Minimal equation
Minimal equation
Simplified equation
$y^2 + y = -x^6 - x^3 - 1$ | (homogenize, simplify) |
$y^2 + z^3y = -x^6 - x^3z^3 - z^6$ | (dehomogenize, simplify) |
$y^2 = -4x^6 - 4x^3 - 3$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(419904\) | \(=\) | \( 2^{6} \cdot 3^{8} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(-839808\) | \(=\) | \( - 2^{7} \cdot 3^{8} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(116\) | \(=\) | \( 2^{2} \cdot 29 \) |
\( I_4 \) | \(=\) | \(513\) | \(=\) | \( 3^{3} \cdot 19 \) |
\( I_6 \) | \(=\) | \(16623\) | \(=\) | \( 3^{2} \cdot 1847 \) |
\( I_{10} \) | \(=\) | \(432\) | \(=\) | \( 2^{4} \cdot 3^{3} \) |
\( J_2 \) | \(=\) | \(348\) | \(=\) | \( 2^{2} \cdot 3 \cdot 29 \) |
\( J_4 \) | \(=\) | \(1968\) | \(=\) | \( 2^{4} \cdot 3 \cdot 41 \) |
\( J_6 \) | \(=\) | \(-3856\) | \(=\) | \( - 2^{4} \cdot 241 \) |
\( J_8 \) | \(=\) | \(-1303728\) | \(=\) | \( - 2^{4} \cdot 3 \cdot 157 \cdot 173 \) |
\( J_{10} \) | \(=\) | \(839808\) | \(=\) | \( 2^{7} \cdot 3^{8} \) |
\( g_1 \) | \(=\) | \(164089192/27\) | ||
\( g_2 \) | \(=\) | \(7999592/81\) | ||
\( g_3 \) | \(=\) | \(-405362/729\) |
Automorphism group
\(\mathrm{Aut}(X)\) | \(\simeq\) | $C_2$ | 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: \(\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 - xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2 - z^3\) | \(1.013495\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 - xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2 - z^3\) | \(1.013495\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 - xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(2xz^2 - z^3\) | \(1.013495\) | \(\infty\) |
2-torsion field: 6.0.13436928.4
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(1\) |
Mordell-Weil rank: | \(1\) |
2-Selmer rank: | \(1\) |
Regulator: | \( 1.013495 \) |
Real period: | \( 3.657605 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 3.706965 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(6\) | \(7\) | \(1\) | \(1\) | |
\(3\) | \(8\) | \(8\) | \(1\) | \(1\) |
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? |
---|---|---|
\(2\) | 2.20.3 | no |
\(3\) | 3.2880.7 | no |
Sato-Tate group
\(\mathrm{ST}\) | \(\simeq\) | $J(E_6)$ |
\(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\) |
Decomposition of the Jacobian
Splits over the number field \(\Q (b) \simeq \) 6.2.45349632.3 with defining polynomial:
\(x^{6} - 18 x^{3} + 6\)
Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
\(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
\(g_4 = -\frac{414}{5} b^{5} + \frac{7506}{5} b^{2}\)
\(g_6 = -\frac{4536}{5} b^{3} + \frac{97524}{5}\)
Conductor norm: 81
\(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
\(g_4 = -\frac{126}{5} b^{5} + \frac{2754}{5} b^{2}\)
\(g_6 = \frac{4536}{5} b^{3} + \frac{15876}{5}\)
Conductor norm: 81
Endomorphisms of the Jacobian
Not of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
\(\End (J_{})\) | \(\simeq\) | \(\Z\) |
\(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
\(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\R\) |
Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a)\) with defining polynomial \(x^{12} + 36\)
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 \(F \simeq \) \(\Q(\sqrt{-3}) \) with generator \(-\frac{1}{24} a^{9} - \frac{1}{4} a^{3} + \frac{1}{2}\) with minimal polynomial \(x^{2} - x + 1\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) \(\Q(\sqrt{-1}) \) with generator \(-\frac{1}{6} a^{6}\) with minimal polynomial \(x^{2} + 1\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R\) |
Not of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) \(\Q(\sqrt{3}) \) with generator \(\frac{1}{12} a^{9} - \frac{1}{2} a^{3}\) with minimal polynomial \(x^{2} - 3\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R\) |
Not of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 3.1.972.2 with generator \(-\frac{1}{24} a^{11} - \frac{1}{12} a^{8} + \frac{1}{4} a^{5}\) with minimal polynomial \(x^{3} - 6\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R\) |
Not of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 3.1.972.2 with generator \(\frac{1}{24} a^{11} - \frac{1}{12} a^{8} - \frac{1}{4} a^{5}\) with minimal polynomial \(x^{3} - 6\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R\) |
Not of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 3.1.972.2 with generator \(\frac{1}{6} a^{8}\) with minimal polynomial \(x^{3} - 6\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R\) |
Not of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) \(\Q(\zeta_{12})\) with generator \(\frac{1}{24} a^{9} + \frac{1}{12} a^{6} - \frac{1}{4} a^{3}\) with minimal polynomial \(x^{4} - x^{2} + 1\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 6.0.2834352.2 with generator \(-\frac{1}{6} a^{7} + a\) with minimal polynomial \(x^{6} + 48\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 6.0.15116544.3 with generator \(-a^{2}\) with minimal polynomial \(x^{6} + 36\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\sqrt{3}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{3}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 6.0.15116544.3 with generator \(\frac{1}{24} a^{11} + \frac{1}{4} a^{5} + \frac{1}{2} a^{2}\) with minimal polynomial \(x^{6} + 36\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\sqrt{3}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{3}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 6.0.15116544.3 with generator \(-\frac{1}{24} a^{11} - \frac{1}{4} a^{5} + \frac{1}{2} a^{2}\) with minimal polynomial \(x^{6} + 36\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\sqrt{3}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{3}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 6.2.45349632.3 with generator \(\frac{1}{24} a^{10} - \frac{1}{6} a^{7} + \frac{1}{4} a^{4} + \frac{1}{2} a\) with minimal polynomial \(x^{6} - 18 x^{3} + 6\):
\(\End (J_{F})\) | \(\simeq\) | an order of index \(2\) in \(\Z \times \Z\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q\) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Of \(\GL_2\)-type, not simple
Over subfield \(F \simeq \) 6.2.45349632.3 with generator \(-\frac{1}{24} a^{10} + \frac{1}{12} a^{7} + \frac{1}{4} a^{4} - a\) with minimal polynomial \(x^{6} - 18 x^{3} + 6\):
\(\End (J_{F})\) | \(\simeq\) | an order of index \(2\) in \(\Z \times \Z\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q\) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Of \(\GL_2\)-type, not simple
Over subfield \(F \simeq \) 6.2.45349632.3 with generator \(-\frac{1}{12} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a\) with minimal polynomial \(x^{6} - 18 x^{3} + 6\):
\(\End (J_{F})\) | \(\simeq\) | an order of index \(2\) in \(\Z \times \Z\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q\) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Of \(\GL_2\)-type, not simple