Minimal equation
Minimal equation
Simplified equation
| $y^2 + (x^3 + x + 1)y = -x^6 + 2x^5 - 4x^4 + 4x^3 - 5x^2 + 2x - 1$ | (homogenize, simplify) |
| $y^2 + (x^3 + xz^2 + z^3)y = -x^6 + 2x^5z - 4x^4z^2 + 4x^3z^3 - 5x^2z^4 + 2xz^5 - z^6$ | (dehomogenize, simplify) |
| $y^2 = -3x^6 + 8x^5 - 14x^4 + 18x^3 - 19x^2 + 10x - 3$ | (homogenize, minimize) |
Invariants
| Conductor: | \( N \) | \(=\) | \(15876\) | \(=\) | \( 2^{2} \cdot 3^{4} \cdot 7^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(-222264\) | \(=\) | \( - 2^{3} \cdot 3^{4} \cdot 7^{3} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(636\) | \(=\) | \( 2^{2} \cdot 3 \cdot 53 \) |
| \( I_4 \) | \(=\) | \(6129\) | \(=\) | \( 3^{3} \cdot 227 \) |
| \( I_6 \) | \(=\) | \(310743\) | \(=\) | \( 3^{3} \cdot 17 \cdot 677 \) |
| \( I_{10} \) | \(=\) | \(28449792\) | \(=\) | \( 2^{10} \cdot 3^{4} \cdot 7^{3} \) |
| \( J_2 \) | \(=\) | \(159\) | \(=\) | \( 3 \cdot 53 \) |
| \( J_4 \) | \(=\) | \(798\) | \(=\) | \( 2 \cdot 3 \cdot 7 \cdot 19 \) |
| \( J_6 \) | \(=\) | \(16268\) | \(=\) | \( 2^{2} \cdot 7^{2} \cdot 83 \) |
| \( J_8 \) | \(=\) | \(487452\) | \(=\) | \( 2^{2} \cdot 3 \cdot 7^{2} \cdot 829 \) |
| \( J_{10} \) | \(=\) | \(222264\) | \(=\) | \( 2^{3} \cdot 3^{4} \cdot 7^{3} \) |
| \( g_1 \) | \(=\) | \(1254586479/2744\) | ||
| \( g_2 \) | \(=\) | \(2828663/196\) | ||
| \( g_3 \) | \(=\) | \(233147/126\) |
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
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: \(\Z/{3}\Z\)
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - D_\infty\) | \(x^2 - xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0\) | \(3\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - D_\infty\) | \(x^2 - xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0\) | \(3\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - D_\infty\) | \(x^2 - xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + xz^2 - z^3\) | \(0\) | \(3\) |
2-torsion field: 6.0.14224896.1
BSD invariants
| Hasse-Weil conjecture: | verified |
| Analytic rank: | \(0\) |
| Mordell-Weil rank: | \(0\) |
| 2-Selmer rank: | \(0\) |
| Regulator: | \( 1 \) |
| Real period: | \( 5.070237 \) |
| Tamagawa product: | \( 3 \) |
| Torsion order: | \( 3 \) |
| Leading coefficient: | \( 1.690079 \) |
| Analytic order of Ш: | \( 1 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(2\) | \(2\) | \(3\) | \(1\) | \(1 - T + T^{2}\) | |
| \(3\) | \(4\) | \(4\) | \(1\) | \(1 + 3 T^{2}\) |  |
| \(7\) | \(2\) | \(3\) | \(3\) | \(1 + T + T^{2}\) |  |
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? |
|---|---|---|
| \(2\) | 2.40.3 | no |
| \(3\) | 3.1920.3 | yes |
Sato-Tate group
| \(\mathrm{ST}\) | \(\simeq\) | $E_3$ |
| \(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\) |
Decomposition of the Jacobian
Splits over the number field \(\Q (b) \simeq \) \(\Q(\zeta_{9})^+\) with defining polynomial:
\(x^{3} - 3 x - 1\)
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
\(g_4 = \frac{41553}{16} b^{2} + \frac{78489}{16} b + \frac{23085}{16}\)
\(g_6 = \frac{29032425}{64} b^{2} + \frac{54580959}{64} b + \frac{1935495}{8}\)
Conductor norm: 2744
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 \(K = \Q (a) \simeq \) \(\Q(\zeta_{9})^+\) with defining polynomial \(x^{3} - 3 x - 1\)
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 \(\End (J_{\overline{\Q}}) \otimes \Q\) |
| \(\End (J_{\overline{\Q}}) \otimes \Q \) | \(\simeq\) | \(\mathrm{M}_2(\)\(\Q\)\()\) |
| \(\End (J_{\overline{\Q}}) \otimes \R\) | \(\simeq\) | \(\mathrm{M}_2 (\R)\) |