Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^2 + x)y = x^5 + 8x^4 + 11x^3 + 3x^2 - x$ | (homogenize, simplify) |
$y^2 + (x^2z + xz^2)y = x^5z + 8x^4z^2 + 11x^3z^3 + 3x^2z^4 - xz^5$ | (dehomogenize, simplify) |
$y^2 = 4x^5 + 33x^4 + 46x^3 + 13x^2 - 4x$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(8281\) | \(=\) | \( 7^{2} \cdot 13^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(405769\) | \(=\) | \( 7^{4} \cdot 13^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
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Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(2596\) | \(=\) | \( 2^{2} \cdot 11 \cdot 59 \) |
\( I_4 \) | \(=\) | \(375193\) | \(=\) | \( 7^{2} \cdot 13 \cdot 19 \cdot 31 \) |
\( I_6 \) | \(=\) | \(248614093\) | \(=\) | \( 7^{2} \cdot 13 \cdot 390289 \) |
\( I_{10} \) | \(=\) | \(51938432\) | \(=\) | \( 2^{7} \cdot 7^{4} \cdot 13^{2} \) |
\( J_2 \) | \(=\) | \(649\) | \(=\) | \( 11 \cdot 59 \) |
\( J_4 \) | \(=\) | \(1917\) | \(=\) | \( 3^{3} \cdot 71 \) |
\( J_6 \) | \(=\) | \(-1907\) | \(=\) | \( -1907 \) |
\( J_8 \) | \(=\) | \(-1228133\) | \(=\) | \( -1228133 \) |
\( J_{10} \) | \(=\) | \(405769\) | \(=\) | \( 7^{4} \cdot 13^{2} \) |
\( g_1 \) | \(=\) | \(115139273278249/405769\) | ||
\( g_2 \) | \(=\) | \(524030063733/405769\) | ||
\( g_3 \) | \(=\) | \(-803230307/405769\) |
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);
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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 | |||||
---|---|---|---|---|---|---|---|---|
\((0 : 0 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
\((-1 : 0 : 1) - (1 : 0 : 0)\) | \(x + z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : 0 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
\((-1 : 0 : 1) - (1 : 0 : 0)\) | \(x + z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : 0 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^2z + xz^2\) | \(0\) | \(2\) |
\((-1 : 0 : 1) - (1 : 0 : 0)\) | \(x + z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^2z + xz^2\) | \(0\) | \(2\) |
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 1 \) |
Real period: | \( 19.78540 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 4 \) |
Leading coefficient: | \( 1.236587 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(7\) | \(2\) | \(4\) | \(1\) | \(1 + 4 T + 7 T^{2}\) | |
\(13\) | \(2\) | \(2\) | \(1\) | \(1 + 2 T + 13 T^{2}\) |
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 \(\Q (b) \simeq \) 6.6.891474493.1 with defining polynomial:
\(x^{6} - x^{5} - 31 x^{4} + 4 x^{3} + 253 x^{2} + 101 x - 391\)
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
\(g_4 = \frac{6900061}{184} b^{5} + \frac{2308663}{368} b^{4} - \frac{213373265}{184} b^{3} - \frac{110121753}{92} b^{2} + \frac{189026334}{23} b + \frac{212819299}{16}\)
\(g_6 = -\frac{14940873857}{1472} b^{5} - \frac{1373744645}{736} b^{4} + \frac{7227770823}{23} b^{3} + \frac{241586544017}{736} b^{2} - \frac{3281620049799}{1472} b - \frac{14496977859}{4}\)
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 \(K = \Q (a) \simeq \) 6.6.891474493.1 with defining polynomial \(x^{6} - x^{5} - 31 x^{4} + 4 x^{3} + 253 x^{2} + 101 x - 391\)
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 \(F \simeq \) \(\Q(\sqrt{13}) \) with generator \(-\frac{1}{368} a^{5} - \frac{3}{92} a^{4} + \frac{59}{368} a^{3} + \frac{211}{368} a^{2} - \frac{227}{184} a - \frac{13}{16}\) with minimal polynomial \(x^{2} - x - 3\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 3.3.8281.1 with generator \(\frac{3}{184} a^{5} - \frac{5}{92} a^{4} - \frac{39}{184} a^{3} + \frac{195}{184} a^{2} - \frac{31}{23} a - \frac{43}{8}\) with minimal polynomial \(x^{3} - x^{2} - 30 x + 64\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple