This curve is isomorphic to the quotient of the modular curve $X_0(125)$ by its Fricke involution $w_{125}$; this quotient is also denoted $X_0^+(125)$.
Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x + 1)y = 2x^5 + 2x^4 + 2x^3 + x^2$ | (homogenize, simplify) |
$y^2 + (x^3 + xz^2 + z^3)y = 2x^5z + 2x^4z^2 + 2x^3z^3 + x^2z^4$ | (dehomogenize, simplify) |
$y^2 = x^6 + 8x^5 + 10x^4 + 10x^3 + 5x^2 + 2x + 1$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(15625\) | \(=\) | \( 5^{6} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(15625\) | \(=\) | \( 5^{6} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(20\) | \(=\) | \( 2^{2} \cdot 5 \) |
\( I_4 \) | \(=\) | \(25\) | \(=\) | \( 5^{2} \) |
\( I_6 \) | \(=\) | \(1565\) | \(=\) | \( 5 \cdot 313 \) |
\( I_{10} \) | \(=\) | \(640\) | \(=\) | \( 2^{7} \cdot 5 \) |
\( J_2 \) | \(=\) | \(25\) | \(=\) | \( 5^{2} \) |
\( J_4 \) | \(=\) | \(0\) | \(=\) | \( 0 \) |
\( J_6 \) | \(=\) | \(-2500\) | \(=\) | \( - 2^{2} \cdot 5^{4} \) |
\( J_8 \) | \(=\) | \(-15625\) | \(=\) | \( - 5^{6} \) |
\( J_{10} \) | \(=\) | \(15625\) | \(=\) | \( 5^{6} \) |
\( g_1 \) | \(=\) | \(625\) | ||
\( g_2 \) | \(=\) | \(0\) | ||
\( g_3 \) | \(=\) | \(-100\) |
Automorphism group
\(\mathrm{Aut}(X)\) | \(\simeq\) | $C_2$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
|
\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $C_2$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
|
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 | |||||
---|---|---|---|---|---|---|---|---|
\(2 \cdot(0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2 - z^3\) | \(0.208772\) | \(\infty\) |
\((0 : -1 : 1) - (1 : 0 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0.234921\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(2 \cdot(0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2 - z^3\) | \(0.208772\) | \(\infty\) |
\((0 : -1 : 1) - (1 : 0 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0.234921\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(2 \cdot(0 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - xz^2 - z^3\) | \(0.208772\) | \(\infty\) |
\((0 : -1 : 1) - (1 : 1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 + xz^2 - z^3\) | \(0.234921\) | \(\infty\) |
2-torsion field: 5.1.1000000.2
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(2\) |
Mordell-Weil rank: | \(2\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 0.048361 \) |
Real period: | \( 13.02621 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 0.629964 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(5\) | \(6\) | \(6\) | \(1\) | \(1\) |
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? |
---|---|---|
\(2\) | 2.12.2 | no |
\(3\) | 3.2592.2 | no |
Sato-Tate group
\(\mathrm{ST}\) | \(\simeq\) | $\mathrm{SU}(2)\times\mathrm{SU}(2)$ |
\(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\times\mathrm{SU}(2)\) |
Decomposition of the Jacobian
Simple over \(\overline{\Q}\)
Endomorphisms of the Jacobian
Of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
\(\End (J_{})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{5}}{2}]\) |
\(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{5}) \) |
\(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).