Minimal equation
Minimal equation
Simplified equation
$y^2 = x^5 - x$ | (homogenize, simplify) |
$y^2 = x^5z - xz^5$ | (dehomogenize, simplify) |
$y^2 = x^5 - x$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(4096\) | \(=\) | \( 2^{12} \) | magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(4096,2),R![1]>*])); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(-65536\) | \(=\) | \( - 2^{16} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(20\) | \(=\) | \( 2^{2} \cdot 5 \) |
\( I_4 \) | \(=\) | \(-20\) | \(=\) | \( - 2^{2} \cdot 5 \) |
\( I_6 \) | \(=\) | \(-40\) | \(=\) | \( - 2^{3} \cdot 5 \) |
\( I_{10} \) | \(=\) | \(8\) | \(=\) | \( 2^{3} \) |
\( J_2 \) | \(=\) | \(80\) | \(=\) | \( 2^{4} \cdot 5 \) |
\( J_4 \) | \(=\) | \(480\) | \(=\) | \( 2^{5} \cdot 3 \cdot 5 \) |
\( J_6 \) | \(=\) | \(-1280\) | \(=\) | \( - 2^{8} \cdot 5 \) |
\( J_8 \) | \(=\) | \(-83200\) | \(=\) | \( - 2^{8} \cdot 5^{2} \cdot 13 \) |
\( J_{10} \) | \(=\) | \(65536\) | \(=\) | \( 2^{16} \) |
\( g_1 \) | \(=\) | \(50000\) | ||
\( g_2 \) | \(=\) | \(3750\) | ||
\( g_3 \) | \(=\) | \(-125\) |
Automorphism group
\(\mathrm{Aut}(X)\) | \(\simeq\) | $C_4$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
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\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $GL(2,3)$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
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Rational points
Number of rational Weierstrass points: \(4\)
This curve is locally solvable everywhere.
Mordell-Weil group of the Jacobian
Group structure: \(\Z/{2}\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 0 : 1) - (1 : 0 : 0)\) | \(x + z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
\((0 : 0 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 0 : 1) - (1 : 0 : 0)\) | \(x + z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
\((0 : 0 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 0 : 1) - (1 : 0 : 0)\) | \(x + z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
\((0 : 0 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
2-torsion field: \(\Q(\sqrt{-1}) \)
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 1 \) |
Real period: | \( 12.68998 \) |
Tamagawa product: | \( 4 \) |
Torsion order: | \( 8 \) |
Leading coefficient: | \( 0.793124 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(12\) | \(16\) | \(4\) | \(1\) |
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? |
---|---|---|
\(2\) | 2.360.2 | yes |
\(3\) | 3.6480.22 | no |
Sato-Tate group
\(\mathrm{ST}\) | \(\simeq\) | $J(C_2)$ |
\(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{U}(1)\) |
Decomposition of the Jacobian
Splits over the number field \(\Q (b) \simeq \) \(\Q(\sqrt{2}) \) with defining polynomial:
\(x^{2} - 2\)
Decomposes up to isogeny as the square of the elliptic curve isogeny class:
Elliptic curve isogeny class 2.2.8.1-64.1-a
Endomorphisms of the Jacobian
Of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
\(\End (J_{})\) | \(\simeq\) | \(\Z [\sqrt{-1}]\) |
\(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-1}) \) |
\(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\C\) |
Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\zeta_{8})\) with defining polynomial \(x^{4} + 1\)
Not of \(\GL_2\)-type over \(\overline{\Q}\)
Endomorphism ring over \(\overline{\Q}\):
\(\End (J_{\overline{\Q}})\) | \(\simeq\) | a maximal order of \(\End (J_{\overline{\Q}}) \otimes \Q\) |
\(\End (J_{\overline{\Q}}) \otimes \Q \) | \(\simeq\) | \(\mathrm{M}_2(\)\(\Q(\sqrt{-2}) \)\()\) |
\(\End (J_{\overline{\Q}}) \otimes \R\) | \(\simeq\) | \(\mathrm{M}_2 (\C)\) |
Remainder of the endomorphism lattice by field
Over subfield \(F \simeq \) \(\Q(\sqrt{-2}) \) with generator \(-a^{3} - a\) with minimal polynomial \(x^{2} + 2\):
\(\End (J_{F})\) | \(\simeq\) | the maximal order of \(\End (J_{F}) \otimes \Q\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\zeta_{8})\) (CM) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C \times \C\) |
Not of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) \(\Q(\sqrt{2}) \) with generator \(a^{3} - a\) with minimal polynomial \(x^{2} - 2\):
\(\End (J_{F})\) | \(\simeq\) | a non-Eichler order of index \(4\) in a maximal order of \(\End (J_{F}) \otimes \Q\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\mathrm{M}_2(\)\(\Q\)\()\) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\mathrm{M}_2 (\R)\) |
Not of \(\GL_2\)-type, not simple
Over subfield \(F \simeq \) \(\Q(\sqrt{-1}) \) with generator \(-a^{2}\) with minimal polynomial \(x^{2} + 1\):
\(\End (J_{F})\) | \(\simeq\) | a maximal order of \(\End (J_{F}) \otimes \Q\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | the quaternion algebra over \(\Q\) of discriminant 2 |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\H\) |
Not of \(\GL_2\)-type, simple