Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x + 1)y = x^5 - 3x^4 - 3x^3 + 6x^2 - 3x$ | (homogenize, simplify) |
$y^2 + (x^3 + xz^2 + z^3)y = x^5z - 3x^4z^2 - 3x^3z^3 + 6x^2z^4 - 3xz^5$ | (dehomogenize, simplify) |
$y^2 = x^6 + 4x^5 - 10x^4 - 10x^3 + 25x^2 - 10x + 1$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(110889\) | \(=\) | \( 3^{4} \cdot 37^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(110889\) | \(=\) | \( 3^{4} \cdot 37^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
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Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(1380\) | \(=\) | \( 2^{2} \cdot 3 \cdot 5 \cdot 23 \) |
\( I_4 \) | \(=\) | \(94905\) | \(=\) | \( 3^{3} \cdot 5 \cdot 19 \cdot 37 \) |
\( I_6 \) | \(=\) | \(34200765\) | \(=\) | \( 3^{3} \cdot 5 \cdot 37 \cdot 41 \cdot 167 \) |
\( I_{10} \) | \(=\) | \(14193792\) | \(=\) | \( 2^{7} \cdot 3^{4} \cdot 37^{2} \) |
\( J_2 \) | \(=\) | \(345\) | \(=\) | \( 3 \cdot 5 \cdot 23 \) |
\( J_4 \) | \(=\) | \(1005\) | \(=\) | \( 3 \cdot 5 \cdot 67 \) |
\( J_6 \) | \(=\) | \(-995\) | \(=\) | \( - 5 \cdot 199 \) |
\( J_8 \) | \(=\) | \(-338325\) | \(=\) | \( - 3 \cdot 5^{2} \cdot 13 \cdot 347 \) |
\( J_{10} \) | \(=\) | \(110889\) | \(=\) | \( 3^{4} \cdot 37^{2} \) |
\( g_1 \) | \(=\) | \(60340715625/1369\) | ||
\( g_2 \) | \(=\) | \(509493125/1369\) | ||
\( g_3 \) | \(=\) | \(-13158875/12321\) |
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: \(0\)
This curve is locally solvable everywhere.
Mordell-Weil group of the Jacobian
Group structure: \(\Z \oplus \Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : -1 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.228168\) | \(\infty\) |
\((1 : -1 : 1) - (1 : 0 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3\) | \(0.228168\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : -1 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.228168\) | \(\infty\) |
\((1 : -1 : 1) - (1 : 0 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3\) | \(0.228168\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : -1 : 1) + (1 : 1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + xz^2 - z^3\) | \(0.228168\) | \(\infty\) |
\((1 : 1 : 1) - (1 : 1 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 + xz^2 + z^3\) | \(0.228168\) | \(\infty\) |
2-torsion field: 9.9.1363532208525369.2
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(2\) |
Mordell-Weil rank: | \(2\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 0.039045 \) |
Real period: | \( 21.98056 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 0.858248 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(3\) | \(4\) | \(4\) | \(1\) | \(1 + 3 T + 3 T^{2}\) | |
\(37\) | \(2\) | \(2\) | \(1\) | \(1 + 10 T + 37 T^{2}\) |
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 \(\Q (b) \simeq \) 6.6.454965701877.2 with defining polynomial:
\(x^{6} - 111 x^{4} - 296 x^{3} + 999 x^{2} + 3663 x + 2331\)
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
\(g_4 = -\frac{69588900}{7686973} b^{5} + \frac{195589215}{7686973} b^{4} + \frac{4154341500}{7686973} b^{3} + \frac{23045260005}{7686973} b^{2} + \frac{53213628105}{7686973} b + \frac{5891307795}{1098139}\)
\(g_6 = \frac{7126269800796}{2636631739} b^{5} - \frac{78121867595337}{2636631739} b^{4} - \frac{220022576402778}{2636631739} b^{3} + \frac{1682832880289313}{2636631739} b^{2} + \frac{5514370690925493}{2636631739} b + \frac{502485786075150}{376661677}\)
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 \(K = \Q (a) \simeq \) 6.6.454965701877.2 with defining polynomial \(x^{6} - 111 x^{4} - 296 x^{3} + 999 x^{2} + 3663 x + 2331\)
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 \(F \simeq \) \(\Q(\sqrt{37}) \) with generator \(-\frac{225}{22411} a^{5} + \frac{690}{22411} a^{4} + \frac{22859}{22411} a^{3} - \frac{4995}{22411} a^{2} - \frac{209457}{22411} a - \frac{87714}{22411}\) with minimal polynomial \(x^{2} - x - 9\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 3.3.110889.1 with generator \(-\frac{8}{921} a^{5} + \frac{15}{307} a^{4} + \frac{250}{307} a^{3} - \frac{2081}{921} a^{2} - \frac{2331}{307} a + \frac{1924}{307}\) with minimal polynomial \(x^{3} - 111 x - 37\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple