Minimal equation
Minimal equation
Simplified equation
$y^2 = x^5 - 2x^3 + 2x$ | (homogenize, simplify) |
$y^2 = x^5z - 2x^3z^3 + 2xz^5$ | (dehomogenize, simplify) |
$y^2 = x^5 - 2x^3 + 2x$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(262144\) | \(=\) | \( 2^{18} \) | magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(262144,2),R![1]>*])); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(524288\) | \(=\) | \( 2^{19} \) | magma: Discriminant(C); Factorization(Integers()!$1);
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Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(26\) | \(=\) | \( 2 \cdot 13 \) |
\( I_4 \) | \(=\) | \(-2\) | \(=\) | \( -2 \) |
\( I_6 \) | \(=\) | \(40\) | \(=\) | \( 2^{3} \cdot 5 \) |
\( I_{10} \) | \(=\) | \(2\) | \(=\) | \( 2 \) |
\( J_2 \) | \(=\) | \(208\) | \(=\) | \( 2^{4} \cdot 13 \) |
\( J_4 \) | \(=\) | \(1888\) | \(=\) | \( 2^{5} \cdot 59 \) |
\( J_6 \) | \(=\) | \(-2304\) | \(=\) | \( - 2^{8} \cdot 3^{2} \) |
\( J_8 \) | \(=\) | \(-1010944\) | \(=\) | \( - 2^{8} \cdot 11 \cdot 359 \) |
\( J_{10} \) | \(=\) | \(524288\) | \(=\) | \( 2^{19} \) |
\( g_1 \) | \(=\) | \(742586\) | ||
\( g_2 \) | \(=\) | \(129623/4\) | ||
\( g_3 \) | \(=\) | \(-1521/8\) |
Automorphism group
\(\mathrm{Aut}(X)\) | \(\simeq\) | $C_2$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
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\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $D_4$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
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Rational points
Number of rational Weierstrass points: \(2\)
This curve is locally solvable everywhere.
Mordell-Weil group of the Jacobian
Group structure: \(\Z \oplus \Z/{2}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((1 : -1 : 1) - (1 : 0 : 0)\) | \(x - z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.759196\) | \(\infty\) |
\((0 : 0 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((1 : -1 : 1) - (1 : 0 : 0)\) | \(x - z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.759196\) | \(\infty\) |
\((0 : 0 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((1 : -1/2 : 1) - (1 : 0 : 0)\) | \(x - z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-1/2z^3\) | \(0.759196\) | \(\infty\) |
\((0 : 0 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(1\) |
Mordell-Weil rank: | \(1\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 0.759196 \) |
Real period: | \( 7.048010 \) |
Tamagawa product: | \( 2 \) |
Torsion order: | \( 2 \) |
Leading coefficient: | \( 2.675411 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(18\) | \(19\) | \(2\) | \(1\) |
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? |
---|---|---|
\(2\) | 2.90.3 | yes |
\(3\) | 3.270.1 | no |
Sato-Tate group
\(\mathrm{ST}\) | \(\simeq\) | $J(E_4)$ |
\(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\) |
Decomposition of the Jacobian
Splits over the number field \(\Q (b) \simeq \) 4.2.2048.1 with defining polynomial:
\(x^{4} - 2\)
Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
\(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
\(g_4 = 160 b^{2} + 96\)
\(g_6 = 1792 b^{3} + 2304 b\)
Conductor norm: 16384
\(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
\(g_4 = 160 b^{2} + 96\)
\(g_6 = -1792 b^{3} - 2304 b\)
Conductor norm: 16384
Endomorphisms of the Jacobian
Not of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
\(\End (J_{})\) | \(\simeq\) | \(\Z\) |
\(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
\(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\R\) |
Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) 8.0.16777216.2 with defining polynomial \(x^{8} - 4 x^{6} + 8 x^{4} - 4 x^{2} + 1\)
Not of \(\GL_2\)-type over \(\overline{\Q}\)
Endomorphism ring over \(\overline{\Q}\):
\(\End (J_{\overline{\Q}})\) | \(\simeq\) | a non-Eichler order of index \(4\) 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 \(F \simeq \) \(\Q(\sqrt{-1}) \) with generator \(\frac{2}{3} a^{6} - \frac{7}{3} a^{4} + \frac{14}{3} a^{2} - \frac{4}{3}\) with minimal polynomial \(x^{2} + 1\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\sqrt{-1}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-1}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) \(\Q(\sqrt{2}) \) with generator \(-\frac{1}{3} a^{6} + \frac{2}{3} a^{4} - \frac{1}{3} a^{2} - \frac{4}{3}\) with minimal polynomial \(x^{2} - 2\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R\) |
Not of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) \(\Q(\sqrt{-2}) \) with generator \(a^{6} - 4 a^{4} + 7 a^{2} - 2\) with minimal polynomial \(x^{2} + 2\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R\) |
Not of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 4.2.2048.1 with generator \(\frac{4}{3} a^{7} - \frac{14}{3} a^{5} + \frac{25}{3} a^{3} - \frac{5}{3} a\) with minimal polynomial \(x^{4} - 2\):
\(\End (J_{F})\) | \(\simeq\) | an order of index \(2\) in \(\Z \times \Z\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q\) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Of \(\GL_2\)-type, not simple
Over subfield \(F \simeq \) 4.0.2048.1 with generator \(-a^{7} + 4 a^{5} - 8 a^{3} + 3 a\) with minimal polynomial \(x^{4} + 2\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\sqrt{2}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{2}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 4.0.2048.1 with generator \(\frac{2}{3} a^{7} - \frac{7}{3} a^{5} + \frac{11}{3} a^{3} + \frac{2}{3} a\) with minimal polynomial \(x^{4} + 2\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\sqrt{2}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{2}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 4.2.2048.1 with generator \(\frac{1}{3} a^{7} - \frac{5}{3} a^{5} + \frac{10}{3} a^{3} - \frac{8}{3} a\) with minimal polynomial \(x^{4} - 2\):
\(\End (J_{F})\) | \(\simeq\) | an order of index \(2\) in \(\Z \times \Z\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q\) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Of \(\GL_2\)-type, not simple
Over subfield \(F \simeq \) \(\Q(\zeta_{8})\) with generator \(\frac{1}{3} a^{6} - \frac{5}{3} a^{4} + \frac{10}{3} a^{2} - \frac{5}{3}\) with minimal polynomial \(x^{4} + 1\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\sqrt{-1}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-1}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple