This is a model for the quotient of the modular curve $X_0(191)$ by its Fricke involution involution $w_{191}$; this quotient is also denoted $X_0^+(191)$. This is the largest level $N \in \mathbb{N}$ such that $X_0(N)/ \langle w_N \rangle$ is of genus $2$.
Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x + 1)y = x^2 - 2x$ | (homogenize, simplify) |
$y^2 + (x^3 + xz^2 + z^3)y = x^2z^4 - 2xz^5$ | (dehomogenize, simplify) |
$y^2 = x^6 + 2x^4 + 2x^3 + 5x^2 - 6x + 1$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(36481\) | \(=\) | \( 191^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(36481\) | \(=\) | \( 191^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(188\) | \(=\) | \( 2^{2} \cdot 47 \) |
\( I_4 \) | \(=\) | \(9409\) | \(=\) | \( 97^{2} \) |
\( I_6 \) | \(=\) | \(508799\) | \(=\) | \( 508799 \) |
\( I_{10} \) | \(=\) | \(-4669568\) | \(=\) | \( - 2^{7} \cdot 191^{2} \) |
\( J_2 \) | \(=\) | \(47\) | \(=\) | \( 47 \) |
\( J_4 \) | \(=\) | \(-300\) | \(=\) | \( - 2^{2} \cdot 3 \cdot 5^{2} \) |
\( J_6 \) | \(=\) | \(-1708\) | \(=\) | \( - 2^{2} \cdot 7 \cdot 61 \) |
\( J_8 \) | \(=\) | \(-42569\) | \(=\) | \( -42569 \) |
\( J_{10} \) | \(=\) | \(-36481\) | \(=\) | \( - 191^{2} \) |
\( g_1 \) | \(=\) | \(-229345007/36481\) | ||
\( g_2 \) | \(=\) | \(31146900/36481\) | ||
\( g_3 \) | \(=\) | \(3772972/36481\) |
Automorphism group
\(\mathrm{Aut}(X)\) | \(\simeq\) | $C_2$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
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\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $C_2$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
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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 | |||||
---|---|---|---|---|---|---|---|---|
\((0 : 0 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.309839\) | \(\infty\) |
\((0 : -1 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.210369\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : 0 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.309839\) | \(\infty\) |
\((0 : -1 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.210369\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : 1 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + xz^2 + z^3\) | \(0.309839\) | \(\infty\) |
\((0 : -1 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + xz^2 - z^3\) | \(0.210369\) | \(\infty\) |
2-torsion field: 5.1.2334784.2
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(2\) |
Mordell-Weil rank: | \(2\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 0.055286 \) |
Real period: | \( 17.35657 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 0.959581 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(191\) | \(2\) | \(2\) | \(1\) | \(( 1 + T )^{2}\) |
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? |
---|---|---|
\(2\) | 2.12.2 | no |
\(3\) | 3.432.4 | 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 \(\Q\).
Additional information
The set of rational points on this genus 2 curve with rank 2 Jacobian was computed by Balakrishnan--Dogra--Müller--Tuitman--Vonk using quadratic Chabauty and the Mordell--Weil sieve.