Minimal equation
Minimal equation
Simplified equation
| $y^2 + (x^3 + x^2 + x + 1)y = -x^6 - x^4 - x^3 - x^2 - x - 1$ | (homogenize, simplify) |
| $y^2 + (x^3 + x^2z + xz^2 + z^3)y = -x^6 - x^4z^2 - x^3z^3 - x^2z^4 - xz^5 - z^6$ | (dehomogenize, simplify) |
| $y^2 = -3x^6 + 2x^5 - x^4 - x^2 - 2x - 3$ | (homogenize, minimize) |
Invariants
| Conductor: | \( N \) | \(=\) | \(193600\) | \(=\) | \( 2^{6} \cdot 5^{2} \cdot 11^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(-968000\) | \(=\) | \( - 2^{6} \cdot 5^{3} \cdot 11^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(292\) | \(=\) | \( 2^{2} \cdot 73 \) |
| \( I_4 \) | \(=\) | \(2380\) | \(=\) | \( 2^{2} \cdot 5 \cdot 7 \cdot 17 \) |
| \( I_6 \) | \(=\) | \(214520\) | \(=\) | \( 2^{3} \cdot 5 \cdot 31 \cdot 173 \) |
| \( I_{10} \) | \(=\) | \(121000\) | \(=\) | \( 2^{3} \cdot 5^{3} \cdot 11^{2} \) |
| \( J_2 \) | \(=\) | \(292\) | \(=\) | \( 2^{2} \cdot 73 \) |
| \( J_4 \) | \(=\) | \(1966\) | \(=\) | \( 2 \cdot 983 \) |
| \( J_6 \) | \(=\) | \(-4356\) | \(=\) | \( - 2^{2} \cdot 3^{2} \cdot 11^{2} \) |
| \( J_8 \) | \(=\) | \(-1284277\) | \(=\) | \( - 163 \cdot 7879 \) |
| \( J_{10} \) | \(=\) | \(968000\) | \(=\) | \( 2^{6} \cdot 5^{3} \cdot 11^{2} \) |
| \( g_1 \) | \(=\) | \(33169145488/15125\) | ||
| \( g_2 \) | \(=\) | \(764807422/15125\) | ||
| \( g_3 \) | \(=\) | \(-47961/125\) |
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
Number of rational Weierstrass points: \(0\)
This curve is locally solvable except over $\R$ and $\Q_{2}$.
Mordell-Weil group of the Jacobian
Group structure: \(\Z/{2}\Z\)
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - D_\infty\) | \(x^2 + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - D_\infty\) | \(x^2 + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - D_\infty\) | \(x^2 + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + x^2z + xz^2 + z^3\) | \(0\) | \(2\) |
2-torsion field: 8.0.19360000.2
BSD invariants
| Hasse-Weil conjecture: | verified |
| Analytic rank: | \(0\) |
| Mordell-Weil rank: | \(0\) |
| 2-Selmer rank: | \(2\) |
| Regulator: | \( 1 \) |
| Real period: | \( 3.888936 \) |
| Tamagawa product: | \( 2 \) |
| Torsion order: | \( 2 \) |
| Leading coefficient: | \( 3.888936 \) |
| Analytic order of Ш: | \( 2 \) (rounded) |
| Order of Ш: | twice a square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(2\) | \(6\) | \(6\) | \(1\) | \(1 - 2 T + 2 T^{2}\) | |
| \(5\) | \(2\) | \(3\) | \(2\) | \(1 - 4 T + 5 T^{2}\) |  |
| \(11\) | \(2\) | \(2\) | \(1\) | \(( 1 + T )^{2}\) |  |
Galois representations
For primes $\ell \ge 5$ the Galois representation data has not been computed for this curve since it is not generic.
For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not all of $\GSp(4,\F_\ell)$.
| Prime \(\ell\) | mod-\(\ell\) image | Is torsion prime? |
|---|---|---|
| \(2\) | 2.45.1 | yes |
| \(3\) | 3.540.6 | no |
Sato-Tate group
| \(\mathrm{ST}\) | \(\simeq\) | $E_4$ |
| \(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\) |
Decomposition of the Jacobian
Splits over the number field \(\Q (b) \simeq \) 4.4.8000.1 with defining polynomial:
\(x^{4} - 10 x^{2} + 20\)
Decomposes up to isogeny as the square of the elliptic curve isogeny class:
\(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
\(g_4 = -880 b^{3} - 2400 b^{2} + 2560 b + 7120\)
\(g_6 = 7067360 b^{3} + 19011200 b^{2} - 19529600 b - 52534400\)
Conductor norm: 14641
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 \(K = \Q (a) \simeq \) 4.4.8000.1 with defining polynomial \(x^{4} - 10 x^{2} + 20\)
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 \(\End (J_{\overline{\Q}}) \otimes \Q\) |
| \(\End (J_{\overline{\Q}}) \otimes \Q \) | \(\simeq\) | \(\mathrm{M}_2(\)\(\Q\)\()\) |
| \(\End (J_{\overline{\Q}}) \otimes \R\) | \(\simeq\) | \(\mathrm{M}_2 (\R)\) |
Remainder of the endomorphism lattice by field
Over subfield \(F \simeq \) \(\Q(\sqrt{5}) \) with generator \(\frac{1}{2} a^{2} - 2\) with minimal polynomial \(x^{2} - x - 1\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z [\sqrt{-1}]\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-1}) \) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple