Minimal equation
Minimal equation
Simplified equation
| $y^2 + (x^3 + x + 1)y = -3x^5 + 9x^4 - 7x^3 - 2x^2 + x$ | (homogenize, simplify) |
| $y^2 + (x^3 + xz^2 + z^3)y = -3x^5z + 9x^4z^2 - 7x^3z^3 - 2x^2z^4 + xz^5$ | (dehomogenize, simplify) |
| $y^2 = x^6 - 12x^5 + 38x^4 - 26x^3 - 7x^2 + 6x + 1$ | (homogenize, minimize) |
Invariants
| Conductor: | \( N \) | \(=\) | \(8281\) | \(=\) | \( 7^{2} \cdot 13^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(405769\) | \(=\) | \( 7^{4} \cdot 13^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(2596\) | \(=\) | \( 2^{2} \cdot 11 \cdot 59 \) |
| \( I_4 \) | \(=\) | \(375193\) | \(=\) | \( 7^{2} \cdot 13 \cdot 19 \cdot 31 \) |
| \( I_6 \) | \(=\) | \(248614093\) | \(=\) | \( 7^{2} \cdot 13 \cdot 390289 \) |
| \( I_{10} \) | \(=\) | \(51938432\) | \(=\) | \( 2^{7} \cdot 7^{4} \cdot 13^{2} \) |
| \( J_2 \) | \(=\) | \(649\) | \(=\) | \( 11 \cdot 59 \) |
| \( J_4 \) | \(=\) | \(1917\) | \(=\) | \( 3^{3} \cdot 71 \) |
| \( J_6 \) | \(=\) | \(-1907\) | \(=\) | \( -1907 \) |
| \( J_8 \) | \(=\) | \(-1228133\) | \(=\) | \( -1228133 \) |
| \( J_{10} \) | \(=\) | \(405769\) | \(=\) | \( 7^{4} \cdot 13^{2} \) |
| \( g_1 \) | \(=\) | \(115139273278249/405769\) | ||
| \( g_2 \) | \(=\) | \(524030063733/405769\) | ||
| \( g_3 \) | \(=\) | \(-803230307/405769\) |
Automorphism group
| \(\mathrm{Aut}(X)\) | \(\simeq\) | $C_6$ | 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 | |||||
|---|---|---|---|---|---|
| \((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((1 : -1 : 1)\) | \((1 : -2 : 1)\) |
| \((1 : -1 : 3)\) | \((3 : -6 : 2)\) | \((-2 : -13 : 1)\) | \((-2 : 22 : 1)\) | \((1 : -36 : 3)\) | \((3 : -41 : 2)\) |
| Known points | |||||
|---|---|---|---|---|---|
| \((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((1 : -1 : 1)\) | \((1 : -2 : 1)\) |
| \((1 : -1 : 3)\) | \((3 : -6 : 2)\) | \((-2 : -13 : 1)\) | \((-2 : 22 : 1)\) | \((1 : -36 : 3)\) | \((3 : -41 : 2)\) |
| Known points | |||||
|---|---|---|---|---|---|
| \((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((1 : -1 : 1)\) | \((1 : 1 : 1)\) |
| \((-2 : -35 : 1)\) | \((-2 : 35 : 1)\) | \((1 : -35 : 3)\) | \((1 : 35 : 3)\) | \((3 : -35 : 2)\) | \((3 : 35 : 2)\) |
Number of rational Weierstrass points: \(0\)
This curve is locally solvable everywhere.
Mordell-Weil group of the Jacobian
Group structure: \(\Z \oplus \Z\)
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2 - 4xz + 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-6xz^2 + 3z^3\) | \(0.086938\) | \(\infty\) |
| \(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(2x^2 - z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-4xz^2 + z^3\) | \(0.086938\) | \(\infty\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2 - 4xz + 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-6xz^2 + 3z^3\) | \(0.086938\) | \(\infty\) |
| \(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(2x^2 - z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-4xz^2 + z^3\) | \(0.086938\) | \(\infty\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2 - 4xz + 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - 11xz^2 + 7z^3\) | \(0.086938\) | \(\infty\) |
| \(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) | \(2x^2 - z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(x^3 - 7xz^2 + 3z^3\) | \(0.086938\) | \(\infty\) |
2-torsion field: 9.9.567869252041.1
BSD invariants
| Hasse-Weil conjecture: | verified |
| Analytic rank: | \(2\) |
| Mordell-Weil rank: | \(2\) |
| 2-Selmer rank: | \(2\) |
| Regulator: | \( 0.005668 \) |
| Real period: | \( 19.78540 \) |
| Tamagawa product: | \( 3 \) |
| Torsion order: | \( 1 \) |
| Leading coefficient: | \( 0.336475 \) |
| Analytic order of Ш: | \( 1 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(7\) | \(2\) | \(4\) | \(3\) | \(1 + 5 T + 7 T^{2}\) |  |
| \(13\) | \(2\) | \(2\) | \(1\) | \(1 + 2 T + 13 T^{2}\) |  |
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? |
|---|---|---|
| \(2\) | 2.80.1 | no |
| \(3\) | 3.480.12 | no |
Sato-Tate group
| \(\mathrm{ST}\) | \(\simeq\) | $E_6$ |
| \(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\) |
Decomposition of the Jacobian
Splits over the number field \(\Q (b) \simeq \) 6.6.891474493.2 with defining polynomial:
\(x^{6} - x^{5} - 31 x^{4} + 4 x^{3} + 162 x^{2} - 81 x - 27\)
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
\(g_4 = \frac{1574069}{6561} b^{5} + \frac{13576892}{6561} b^{4} - \frac{56218393}{6561} b^{3} - \frac{17230402}{243} b^{2} + \frac{2496578}{243} b + \frac{81063052}{243}\)
\(g_6 = -\frac{655459038871}{531441} b^{5} + \frac{100362384668}{531441} b^{4} + \frac{20455326360833}{531441} b^{3} + \frac{543846776200}{19683} b^{2} - \frac{3528969869152}{19683} b - \frac{1057160135213}{19683}\)
Conductor norm: 1
Endomorphisms of the Jacobian
Of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
| \(\End (J_{})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
| \(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
| \(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\C\) |
Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) 6.6.891474493.2 with defining polynomial \(x^{6} - x^{5} - 31 x^{4} + 4 x^{3} + 162 x^{2} - 81 x - 27\)
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 \(\End (J_{\overline{\Q}}) \otimes \Q\) |
| \(\End (J_{\overline{\Q}}) \otimes \Q \) | \(\simeq\) | \(\mathrm{M}_2(\)\(\Q\)\()\) |
| \(\End (J_{\overline{\Q}}) \otimes \R\) | \(\simeq\) | \(\mathrm{M}_2 (\R)\) |
Remainder of the endomorphism lattice by field
Over subfield \(F \simeq \) \(\Q(\sqrt{13}) \) with generator \(\frac{4}{243} a^{5} - \frac{1}{243} a^{4} - \frac{145}{243} a^{3} - \frac{32}{243} a^{2} + \frac{316}{81} a - \frac{11}{27}\) with minimal polynomial \(x^{2} - x - 3\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 3.3.8281.2 with generator \(-\frac{1}{27} a^{5} - \frac{2}{27} a^{4} + \frac{34}{27} a^{3} + \frac{71}{27} a^{2} - \frac{49}{9} a - \frac{7}{3}\) with minimal polynomial \(x^{3} - x^{2} - 30 x - 27\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple