Minimal equation
Minimal equation
Simplified equation
| $y^2 + (x^3 + x + 1)y = x^5 + 10x^2 + 19x + 10$ | (homogenize, simplify) |
| $y^2 + (x^3 + xz^2 + z^3)y = x^5z + 10x^2z^4 + 19xz^5 + 10z^6$ | (dehomogenize, simplify) |
| $y^2 = x^6 + 4x^5 + 2x^4 + 2x^3 + 41x^2 + 78x + 41$ | (homogenize, minimize) |
Invariants
| Conductor: | \( N \) | \(=\) | \(32761\) | \(=\) | \( 181^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(-32761\) | \(=\) | \( - 181^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(676\) | \(=\) | \( 2^{2} \cdot 13^{2} \) |
| \( I_4 \) | \(=\) | \(41449\) | \(=\) | \( 181 \cdot 229 \) |
| \( I_6 \) | \(=\) | \(6681253\) | \(=\) | \( 181 \cdot 36913 \) |
| \( I_{10} \) | \(=\) | \(-4193408\) | \(=\) | \( - 2^{7} \cdot 181^{2} \) |
| \( J_2 \) | \(=\) | \(169\) | \(=\) | \( 13^{2} \) |
| \( J_4 \) | \(=\) | \(-537\) | \(=\) | \( - 3 \cdot 179 \) |
| \( J_6 \) | \(=\) | \(-547\) | \(=\) | \( -547 \) |
| \( J_8 \) | \(=\) | \(-95203\) | \(=\) | \( -95203 \) |
| \( J_{10} \) | \(=\) | \(-32761\) | \(=\) | \( - 181^{2} \) |
| \( g_1 \) | \(=\) | \(-137858491849/32761\) | ||
| \( g_2 \) | \(=\) | \(2591996433/32761\) | ||
| \( g_3 \) | \(=\) | \(15622867/32761\) |
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 | |||||
|---|---|---|---|---|---|
| \((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((-1 : 0 : 1)\) | \((-1 : 1 : 1)\) | \((-2 : 4 : 1)\) | \((1 : 5 : 1)\) |
| \((-2 : 5 : 1)\) | \((1 : -8 : 1)\) | \((-4 : 30 : 3)\) | \((-4 : 43 : 3)\) | \((-5 : 62 : 2)\) | \((-5 : 75 : 2)\) |
| Known points | |||||
|---|---|---|---|---|---|
| \((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((-1 : 0 : 1)\) | \((-1 : 1 : 1)\) | \((-2 : 4 : 1)\) | \((1 : 5 : 1)\) |
| \((-2 : 5 : 1)\) | \((1 : -8 : 1)\) | \((-4 : 30 : 3)\) | \((-4 : 43 : 3)\) | \((-5 : 62 : 2)\) | \((-5 : 75 : 2)\) |
| Known points | |||||
|---|---|---|---|---|---|
| \((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((-1 : -1 : 1)\) | \((-1 : 1 : 1)\) | \((-2 : -1 : 1)\) | \((-2 : 1 : 1)\) |
| \((1 : -13 : 1)\) | \((1 : 13 : 1)\) | \((-5 : -13 : 2)\) | \((-5 : 13 : 2)\) | \((-4 : -13 : 3)\) | \((-4 : 13 : 3)\) |
Number of rational Weierstrass points: \(0\)
This curve is locally solvable everywhere.
Mordell-Weil group of the Jacobian
Group structure: \(\Z \oplus \Z\)
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \((-2 : 5 : 1) - (1 : -1 : 0)\) | \(z (x + 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(5z^3\) | \(0.167303\) | \(\infty\) |
| \((-2 : 5 : 1) + (-1 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \((x + z) (x + 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-5xz^2 - 5z^3\) | \(0.167303\) | \(\infty\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \((-2 : 5 : 1) - (1 : -1 : 0)\) | \(z (x + 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(5z^3\) | \(0.167303\) | \(\infty\) |
| \((-2 : 5 : 1) + (-1 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \((x + z) (x + 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-5xz^2 - 5z^3\) | \(0.167303\) | \(\infty\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \((-2 : 1 : 1) - (1 : -1 : 0)\) | \(z (x + 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + xz^2 + 11z^3\) | \(0.167303\) | \(\infty\) |
| \((-2 : 1 : 1) + (-1 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \((x + z) (x + 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - 9xz^2 - 9z^3\) | \(0.167303\) | \(\infty\) |
2-torsion field: 6.0.2096704.1
BSD invariants
| Hasse-Weil conjecture: | verified |
| Analytic rank: | \(2\) |
| Mordell-Weil rank: | \(2\) |
| 2-Selmer rank: | \(2\) |
| Regulator: | \( 0.020992 \) |
| Real period: | \( 24.53325 \) |
| Tamagawa product: | \( 1 \) |
| Torsion order: | \( 1 \) |
| Leading coefficient: | \( 0.515024 \) |
| Analytic order of Ш: | \( 1 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(181\) | \(2\) | \(2\) | \(1\) | \(1 - 7 T + 181 T^{2}\) |  |
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 \(\Q (b) \simeq \) 6.6.194264244901.1 with defining polynomial:
\(x^{6} - x^{5} - 75 x^{4} - 104 x^{3} + 918 x^{2} + 2509 x + 1685\)
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
\(g_4 = -\frac{798799}{209375} b^{5} - \frac{3129874}{209375} b^{4} + \frac{47535977}{209375} b^{3} + \frac{12936296}{8375} b^{2} + \frac{660966068}{209375} b + \frac{84325344}{41875}\)
\(g_6 = \frac{10121301533}{5234375} b^{5} + \frac{2789901058}{5234375} b^{4} - \frac{662632165209}{5234375} b^{3} - \frac{69969753182}{209375} b^{2} + \frac{1174801020694}{5234375} b + \frac{604527106827}{1046875}\)
Conductor norm: 1
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 \(K = \Q (a) \simeq \) 6.6.194264244901.1 with defining polynomial \(x^{6} - x^{5} - 75 x^{4} - 104 x^{3} + 918 x^{2} + 2509 x + 1685\)
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 \(F \simeq \) \(\Q(\sqrt{181}) \) with generator \(-\frac{4}{1675} a^{5} + \frac{13}{1675} a^{4} + \frac{187}{1675} a^{3} + \frac{414}{1675} a^{2} + \frac{254}{1675} a - \frac{2088}{335}\) with minimal polynomial \(x^{2} - x - 45\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 3.3.32761.1 with generator \(-\frac{16}{335} a^{5} + \frac{52}{335} a^{4} + \frac{1083}{335} a^{3} - \frac{689}{335} a^{2} - \frac{13389}{335} a - \frac{2724}{67}\) with minimal polynomial \(x^{3} - x^{2} - 60 x + 67\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple