Minimal equation
Minimal equation
Simplified equation
| $y^2 + (x^3 + x + 1)y = -x^6 - x^5 + 3x^4 + 4x^3 - 3x^2 - 4x - 1$ | (homogenize, simplify) |
| $y^2 + (x^3 + xz^2 + z^3)y = -x^6 - x^5z + 3x^4z^2 + 4x^3z^3 - 3x^2z^4 - 4xz^5 - z^6$ | (dehomogenize, simplify) |
| $y^2 = -3x^6 - 4x^5 + 14x^4 + 18x^3 - 11x^2 - 14x - 3$ | (homogenize, minimize) |
Invariants
| Conductor: | \( N \) | \(=\) | \(301401\) | \(=\) | \( 3^{4} \cdot 61^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(301401\) | \(=\) | \( 3^{4} \cdot 61^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(2244\) | \(=\) | \( 2^{2} \cdot 3 \cdot 11 \cdot 17 \) |
| \( I_4 \) | \(=\) | \(275049\) | \(=\) | \( 3^{3} \cdot 61 \cdot 167 \) |
| \( I_6 \) | \(=\) | \(158130117\) | \(=\) | \( 3^{3} \cdot 61 \cdot 67 \cdot 1433 \) |
| \( I_{10} \) | \(=\) | \(38579328\) | \(=\) | \( 2^{7} \cdot 3^{4} \cdot 61^{2} \) |
| \( J_2 \) | \(=\) | \(561\) | \(=\) | \( 3 \cdot 11 \cdot 17 \) |
| \( J_4 \) | \(=\) | \(1653\) | \(=\) | \( 3 \cdot 19 \cdot 29 \) |
| \( J_6 \) | \(=\) | \(-1643\) | \(=\) | \( - 31 \cdot 53 \) |
| \( J_8 \) | \(=\) | \(-913533\) | \(=\) | \( - 3 \cdot 304511 \) |
| \( J_{10} \) | \(=\) | \(301401\) | \(=\) | \( 3^{4} \cdot 61^{2} \) |
| \( g_1 \) | \(=\) | \(686008169121/3721\) | ||
| \( g_2 \) | \(=\) | \(3603100853/3721\) | ||
| \( g_3 \) | \(=\) | \(-57454067/33489\) |
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: \(0\)
This curve is locally solvable except over $\Q_{2}$ and $\Q_{5}$.
Mordell-Weil group of the Jacobian
Group structure: trivial
2-torsion field: 9.9.27380039270784201.1
BSD invariants
| Hasse-Weil conjecture: | verified |
| Analytic rank: | \(0\) |
| Mordell-Weil rank: | \(0\) |
| 2-Selmer rank: | \(0\) |
| Regulator: | \( 1 \) |
| Real period: | \( 6.900558 \) |
| Tamagawa product: | \( 1 \) |
| Torsion order: | \( 1 \) |
| Leading coefficient: | \( 6.900558 \) |
| Analytic order of Ш: | \( 1 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(3\) | \(4\) | \(4\) | \(1\) | \(1 + 3 T^{2}\) |  |
| \(61\) | \(2\) | \(2\) | \(1\) | \(1 + 13 T + 61 T^{2}\) |  |
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? |
|---|---|---|
| \(2\) | 2.80.1 | 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 \(\Q (b) \simeq \) 6.6.5541396330861.1 with defining polynomial:
\(x^{6} - 183 x^{4} - 244 x^{3} + 1647 x^{2} + 549 x - 2745\)
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
\(g_4 = -\frac{448847426817}{507012500} b^{5} - \frac{3717968867361}{2028050000} b^{4} + \frac{40106532132459}{253506250} b^{3} + \frac{1102504970084661}{2028050000} b^{2} - \frac{673019993223549}{2028050000} b - \frac{475445034877491}{405610000}\)
\(g_6 = -\frac{60422321890130337}{50701250000} b^{5} - \frac{500673379794933771}{202805000000} b^{4} + \frac{21595984719044897871}{101402500000} b^{3} + \frac{148446854122001801571}{202805000000} b^{2} - \frac{90546645136674896739}{202805000000} b - \frac{32026060517213501163}{20280500000}\)
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 \(K = \Q (a) \simeq \) 6.6.5541396330861.1 with defining polynomial \(x^{6} - 183 x^{4} - 244 x^{3} + 1647 x^{2} + 549 x - 2745\)
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 \(F \simeq \) \(\Q(\sqrt{61}) \) with generator \(\frac{441}{202805} a^{5} - \frac{714}{202805} a^{4} - \frac{79547}{202805} a^{3} + \frac{11529}{202805} a^{2} + \frac{707661}{202805} a + \frac{16285}{40561}\) with minimal polynomial \(x^{2} - x - 15\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 3.3.301401.2 with generator \(-\frac{88}{12945} a^{5} - \frac{21}{4315} a^{4} + \frac{5402}{4315} a^{3} + \frac{31603}{12945} a^{2} - \frac{48861}{4315} a - \frac{2440}{863}\) with minimal polynomial \(x^{3} - 183 x - 793\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple