Minimal equation
Minimal equation
Simplified equation
| $y^2 + (x^3 + x + 1)y = x^5 + 3x^4 + 4x^3 + 2x^2$ | (homogenize, simplify) |
| $y^2 + (x^3 + xz^2 + z^3)y = x^5z + 3x^4z^2 + 4x^3z^3 + 2x^2z^4$ | (dehomogenize, simplify) |
| $y^2 = x^6 + 4x^5 + 14x^4 + 18x^3 + 9x^2 + 2x + 1$ | (minimize, homogenize) |
Invariants
| Conductor: | \( N \) | \(=\) | \(12321\) | \(=\) | \( 3^{2} \cdot 37^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(-36963\) | \(=\) | \( - 3^{3} \cdot 37^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(4\) | \(=\) | \( 2^{2} \) |
| \( I_4 \) | \(=\) | \(6697\) | \(=\) | \( 37 \cdot 181 \) |
| \( I_6 \) | \(=\) | \(85285\) | \(=\) | \( 5 \cdot 37 \cdot 461 \) |
| \( I_{10} \) | \(=\) | \(-4731264\) | \(=\) | \( - 2^{7} \cdot 3^{3} \cdot 37^{2} \) |
| \( J_2 \) | \(=\) | \(1\) | \(=\) | \( 1 \) |
| \( J_4 \) | \(=\) | \(-279\) | \(=\) | \( - 3^{2} \cdot 31 \) |
| \( J_6 \) | \(=\) | \(-1107\) | \(=\) | \( - 3^{3} \cdot 41 \) |
| \( J_8 \) | \(=\) | \(-19737\) | \(=\) | \( - 3^{3} \cdot 17 \cdot 43 \) |
| \( J_{10} \) | \(=\) | \(-36963\) | \(=\) | \( - 3^{3} \cdot 37^{2} \) |
| \( g_1 \) | \(=\) | \(-1/36963\) | ||
| \( g_2 \) | \(=\) | \(31/4107\) | ||
| \( g_3 \) | \(=\) | \(41/1369\) |
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
| Known points | |||||
|---|---|---|---|---|---|
| \((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((-1 : 0 : 1)\) | \((0 : -1 : 1)\) | \((-1 : 1 : 1)\) |
| \((-2 : 1 : 1)\) | \((1 : 2 : 1)\) | \((-1 : 2 : 2)\) | \((1 : -5 : 1)\) | \((-1 : -5 : 2)\) | \((-2 : 8 : 1)\) |
Number of rational Weierstrass points: \(0\)
This curve is locally solvable everywhere.
Mordell-Weil group of the Jacobian
Group structure: \(\Z \times \Z\)
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \((0 : -1 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.098911\) | \(\infty\) |
| \((-1 : 0 : 1) + (0 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.098911\) | \(\infty\) |
BSD invariants
| Hasse-Weil conjecture: | verified |
| Analytic rank: | \(2\) |
| Mordell-Weil rank: | \(2\) |
| 2-Selmer rank: | \(2\) |
| Regulator: | \( 0.007337 \) |
| Real period: | \( 18.95274 \) |
| Tamagawa product: | \( 3 \) |
| Torsion order: | \( 1 \) |
| Leading coefficient: | \( 0.417205 \) |
| Analytic order of Ш: | \( 1 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(3\) | \(2\) | \(3\) | \(3\) | \(1 + T + T^{2}\) |  |
| \(37\) | \(2\) | \(2\) | \(1\) | \(1 + 10 T + 37 T^{2}\) |  |
Sato-Tate group
| \(\mathrm{ST}\) | \(\simeq\) | $E_6$ |
| \(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\) |
Decomposition of the Jacobian
Splits over the number field \(\Q (b) \simeq \) 6.6.69343957.1 with defining polynomial:
\(x^{6} - x^{5} - 15 x^{4} + 28 x^{3} + 15 x^{2} - 38 x - 1\)
Decomposes up to isogeny as the square of the elliptic curve:
\(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
\(g_4 = \frac{27938}{11} b^{5} - \frac{102400}{11} b^{4} + \frac{41669}{11} b^{3} + \frac{90470}{11} b^{2} - \frac{45088}{11} b + \frac{110611}{11}\)
\(g_6 = \frac{32589452}{11} b^{5} - \frac{166685962}{11} b^{4} + \frac{182285865}{11} b^{3} + \frac{203348152}{11} b^{2} - \frac{315712416}{11} b - \frac{19805878}{11}\)
Conductor norm: 729
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 \(K = \Q (a) \simeq \) 6.6.69343957.1 with defining polynomial \(x^{6} - x^{5} - 15 x^{4} + 28 x^{3} + 15 x^{2} - 38 x - 1\)
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 \(\End (J_{\overline{\Q}}) \otimes \Q\) |
| \(\End (J_{\overline{\Q}}) \otimes \Q \) | \(\simeq\) | \(\mathrm{M}_2(\)\(\Q\)\()\) |
| \(\End (J_{\overline{\Q}}) \otimes \R\) | \(\simeq\) | \(\mathrm{M}_2 (\R)\) |
Remainder of the endomorphism lattice by field
Over subfield \(F \simeq \) \(\Q(\sqrt{37}) \) with generator \(\frac{2}{11} a^{5} + \frac{1}{11} a^{4} - \frac{23}{11} a^{3} + \frac{27}{11} a^{2} - \frac{1}{11} a - \frac{28}{11}\) with minimal polynomial \(x^{2} - x - 9\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 3.3.1369.1 with generator \(-\frac{4}{11} a^{5} - \frac{2}{11} a^{4} + \frac{57}{11} a^{3} - \frac{32}{11} a^{2} - \frac{108}{11} a + \frac{45}{11}\) with minimal polynomial \(x^{3} - x^{2} - 12 x - 11\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple