Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^2 + x)y = 9x^5 + 2x^4 - 21x^3 - 22x^2 - 8x - 1$ | (homogenize, simplify) |
$y^2 + (x^2z + xz^2)y = 9x^5z + 2x^4z^2 - 21x^3z^3 - 22x^2z^4 - 8xz^5 - z^6$ | (dehomogenize, simplify) |
$y^2 = 36x^5 + 9x^4 - 82x^3 - 87x^2 - 32x - 4$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(8649\) | \(=\) | \( 3^{2} \cdot 31^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(700569\) | \(=\) | \( 3^{6} \cdot 31^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(1132\) | \(=\) | \( 2^{2} \cdot 283 \) |
\( I_4 \) | \(=\) | \(73377\) | \(=\) | \( 3^{2} \cdot 31 \cdot 263 \) |
\( I_6 \) | \(=\) | \(21088959\) | \(=\) | \( 3 \cdot 17 \cdot 31 \cdot 13339 \) |
\( I_{10} \) | \(=\) | \(369024\) | \(=\) | \( 2^{7} \cdot 3 \cdot 31^{2} \) |
\( J_2 \) | \(=\) | \(849\) | \(=\) | \( 3 \cdot 283 \) |
\( J_4 \) | \(=\) | \(2517\) | \(=\) | \( 3 \cdot 839 \) |
\( J_6 \) | \(=\) | \(-2507\) | \(=\) | \( - 23 \cdot 109 \) |
\( J_8 \) | \(=\) | \(-2115933\) | \(=\) | \( - 3 \cdot 823 \cdot 857 \) |
\( J_{10} \) | \(=\) | \(700569\) | \(=\) | \( 3^{6} \cdot 31^{2} \) |
\( g_1 \) | \(=\) | \(1815232161643/2883\) | ||
\( g_2 \) | \(=\) | \(19016091893/8649\) | ||
\( g_3 \) | \(=\) | \(-200783123/77841\) |
Automorphism group
\(\mathrm{Aut}(X)\) | \(\simeq\) | $C_6$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
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\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $D_6$ | 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 | |||||
---|---|---|---|---|---|---|---|---|
\((-2 : 3 : 3) + (-1 : 3 : 3) - 2 \cdot(1 : 0 : 0)\) | \((3x + z) (3x + 2z)\) | \(=\) | \(0,\) | \(9y\) | \(=\) | \(z^3\) | \(0\) | \(2\) |
\((-1 : 3 : 3) - (1 : 0 : 0)\) | \(3x + z\) | \(=\) | \(0,\) | \(9y\) | \(=\) | \(z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-2 : 3 : 3) + (-1 : 3 : 3) - 2 \cdot(1 : 0 : 0)\) | \((3x + z) (3x + 2z)\) | \(=\) | \(0,\) | \(9y\) | \(=\) | \(z^3\) | \(0\) | \(2\) |
\((-1 : 3 : 3) - (1 : 0 : 0)\) | \(3x + z\) | \(=\) | \(0,\) | \(9y\) | \(=\) | \(z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \((3x + z) (3x + 2z)\) | \(=\) | \(0,\) | \(9y\) | \(=\) | \(x^2z + xz^2 + 2z^3\) | \(0\) | \(2\) |
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(3x + z\) | \(=\) | \(0,\) | \(9y\) | \(=\) | \(x^2z + xz^2 + 2z^3\) | \(0\) | \(2\) |
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 1 \) |
Real period: | \( 5.541099 \) |
Tamagawa product: | \( 4 \) |
Torsion order: | \( 4 \) |
Leading coefficient: | \( 1.385274 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(3\) | \(2\) | \(6\) | \(4\) | \(1 - 3 T + 3 T^{2}\) | |
\(31\) | \(2\) | \(2\) | \(1\) | \(1 - 4 T + 31 T^{2}\) |
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 \(\Q (b) \simeq \) 6.6.772987077.1 with defining polynomial:
\(x^{6} - x^{5} - 28 x^{4} + 51 x^{3} + 75 x^{2} - 98 x - 92\)
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
\(g_4 = -\frac{16934823333}{3104} b^{5} + \frac{15604267599}{1552} b^{4} + \frac{223933318713}{1552} b^{3} - \frac{1241168535621}{3104} b^{2} - \frac{111941475201}{1552} b + \frac{462072237363}{776}\)
\(g_6 = -\frac{579872195059509}{6208} b^{5} + \frac{1068664047901587}{6208} b^{4} + \frac{15335612391671001}{6208} b^{3} - \frac{2656270916385201}{388} b^{2} - \frac{958200176670183}{776} b + \frac{7911131565422907}{776}\)
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 \(K = \Q (a) \simeq \) 6.6.772987077.1 with defining polynomial \(x^{6} - x^{5} - 28 x^{4} + 51 x^{3} + 75 x^{2} - 98 x - 92\)
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 \(F \simeq \) \(\Q(\sqrt{93}) \) with generator \(-\frac{1}{2} a^{4} + 13 a^{2} - \frac{23}{2} a - 23\) with minimal polynomial \(x^{2} - x - 23\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 3.3.961.1 with generator \(-\frac{17}{97} a^{5} + \frac{1}{97} a^{4} + \frac{437}{97} a^{3} - \frac{450}{97} a^{2} - \frac{700}{97} a + \frac{60}{97}\) with minimal polynomial \(x^{3} - x^{2} - 10 x + 8\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple