Minimal equation
Minimal equation
Simplified equation
| $y^2 + (x^3 + x + 1)y = -x^6 - 2x^5 + x^4 + 5x^3 - 3x - 1$ | (homogenize, simplify) |
| $y^2 + (x^3 + xz^2 + z^3)y = -x^6 - 2x^5z + x^4z^2 + 5x^3z^3 - 3xz^5 - z^6$ | (dehomogenize, simplify) |
| $y^2 = -3x^6 - 8x^5 + 6x^4 + 22x^3 + x^2 - 10x - 3$ | (homogenize, minimize) |
Invariants
| Conductor: | \( N \) | \(=\) | \(219961\) | \(=\) | \( 7^{2} \cdot 67^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(219961\) | \(=\) | \( 7^{2} \cdot 67^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(1924\) | \(=\) | \( 2^{2} \cdot 13 \cdot 37 \) |
| \( I_4 \) | \(=\) | \(197449\) | \(=\) | \( 7 \cdot 67 \cdot 421 \) |
| \( I_6 \) | \(=\) | \(97792597\) | \(=\) | \( 7 \cdot 67 \cdot 208513 \) |
| \( I_{10} \) | \(=\) | \(28155008\) | \(=\) | \( 2^{7} \cdot 7^{2} \cdot 67^{2} \) |
| \( J_2 \) | \(=\) | \(481\) | \(=\) | \( 13 \cdot 37 \) |
| \( J_4 \) | \(=\) | \(1413\) | \(=\) | \( 3^{2} \cdot 157 \) |
| \( J_6 \) | \(=\) | \(-1403\) | \(=\) | \( - 23 \cdot 61 \) |
| \( J_8 \) | \(=\) | \(-667853\) | \(=\) | \( - 53 \cdot 12601 \) |
| \( J_{10} \) | \(=\) | \(219961\) | \(=\) | \( 7^{2} \cdot 67^{2} \) |
| \( g_1 \) | \(=\) | \(25746925826401/219961\) | ||
| \( g_2 \) | \(=\) | \(157245197733/219961\) | ||
| \( g_3 \) | \(=\) | \(-324599483/219961\) |
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 $\Q_{2}$.
Mordell-Weil group of the Jacobian
Group structure: trivial
2-torsion field: 9.9.10642338203800681.1
BSD invariants
| Hasse-Weil conjecture: | verified |
| Analytic rank: | \(0\) |
| Mordell-Weil rank: | \(0\) |
| 2-Selmer rank: | \(0\) |
| Regulator: | \( 1 \) |
| Real period: | \( 7.499062 \) |
| Tamagawa product: | \( 1 \) |
| Torsion order: | \( 1 \) |
| Leading coefficient: | \( 7.499062 \) |
| Analytic order of Ш: | \( 1 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(7\) | \(2\) | \(2\) | \(1\) | \(1 + T + 7 T^{2}\) |  |
| \(67\) | \(2\) | \(2\) | \(1\) | \(1 + 5 T + 67 T^{2}\) | |
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 \(\Q (b) \simeq \) 6.6.22691552673349.1 with defining polynomial:
\(x^{6} - x^{5} - 195 x^{4} - 582 x^{3} + 5818 x^{2} + 12963 x - 60353\)
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
\(g_4 = -\frac{84853462223}{12275} b^{5} + \frac{724259593494}{12275} b^{4} + \frac{11088828149272}{12275} b^{3} - \frac{34174220408483}{12275} b^{2} - \frac{236160504623173}{12275} b + \frac{679612333837747}{12275}\)
\(g_6 = -\frac{2001828812701473}{2455} b^{5} + \frac{3417287443494470}{491} b^{4} + \frac{261603244143343802}{2455} b^{3} - \frac{806224319730936827}{2455} b^{2} - \frac{5571406188836417069}{2455} b + \frac{3206628805547907390}{491}\)
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 \(K = \Q (a) \simeq \) 6.6.22691552673349.1 with defining polynomial \(x^{6} - x^{5} - 195 x^{4} - 582 x^{3} + 5818 x^{2} + 12963 x - 60353\)
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 \(F \simeq \) \(\Q(\sqrt{469}) \) with generator \(\frac{6}{491} a^{5} - \frac{43}{491} a^{4} - \frac{823}{491} a^{3} + \frac{1174}{491} a^{2} + \frac{15066}{491} a - \frac{25931}{491}\) with minimal polynomial \(x^{2} - x - 117\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 3.3.219961.1 with generator \(-\frac{36}{12275} a^{5} + \frac{258}{12275} a^{4} + \frac{5429}{12275} a^{3} - \frac{10481}{12275} a^{2} - \frac{146861}{12275} a + \frac{203704}{12275}\) with minimal polynomial \(x^{3} - x^{2} - 156 x + 799\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple