Minimal equation
Minimal equation
Simplified equation
| $y^2 = x^6 - 3x^5 - x^4 + 7x^3 - x^2 - 3x + 1$ | (homogenize, simplify) |
| $y^2 = x^6 - 3x^5z - x^4z^2 + 7x^3z^3 - x^2z^4 - 3xz^5 + z^6$ | (dehomogenize, simplify) |
| $y^2 = x^6 - 3x^5 - x^4 + 7x^3 - x^2 - 3x + 1$ | (minimize, homogenize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, -3, -1, 7, -1, -3, 1]), R([]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![1, -3, -1, 7, -1, -3, 1], R![]);
sage: X = HyperellipticCurve(R([1, -3, -1, 7, -1, -3, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
| Conductor: | \( N \) | \(=\) | \(38416\) | \(=\) | \( 2^{4} \cdot 7^{4} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(614656\) | \(=\) | \( 2^{8} \cdot 7^{4} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(398\) | \(=\) | \( 2 \cdot 199 \) |
| \( I_4 \) | \(=\) | \(9016\) | \(=\) | \( 2^{3} \cdot 7^{2} \cdot 23 \) |
| \( I_6 \) | \(=\) | \(912086\) | \(=\) | \( 2 \cdot 7^{2} \cdot 41 \cdot 227 \) |
| \( I_{10} \) | \(=\) | \(2401\) | \(=\) | \( 7^{4} \) |
| \( J_2 \) | \(=\) | \(796\) | \(=\) | \( 2^{2} \cdot 199 \) |
| \( J_4 \) | \(=\) | \(2358\) | \(=\) | \( 2 \cdot 3^{2} \cdot 131 \) |
| \( J_6 \) | \(=\) | \(-2348\) | \(=\) | \( - 2^{2} \cdot 587 \) |
| \( J_8 \) | \(=\) | \(-1857293\) | \(=\) | \( -1857293 \) |
| \( J_{10} \) | \(=\) | \(614656\) | \(=\) | \( 2^{8} \cdot 7^{4} \) |
| \( g_1 \) | \(=\) | \(1248318403996/2401\) | ||
| \( g_2 \) | \(=\) | \(9291226221/4802\) | ||
| \( g_3 \) | \(=\) | \(-23245787/9604\) |
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
Automorphism group
| \(\mathrm{Aut}(X)\) | \(\simeq\) | $D_6$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
|
| \(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $D_6$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
|
Rational points
Known points: \((1 : -1 : 0),\, (1 : 1 : 0),\, (0 : -1 : 1),\, (0 : 1 : 1),\, (1 : -1 : 1),\, (1 : 1 : 1)\)
magma: [C![0,-1,1],C![0,1,1],C![1,-1,0],C![1,-1,1],C![1,1,0],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: \(\Z \times \Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2 - 3xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2xz^2 + z^3\) | \(0.172070\) | \(\infty\) |
| \(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2 + xz - z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2xz^2 + z^3\) | \(0.172070\) | \(\infty\) |
2-torsion field: \(\Q(\zeta_{7})^+\)
BSD invariants
| Hasse-Weil conjecture: | verified |
| Analytic rank: | \(2\) |
| Mordell-Weil rank: | \(2\) |
| 2-Selmer rank: | \(2\) |
| Regulator: | \( 0.022206 \) |
| Real period: | \( 19.01761 \) |
| Tamagawa product: | \( 3 \) |
| Torsion order: | \( 1 \) |
| Leading coefficient: | \( 1.266932 \) |
| Analytic order of Ш: | \( 1 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(2\) | \(4\) | \(8\) | \(3\) | \(1\) | |
| \(7\) | \(4\) | \(4\) | \(1\) | \(1\) |  |
Sato-Tate group
| \(\mathrm{ST}\) | \(\simeq\) | $E_1$ |
| \(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\) |
Decomposition of the Jacobian
Splits over \(\Q\)
Decomposes up to isogeny as the square of the elliptic curve:
Elliptic curve 196.a1
Endomorphisms of the Jacobian
Not of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
| \(\End (J_{})\) | \(\simeq\) | an Eichler order of index \(3\) in a maximal order of \(\End (J_{}) \otimes \Q\) |
| \(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\mathrm{M}_2(\)\(\Q\)\()\) |
| \(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\mathrm{M}_2 (\R)\) |
All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).