Minimal equation
Minimal equation
Simplified equation
| $y^2 + (x^2 + x)y = -5x^6 + 11x^5 - 20x^4 + 20x^3 - 20x^2 + 11x - 5$ | (homogenize, simplify) |
| $y^2 + (x^2z + xz^2)y = -5x^6 + 11x^5z - 20x^4z^2 + 20x^3z^3 - 20x^2z^4 + 11xz^5 - 5z^6$ | (dehomogenize, simplify) |
| $y^2 = -20x^6 + 44x^5 - 79x^4 + 82x^3 - 79x^2 + 44x - 20$ | (minimize, homogenize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-5, 11, -20, 20, -20, 11, -5]), R([0, 1, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-5, 11, -20, 20, -20, 11, -5], R![0, 1, 1]);
sage: X = HyperellipticCurve(R([-20, 44, -79, 82, -79, 44, -20]))
magma: X,pi:= SimplifiedModel(C);
Invariants
| Conductor: | \( N \) | \(=\) | \(644\) | \(=\) | \( 2^{2} \cdot 7 \cdot 23 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(-2576\) | \(=\) | \( - 2^{4} \cdot 7 \cdot 23 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(39036\) | \(=\) | \( 2^{2} \cdot 3 \cdot 3253 \) |
| \( I_4 \) | \(=\) | \(4124865\) | \(=\) | \( 3 \cdot 5 \cdot 73 \cdot 3767 \) |
| \( I_6 \) | \(=\) | \(50880984159\) | \(=\) | \( 3 \cdot 73 \cdot 643 \cdot 361327 \) |
| \( I_{10} \) | \(=\) | \(329728\) | \(=\) | \( 2^{11} \cdot 7 \cdot 23 \) |
| \( J_2 \) | \(=\) | \(9759\) | \(=\) | \( 3 \cdot 3253 \) |
| \( J_4 \) | \(=\) | \(3796384\) | \(=\) | \( 2^{5} \cdot 31 \cdot 43 \cdot 89 \) |
| \( J_6 \) | \(=\) | \(1910683600\) | \(=\) | \( 2^{4} \cdot 5^{2} \cdot 7 \cdot 23 \cdot 29669 \) |
| \( J_8 \) | \(=\) | \(1058457444236\) | \(=\) | \( 2^{2} \cdot 269891 \cdot 980449 \) |
| \( J_{10} \) | \(=\) | \(2576\) | \(=\) | \( 2^{4} \cdot 7 \cdot 23 \) |
| \( g_1 \) | \(=\) | \(88516980336138032799/2576\) | ||
| \( g_2 \) | \(=\) | \(220529201888022246/161\) | ||
| \( g_3 \) | \(=\) | \(70640465629725\) |
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: [];
Number of rational Weierstrass points: \(0\)
magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));
This curve is locally solvable except over $\R$.
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: \(\Z/{6}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - D_\infty\) | \(x^2 - xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2\) | \(0\) | \(6\) |
2-torsion field: 8.0.1698758656.7
BSD invariants
| Hasse-Weil conjecture: | verified |
| Analytic rank: | \(0\) |
| Mordell-Weil rank: | \(0\) |
| 2-Selmer rank: | \(2\) |
| Regulator: | \( 1 \) |
| Real period: | \( 3.928431 \) |
| Tamagawa product: | \( 1 \) |
| Torsion order: | \( 6 \) |
| Leading coefficient: | \( 0.218246 \) |
| Analytic order of Ш: | \( 2 \) (rounded) |
| Order of Ш: | twice a square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(2\) | \(2\) | \(4\) | \(1\) | \(( 1 + T )^{2}\) | |
| \(7\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 4 T + 7 T^{2} )\) |  |
| \(23\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 23 T^{2} )\) | |
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 14.a4
Elliptic curve 46.a1
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 \(\Q\).