Minimal equation
Minimal equation
Simplified equation
| $y^2 + (x^3 + 1)y = -1$ | (homogenize, simplify) |
| $y^2 + (x^3 + z^3)y = -z^6$ | (dehomogenize, simplify) |
| $y^2 = x^6 + 2x^3 - 3$ | (homogenize, minimize) |
Invariants
| Conductor: | \( N \) | \(=\) | \(2187\) | \(=\) | \( 3^{7} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(6561\) | \(=\) | \( 3^{8} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(124\) | \(=\) | \( 2^{2} \cdot 31 \) |
| \( I_4 \) | \(=\) | \(297\) | \(=\) | \( 3^{3} \cdot 11 \) |
| \( I_6 \) | \(=\) | \(13275\) | \(=\) | \( 3^{2} \cdot 5^{2} \cdot 59 \) |
| \( I_{10} \) | \(=\) | \(3456\) | \(=\) | \( 2^{7} \cdot 3^{3} \) |
| \( J_2 \) | \(=\) | \(93\) | \(=\) | \( 3 \cdot 31 \) |
| \( J_4 \) | \(=\) | \(249\) | \(=\) | \( 3 \cdot 83 \) |
| \( J_6 \) | \(=\) | \(-239\) | \(=\) | \( -239 \) |
| \( J_8 \) | \(=\) | \(-21057\) | \(=\) | \( - 3 \cdot 7019 \) |
| \( J_{10} \) | \(=\) | \(6561\) | \(=\) | \( 3^{8} \) |
| \( g_1 \) | \(=\) | \(28629151/27\) | ||
| \( g_2 \) | \(=\) | \(2472653/81\) | ||
| \( g_3 \) | \(=\) | \(-229679/729\) |
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: \(1\)
This curve is locally solvable everywhere.
Mordell-Weil group of the Jacobian
Group structure: \(\Z/{6}\Z\)
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \((1 : -1 : 1) - (1 : 0 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3\) | \(0\) | \(6\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \((1 : -1 : 1) - (1 : 0 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3\) | \(0\) | \(6\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \((1 : 0 : 1) - (1 : 1 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 + z^3\) | \(0\) | \(6\) |
BSD invariants
| Hasse-Weil conjecture: | unverified |
| Analytic rank: | \(0\) |
| Mordell-Weil rank: | \(0\) |
| 2-Selmer rank: | \(1\) |
| Regulator: | \( 1 \) |
| Real period: | \( 10.92567 \) |
| Tamagawa product: | \( 2 \) |
| Torsion order: | \( 6 \) |
| Leading coefficient: | \( 0.606982 \) |
| Analytic order of Ш: | \( 1 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(3\) | \(7\) | \(8\) | \(2\) | \(1\) |  |
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? |
|---|---|---|
| \(2\) | 2.120.1 | yes |
| \(3\) | 3.5760.5 | yes |
Sato-Tate group
| \(\mathrm{ST}\) | \(\simeq\) | $J(E_3)$ |
| \(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\) |
Decomposition of the Jacobian
Splits over the number field \(\Q (b) \simeq \) 6.0.177147.2 with defining polynomial:
\(x^{6} + 3\)
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
\(g_4 = -\frac{891}{32} b^{5} + \frac{2187}{32} b^{2}\)
\(g_6 = \frac{15309}{32} b^{3} + \frac{67797}{64}\)
Conductor norm: 9
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 \(K = \Q (a) \simeq \) 6.0.177147.2 with defining polynomial \(x^{6} + 3\)
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 \(-\frac{1}{2} a^{3} + \frac{1}{2}\) 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 \(F \simeq \) 3.1.243.1 with generator \(\frac{1}{2} a^{5} + \frac{1}{2} a^{2}\) with minimal polynomial \(x^{3} - 3\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z [\sqrt{3}]\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{3}) \) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 3.1.243.1 with generator \(-\frac{1}{2} a^{5} + \frac{1}{2} a^{2}\) with minimal polynomial \(x^{3} - 3\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z [\sqrt{3}]\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{3}) \) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 3.1.243.1 with generator \(-a^{2}\) with minimal polynomial \(x^{3} - 3\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z [\sqrt{3}]\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{3}) \) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Of \(\GL_2\)-type, simple