Minimal equation
Minimal equation
Simplified equation
| $y^2 + (x^3 + x + 1)y = -x^6 - 2x^5 + 3x^3 - x^2 - 3x - 1$ | (homogenize, simplify) |
| $y^2 + (x^3 + xz^2 + z^3)y = -x^6 - 2x^5z + 3x^3z^3 - x^2z^4 - 3xz^5 - z^6$ | (dehomogenize, simplify) |
| $y^2 = -3x^6 - 8x^5 + 2x^4 + 14x^3 - 3x^2 - 10x - 3$ | (homogenize, minimize) |
Invariants
| Conductor: | \( N \) | \(=\) | \(391876\) | \(=\) | \( 2^{2} \cdot 313^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(-783752\) | \(=\) | \( - 2^{3} \cdot 313^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(1156\) | \(=\) | \( 2^{2} \cdot 17^{2} \) |
| \( I_4 \) | \(=\) | \(128017\) | \(=\) | \( 313 \cdot 409 \) |
| \( I_6 \) | \(=\) | \(35126425\) | \(=\) | \( 5^{2} \cdot 67^{2} \cdot 313 \) |
| \( I_{10} \) | \(=\) | \(-100320256\) | \(=\) | \( - 2^{10} \cdot 313^{2} \) |
| \( J_2 \) | \(=\) | \(289\) | \(=\) | \( 17^{2} \) |
| \( J_4 \) | \(=\) | \(-1854\) | \(=\) | \( - 2 \cdot 3^{2} \cdot 103 \) |
| \( J_6 \) | \(=\) | \(-3788\) | \(=\) | \( - 2^{2} \cdot 947 \) |
| \( J_8 \) | \(=\) | \(-1133012\) | \(=\) | \( - 2^{2} \cdot 191 \cdot 1483 \) |
| \( J_{10} \) | \(=\) | \(-783752\) | \(=\) | \( - 2^{3} \cdot 313^{2} \) |
| \( g_1 \) | \(=\) | \(-2015993900449/783752\) | ||
| \( g_2 \) | \(=\) | \(22375526463/391876\) | ||
| \( g_3 \) | \(=\) | \(79094387/195938\) |
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: trivial
2-torsion field: 6.0.50160128.1
BSD invariants
| Hasse-Weil conjecture: | verified |
| Analytic rank: | \(0\) |
| Mordell-Weil rank: | \(0\) |
| 2-Selmer rank: | \(0\) |
| Regulator: | \( 1 \) |
| Real period: | \( 2.096035 \) |
| Tamagawa product: | \( 1 \) |
| Torsion order: | \( 1 \) |
| Leading coefficient: | \( 2.096035 \) |
| Analytic order of Ш: | \( 1 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(2\) | \(2\) | \(3\) | \(1\) | \(1 - T + T^{2}\) | |
| \(313\) | \(2\) | \(2\) | \(1\) | \(1 + 13 T + 313 T^{2}\) |  |
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 |
| \(3\) | 3.480.12 | 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 \(\Q (b) \simeq \) 6.6.3004150512793.1 with defining polynomial:
\(x^{6} - x^{5} - 130 x^{4} + 481 x^{3} + 2170 x^{2} - 6196 x - 15304\)
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
\(g_4 = \frac{565236}{269} b^{5} - \frac{6779127}{1076} b^{4} - \frac{280105729}{1076} b^{3} + \frac{824114745}{538} b^{2} + \frac{1585191241}{1076} b - \frac{8578763427}{538}\)
\(g_6 = -\frac{631755361055}{2152} b^{5} + \frac{1903087063619}{2152} b^{4} + \frac{39149248059607}{1076} b^{3} - \frac{461440519641675}{2152} b^{2} - \frac{221159457020257}{1076} b + \frac{600559022448280}{269}\)
Conductor norm: 64
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 \(K = \Q (a) \simeq \) 6.6.3004150512793.1 with defining polynomial \(x^{6} - x^{5} - 130 x^{4} + 481 x^{3} + 2170 x^{2} - 6196 x - 15304\)
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 \(F \simeq \) \(\Q(\sqrt{313}) \) with generator \(-\frac{5}{269} a^{5} - \frac{21}{269} a^{4} + \frac{487}{269} a^{3} - \frac{34}{269} a^{2} - \frac{5324}{269} a - \frac{3914}{269}\) with minimal polynomial \(x^{2} - x - 78\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 3.3.97969.1 with generator \(-\frac{25}{4304} a^{5} - \frac{105}{4304} a^{4} + \frac{169}{269} a^{3} + \frac{1175}{4304} a^{2} - \frac{12035}{1076} a - \frac{6641}{1076}\) with minimal polynomial \(x^{3} - x^{2} - 104 x - 371\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple