Minimal equation
Minimal equation
Simplified equation
| $y^2 = x^5 + 2x^4 - 9x^3 + 7x^2 - x$ | (homogenize, simplify) |
| $y^2 = x^5z + 2x^4z^2 - 9x^3z^3 + 7x^2z^4 - xz^5$ | (dehomogenize, simplify) |
| $y^2 = x^5 + 2x^4 - 9x^3 + 7x^2 - x$ | (homogenize, minimize) |
Invariants
| Conductor: | \( N \) | \(=\) | \(20736\) | \(=\) | \( 2^{8} \cdot 3^{4} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(186624\) | \(=\) | \( 2^{8} \cdot 3^{6} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(74\) | \(=\) | \( 2 \cdot 37 \) |
| \( I_4 \) | \(=\) | \(288\) | \(=\) | \( 2^{5} \cdot 3^{2} \) |
| \( I_6 \) | \(=\) | \(5502\) | \(=\) | \( 2 \cdot 3 \cdot 7 \cdot 131 \) |
| \( I_{10} \) | \(=\) | \(3\) | \(=\) | \( 3 \) |
| \( J_2 \) | \(=\) | \(444\) | \(=\) | \( 2^{2} \cdot 3 \cdot 37 \) |
| \( J_4 \) | \(=\) | \(1302\) | \(=\) | \( 2 \cdot 3 \cdot 7 \cdot 31 \) |
| \( J_6 \) | \(=\) | \(-1292\) | \(=\) | \( - 2^{2} \cdot 17 \cdot 19 \) |
| \( J_8 \) | \(=\) | \(-567213\) | \(=\) | \( - 3 \cdot 43 \cdot 4397 \) |
| \( J_{10} \) | \(=\) | \(186624\) | \(=\) | \( 2^{8} \cdot 3^{6} \) |
| \( g_1 \) | \(=\) | \(277375828/3\) | ||
| \( g_2 \) | \(=\) | \(10991701/18\) | ||
| \( g_3 \) | \(=\) | \(-442187/324\) |
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: \(3\)
This curve is locally solvable everywhere.
Mordell-Weil group of the Jacobian
Group structure: \(\Z/{2}\Z \oplus \Z/{2}\Z\)
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \((0 : 0 : 1) + (1 : 0 : 1) - 2 \cdot(1 : 0 : 0)\) | \(x (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
| \((1 : 0 : 1) - (1 : 0 : 0)\) | \(x - z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \((0 : 0 : 1) + (1 : 0 : 1) - 2 \cdot(1 : 0 : 0)\) | \(x (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
| \((1 : 0 : 1) - (1 : 0 : 0)\) | \(x - z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \((0 : 0 : 1) + (1 : 0 : 1) - 2 \cdot(1 : 0 : 0)\) | \(x (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
| \((1 : 0 : 1) - (1 : 0 : 0)\) | \(x - z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
2-torsion field: \(\Q(\zeta_{9})^+\)
BSD invariants
| Hasse-Weil conjecture: | verified |
| Analytic rank: | \(0\) |
| Mordell-Weil rank: | \(0\) |
| 2-Selmer rank: | \(2\) |
| Regulator: | \( 1 \) |
| Real period: | \( 21.15344 \) |
| Tamagawa product: | \( 1 \) |
| Torsion order: | \( 4 \) |
| Leading coefficient: | \( 1.322090 \) |
| Analytic order of Ш: | \( 1 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(2\) | \(8\) | \(8\) | \(1\) | \(1\) | |
| \(3\) | \(4\) | \(6\) | \(1\) | \(1 + 3 T^{2}\) |  |
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? |
|---|---|---|
| \(2\) | 2.240.1 | yes |
| \(3\) | 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 \(\Q (b) \simeq \) \(\Q(\zeta_{36})^+\) with defining polynomial:
\(x^{6} - 6 x^{4} + 9 x^{2} - 3\)
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
\(g_4 = 1174176 b^{5} + 800928 b^{4} - 6485184 b^{3} - 4420656 b^{2} + 7503840 b + 5128272\)
\(g_6 = 586185984 b^{5} + 401008320 b^{4} - 3245158080 b^{3} - 2221502112 b^{2} + 3763179648 b + 2574851328\)
Conductor norm: 4096
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 \(K = \Q (a) \simeq \) \(\Q(\zeta_{36})^+\) with defining polynomial \(x^{6} - 6 x^{4} + 9 x^{2} - 3\)
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 \(a^{3} - 3 a\) with minimal polynomial \(x^{2} - 3\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) \(\Q(\zeta_{9})^+\) with generator \(a^{2} - 2\) with minimal polynomial \(x^{3} - 3 x - 1\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple