | \(34\) | \(=\) | \( 2 \cdot 17 \) |
| \( J_6 \) | \(=\) | \(20\) | \(=\) | \( 2^{2} \cdot 5 \) |
| \( J_8 \) | \(=\) | \(56\) | \(=\) | \( 2^{3} \cdot 7 \) |
| \( J_{10} \) | \(=\) | \(7840\) | \(=\) | \( 2^{5} \cdot 5 \cdot 7^{2} \) |
| \( g_1 \) | \(=\) | \(1564031349/7840\) | ||
| \( g_2 \) | \(=\) | \(5584653/3920\) | ||
| \( g_3 \) | \(=\) | \(4761/392\) |
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: \((0 : 0 : 1),\, (0 : -1 : 1),\, (1 : -1 : 1),\, (1 : -2 : 1)\)
magma: [C![0,-1,1],C![0,0,1],C![1,-2,1],C![1,-1,1]];
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: \(\Z/{12}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(2 \cdot(1 : -1 : 1) - D_\infty\) | \((x - z)^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2\) | \(0\) | \(12\) |
2-torsion field: 8.0.96040000.1
BSD invariants
| Hasse-Weil conjecture: | verified |
| Analytic rank: | \(0\) |
| Mordell-Weil rank: | \(0\) |
| 2-Selmer rank: | \(1\) |
| Regulator: | \( 1 \) |
| Real period: | \( 14.03151 \) |
| Tamagawa product: | \( 4 \) |
| Torsion order: | \( 12 \) |
| Leading coefficient: | \( 0.389764 \) |
| Analytic order of Ш: | \( 1 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(2\) | \(2\) | \(5\) | \(4\) | \(( 1 - T )( 1 + T )\) | |
| \(5\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 5 T^{2} )\) |  |
| \(7\) | \(2\) | \(2\) | \(1\) | \(( 1 - T )( 1 + T )\) |  |
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 70.a4
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 \(\Q\).