Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^2 + x)y = x^5 - 10x^4 + 14x^3 - 5x^2 - x$ | (homogenize, simplify) |
$y^2 + (x^2z + xz^2)y = x^5z - 10x^4z^2 + 14x^3z^3 - 5x^2z^4 - xz^5$ | (dehomogenize, simplify) |
$y^2 = 4x^5 - 39x^4 + 58x^3 - 19x^2 - 4x$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(900601\) | \(=\) | \( 13^{2} \cdot 73^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(900601\) | \(=\) | \( 13^{2} \cdot 73^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(3844\) | \(=\) | \( 2^{2} \cdot 31^{2} \) |
\( I_4 \) | \(=\) | \(855049\) | \(=\) | \( 13 \cdot 17 \cdot 53 \cdot 73 \) |
\( I_6 \) | \(=\) | \(832873717\) | \(=\) | \( 13 \cdot 67 \cdot 73 \cdot 13099 \) |
\( I_{10} \) | \(=\) | \(115276928\) | \(=\) | \( 2^{7} \cdot 13^{2} \cdot 73^{2} \) |
\( J_2 \) | \(=\) | \(961\) | \(=\) | \( 31^{2} \) |
\( J_4 \) | \(=\) | \(2853\) | \(=\) | \( 3^{2} \cdot 317 \) |
\( J_6 \) | \(=\) | \(-2843\) | \(=\) | \( -2843 \) |
\( J_8 \) | \(=\) | \(-2717933\) | \(=\) | \( - 23 \cdot 118171 \) |
\( J_{10} \) | \(=\) | \(900601\) | \(=\) | \( 13^{2} \cdot 73^{2} \) |
\( g_1 \) | \(=\) | \(819628286980801/900601\) | ||
\( g_2 \) | \(=\) | \(2532048001893/900601\) | ||
\( g_3 \) | \(=\) | \(-2625570203/900601\) |
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 | |||||
---|---|---|---|---|---|---|---|---|
\((0 : 0 : 1) + (1 : -1 : 1) - 2 \cdot(1 : 0 : 0)\) | \(x (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2\) | \(0\) | \(2\) |
\((0 : 0 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : 0 : 1) + (1 : -1 : 1) - 2 \cdot(1 : 0 : 0)\) | \(x (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2\) | \(0\) | \(2\) |
\((0 : 0 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : 0 : 1) + (1 : 0 : 1) - 2 \cdot(1 : 0 : 0)\) | \(x (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^2z - xz^2\) | \(0\) | \(2\) |
\((0 : 0 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^2z + xz^2\) | \(0\) | \(2\) |
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(4\) |
Regulator: | \( 1 \) |
Real period: | \( 5.194205 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 4 \) |
Leading coefficient: | \( 1.298551 \) |
Analytic order of Ш: | \( 4 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(13\) | \(2\) | \(2\) | \(1\) | \(1 + 2 T + 13 T^{2}\) | |
\(73\) | \(2\) | \(2\) | \(1\) | \(1 - 10 T + 73 T^{2}\) |
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 \(\Q (b)\) with defining polynomial:
\(x^{6} - x^{5} - 395 x^{4} + 5360 x^{3} - 29140 x^{2} + 71536 x - 64832\)
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
\(g_4 = -\frac{35833505}{4096} b^{5} - \frac{112070395}{4096} b^{4} + \frac{13691736455}{4096} b^{3} - \frac{33888492095}{1024} b^{2} + \frac{121164642869}{1024} b - \frac{4396582523}{32}\)
\(g_6 = -\frac{41591954911}{8192} b^{5} - \frac{260222409785}{16384} b^{4} + \frac{31783394364189}{16384} b^{3} - \frac{314654921803391}{16384} b^{2} + \frac{281247209441965}{4096} b - \frac{326581689584709}{4096}\)
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 \(K = \Q (a)\) with defining polynomial \(x^{6} - x^{5} - 395 x^{4} + 5360 x^{3} - 29140 x^{2} + 71536 x - 64832\)
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 \(F \simeq \) \(\Q(\sqrt{949}) \) with generator \(-\frac{1}{8} a^{5} - \frac{1}{2} a^{4} + \frac{187}{4} a^{3} - \frac{3499}{8} a^{2} + \frac{5983}{4} a - 1699\) with minimal polynomial \(x^{2} - x - 237\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 3.3.900601.1 with generator \(\frac{1}{18} a^{5} + \frac{11}{36} a^{4} - \frac{241}{12} a^{3} + \frac{5989}{36} a^{2} - \frac{1481}{3} a + \frac{4220}{9}\) with minimal polynomial \(x^{3} - x^{2} - 316 x + 2144\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple