Minimal equation
Minimal equation
Simplified equation
| $y^2 + (x^3 + x^2 + x + 1)y = -x^3 - x$ | (homogenize, simplify) |
| $y^2 + (x^3 + x^2z + xz^2 + z^3)y = -x^3z^3 - xz^5$ | (dehomogenize, simplify) |
| $y^2 = x^6 + 2x^5 + 3x^4 + 3x^2 - 2x + 1$ | (minimize, homogenize) |
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, 0, -1], R![1, 1, 1, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, 0, -1]), R([1, 1, 1, 1]));
magma: X,pi:= SimplifiedModel(C);
sage: X = HyperellipticCurve(R([1, -2, 3, 0, 3, 2, 1]))
Invariants
| Conductor: | \( N \) | = | \(576\) | = | \( 2^{6} \cdot 3^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | = | \(-576\) | = | \( - 2^{6} \cdot 3^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | = | \(-544\) | = | \( - 2^{5} \cdot 17 \) |
| \( I_4 \) | = | \(7936\) | = | \( 2^{8} \cdot 31 \) |
| \( I_6 \) | = | \(-1339392\) | = | \( - 2^{12} \cdot 3 \cdot 109 \) |
| \( I_{10} \) | = | \(-2359296\) | = | \( - 2^{18} \cdot 3^{2} \) |
| \( J_2 \) | = | \(-68\) | = | \( - 2^{2} \cdot 17 \) |
| \( J_4 \) | = | \(110\) | = | \( 2 \cdot 5 \cdot 11 \) |
| \( J_6 \) | = | \(36\) | = | \( 2^{2} \cdot 3^{2} \) |
| \( J_8 \) | = | \(-3637\) | = | \( - 3637 \) |
| \( J_{10} \) | = | \(-576\) | = | \( - 2^{6} \cdot 3^{2} \) |
| \( g_1 \) | = | \(22717712/9\) | ||
| \( g_2 \) | = | \(540430/9\) | ||
| \( g_3 \) | = | \(-289\) |
magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
Automorphism group
| \(\mathrm{Aut}(X)\) | \(\simeq\) | $C_4$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
|
| \(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $D_4$ | 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: \(\Z/{10}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \((0 : -1 : 1) - (1 : 0 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0\) | \(10\) |
2-torsion field: \(\Q(\zeta_{12})\)
BSD invariants
| Hasse-Weil conjecture: | verified |
| Analytic rank: | \(0\) |
| Mordell-Weil rank: | \(0\) |
| 2-Selmer rank: | \(1\) |
| Regulator: | \( 1 \) |
| Real period: | \( 22.39625 \) |
| Tamagawa product: | \( 1 \) |
| Torsion order: | \( 10 \) |
| Leading coefficient: | \( 0.223962 \) |
| Analytic order of Ш: | \( 1 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor |
|---|---|---|---|---|
| \(2\) | \(6\) | \(6\) | \(1\) | \(1 + 2 T + 2 T^{2}\) |
| \(3\) | \(2\) | \(2\) | \(1\) | \(1 + T^{2}\) |
Sato-Tate group
| \(\mathrm{ST}\) | \(\simeq\) | $E_2$ |
| \(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\) |
Decomposition of the Jacobian
Splits over the number field \(\Q (b) \simeq \) \(\Q(\sqrt{2}) \) with defining polynomial:
\(x^{2} - 2\)
Decomposes up to isogeny as the square of the elliptic curve:
Elliptic curve 2.2.8.1-9.1-a3
Endomorphisms of the Jacobian
Of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
| \(\End (J_{})\) | \(\simeq\) | \(\Z [\sqrt{-1}]\) |
| \(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-1}) \) |
| \(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\C\) |
Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\sqrt{2}) \) with defining polynomial \(x^{2} - 2\)
Not of \(\GL_2\)-type over \(\overline{\Q}\)
Endomorphism ring over \(\overline{\Q}\):
| \(\End (J_{\overline{\Q}})\) | \(\simeq\) | a non-Eichler order of index \(4\) in a maximal order of \(\End (J_{\overline{\Q}}) \otimes \Q\) |
| \(\End (J_{\overline{\Q}}) \otimes \Q \) | \(\simeq\) | \(\mathrm{M}_2(\)\(\Q\)\()\) |
| \(\End (J_{\overline{\Q}}) \otimes \R\) | \(\simeq\) | \(\mathrm{M}_2 (\R)\) |