Minimal equation
Minimal equation
Simplified equation
$y^2 + y = x^5$ | (homogenize, simplify) |
$y^2 + z^3y = x^5z$ | (dehomogenize, simplify) |
$y^2 = 4x^5 + 1$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(3125\) | \(=\) | \( 5^{5} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(3125\) | \(=\) | \( 5^{5} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(0\) | \(=\) | \( 0 \) |
\( I_4 \) | \(=\) | \(0\) | \(=\) | \( 0 \) |
\( I_6 \) | \(=\) | \(0\) | \(=\) | \( 0 \) |
\( I_{10} \) | \(=\) | \(4\) | \(=\) | \( 2^{2} \) |
\( J_2 \) | \(=\) | \(0\) | \(=\) | \( 0 \) |
\( J_4 \) | \(=\) | \(0\) | \(=\) | \( 0 \) |
\( J_6 \) | \(=\) | \(0\) | \(=\) | \( 0 \) |
\( J_8 \) | \(=\) | \(0\) | \(=\) | \( 0 \) |
\( J_{10} \) | \(=\) | \(3125\) | \(=\) | \( 5^{5} \) |
\( g_1 \) | \(=\) | \(0\) | ||
\( g_2 \) | \(=\) | \(0\) | ||
\( g_3 \) | \(=\) | \(0\) |
Automorphism group
\(\mathrm{Aut}(X)\) | \(\simeq\) | $C_2$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
|
\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $C_{10}$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
|
Rational points
Number of rational Weierstrass points: \(1\)
This curve is locally solvable everywhere.
Mordell-Weil group of the Jacobian
Group structure: \(\Z/{5}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : 0 : 1) - (1 : 0 : 0)\) | \(x^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(5\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : 0 : 1) - (1 : 0 : 0)\) | \(x^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(5\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : 1 : 1) - (1 : 0 : 0)\) | \(x^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(z^3\) | \(0\) | \(5\) |
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(0\) |
Regulator: | \( 1 \) |
Real period: | \( 17.95784 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 5 \) |
Leading coefficient: | \( 0.718313 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(5\) | \(5\) | \(5\) | \(1\) | \(1\) |
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? |
---|---|---|
\(2\) | 2.36.1 | no |
\(3\) | 3.1296.1 | no |
Sato-Tate group
\(\mathrm{ST}\) | \(\simeq\) | $F_{ac}$ |
\(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{U}(1)\times\mathrm{U}(1)\) |
Decomposition of the Jacobian
Simple over \(\overline{\Q}\)
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 \(K = \Q (a) \simeq \) \(\Q(\zeta_{5})\) with defining polynomial \(x^{4} - x^{3} + x^{2} - x + 1\)
Not of \(\GL_2\)-type over \(\overline{\Q}\)
Endomorphism ring over \(\overline{\Q}\):
\(\End (J_{\overline{\Q}})\) | \(\simeq\) | the maximal order of \(\End (J_{\overline{\Q}}) \otimes \Q\) |
\(\End (J_{\overline{\Q}}) \otimes \Q \) | \(\simeq\) | \(\Q(\zeta_{5})\) (CM) |
\(\End (J_{\overline{\Q}}) \otimes \R\) | \(\simeq\) | \(\C \times \C\) |
Remainder of the endomorphism lattice by field
Over subfield \(F \simeq \) \(\Q(\sqrt{5}) \) with generator \(a^{3} - a^{2}\) with minimal polynomial \(x^{2} - x - 1\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{5}}{2}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{5}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Of \(\GL_2\)-type, simple