Minimal equation
Minimal equation
Simplified equation
| $y^2 + (x^3 + 1)y = 2x^3$ | (homogenize, simplify) |
| $y^2 + (x^3 + z^3)y = 2x^3z^3$ | (dehomogenize, simplify) |
| $y^2 = x^6 + 10x^3 + 1$ | (minimize, homogenize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, 0, 2]), R([1, 0, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, 0, 2], R![1, 0, 0, 1]);
sage: X = HyperellipticCurve(R([1, 0, 0, 10, 0, 0, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
| Conductor: | \( N \) | \(=\) | \(26244\) | \(=\) | \( 2^{2} \cdot 3^{8} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(157464\) | \(=\) | \( 2^{3} \cdot 3^{9} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(60\) | \(=\) | \( 2^{2} \cdot 3 \cdot 5 \) |
| \( I_4 \) | \(=\) | \(945\) | \(=\) | \( 3^{3} \cdot 5 \cdot 7 \) |
| \( I_6 \) | \(=\) | \(2295\) | \(=\) | \( 3^{3} \cdot 5 \cdot 17 \) |
| \( I_{10} \) | \(=\) | \(82944\) | \(=\) | \( 2^{10} \cdot 3^{4} \) |
| \( J_2 \) | \(=\) | \(45\) | \(=\) | \( 3^{2} \cdot 5 \) |
| \( J_4 \) | \(=\) | \(-270\) | \(=\) | \( - 2 \cdot 3^{3} \cdot 5 \) |
| \( J_6 \) | \(=\) | \(3780\) | \(=\) | \( 2^{2} \cdot 3^{3} \cdot 5 \cdot 7 \) |
| \( J_8 \) | \(=\) | \(24300\) | \(=\) | \( 2^{2} \cdot 3^{5} \cdot 5^{2} \) |
| \( J_{10} \) | \(=\) | \(157464\) | \(=\) | \( 2^{3} \cdot 3^{9} \) |
| \( g_1 \) | \(=\) | \(9375/8\) | ||
| \( g_2 \) | \(=\) | \(-625/4\) | ||
| \( g_3 \) | \(=\) | \(875/18\) |
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);
|
| \(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $D_6$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
|
Rational points
All points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1)\)
magma: [C![0,-1,1],C![0,0,1],C![1,-1,0],C![1,0,0]];
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: \(\Z/{3}\Z \times \Z/{3}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \((0 : -1 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0\) | \(3\) |
| \((0 : -1 : 1) - (1 : 0 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0\) | \(3\) |
2-torsion field: 6.0.1259712.1
BSD invariants
| Hasse-Weil conjecture: | verified |
| Analytic rank: | \(0\) |
| Mordell-Weil rank: | \(0\) |
| 2-Selmer rank: | \(0\) |
| Regulator: | \( 1 \) |
| Real period: | \( 14.14876 \) |
| Tamagawa product: | \( 9 \) |
| Torsion order: | \( 9 \) |
| Leading coefficient: | \( 1.572084 \) |
| Analytic order of Ш: | \( 1 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(2\) | \(2\) | \(3\) | \(3\) | \(( 1 - T )( 1 + T )\) | |
| \(3\) | \(8\) | \(9\) | \(3\) | \(1\) |  |
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 curves:
Elliptic curve 162.b4
Elliptic curve 162.c3
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 \(K = \Q (a) \simeq \) \(\Q(\sqrt{-3}) \) 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 \(3\) in a maximal order of \(\End (J_{\overline{\Q}}) \otimes \Q\) |
| \(\End (J_{\overline{\Q}}) \otimes \Q \) | \(\simeq\) | \(\mathrm{M}_2(\)\(\Q\)\()\) |
| \(\End (J_{\overline{\Q}}) \otimes \R\) | \(\simeq\) | \(\mathrm{M}_2 (\R)\) |