Minimal equation
Minimal equation
Simplified equation
$y^2 = x^5 - x^4 + x^3 + x^2 - 2x + 1$ | (homogenize, simplify) |
$y^2 = x^5z - x^4z^2 + x^3z^3 + x^2z^4 - 2xz^5 + z^6$ | (dehomogenize, simplify) |
$y^2 = x^5 - x^4 + x^3 + x^2 - 2x + 1$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(35344\) | \(=\) | \( 2^{4} \cdot 47^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(565504\) | \(=\) | \( 2^{8} \cdot 47^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(58\) | \(=\) | \( 2 \cdot 29 \) |
\( I_4 \) | \(=\) | \(64\) | \(=\) | \( 2^{6} \) |
\( I_6 \) | \(=\) | \(562\) | \(=\) | \( 2 \cdot 281 \) |
\( I_{10} \) | \(=\) | \(-2209\) | \(=\) | \( - 47^{2} \) |
\( J_2 \) | \(=\) | \(116\) | \(=\) | \( 2^{2} \cdot 29 \) |
\( J_4 \) | \(=\) | \(390\) | \(=\) | \( 2 \cdot 3 \cdot 5 \cdot 13 \) |
\( J_6 \) | \(=\) | \(5116\) | \(=\) | \( 2^{2} \cdot 1279 \) |
\( J_8 \) | \(=\) | \(110339\) | \(=\) | \( 110339 \) |
\( J_{10} \) | \(=\) | \(-565504\) | \(=\) | \( - 2^{8} \cdot 47^{2} \) |
\( g_1 \) | \(=\) | \(-82044596/2209\) | ||
\( g_2 \) | \(=\) | \(-4755855/4418\) | ||
\( g_3 \) | \(=\) | \(-1075639/8836\) |
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
All points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((-1 : -1 : 1)\) | \((-1 : 1 : 1)\) | \((1 : -1 : 1)\) |
\((1 : 1 : 1)\) | \((2 : -5 : 1)\) | \((2 : 5 : 1)\) | \((4 : -29 : 1)\) | \((4 : 29 : 1)\) |
All points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((-1 : -1 : 1)\) | \((-1 : 1 : 1)\) | \((1 : -1 : 1)\) |
\((1 : 1 : 1)\) | \((2 : -5 : 1)\) | \((2 : 5 : 1)\) | \((4 : -29 : 1)\) | \((4 : 29 : 1)\) |
All points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((0 : -1/2 : 1)\) | \((0 : 1/2 : 1)\) | \((-1 : -1/2 : 1)\) | \((-1 : 1/2 : 1)\) | \((1 : -1/2 : 1)\) |
\((1 : 1/2 : 1)\) | \((2 : -5/2 : 1)\) | \((2 : 5/2 : 1)\) | \((4 : -29/2 : 1)\) | \((4 : 29/2 : 1)\) |
Number of rational Weierstrass points: \(1\)
This curve is locally solvable everywhere.
Mordell-Weil group of the Jacobian
Group structure: \(\Z \oplus \Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : 1 : 1) + (1 : 1 : 1) - 2 \cdot(1 : 0 : 0)\) | \(x (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(z^3\) | \(0.127053\) | \(\infty\) |
\((0 : -1 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.096317\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : 1 : 1) + (1 : 1 : 1) - 2 \cdot(1 : 0 : 0)\) | \(x (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(z^3\) | \(0.127053\) | \(\infty\) |
\((0 : -1 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.096317\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : 1/2 : 1) + (1 : 1/2 : 1) - 2 \cdot(1 : 0 : 0)\) | \(x (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(1/2z^3\) | \(0.127053\) | \(\infty\) |
\((0 : -1/2 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-1/2z^3\) | \(0.096317\) | \(\infty\) |
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(2\) |
Mordell-Weil rank: | \(2\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 0.011292 \) |
Real period: | \( 11.51929 \) |
Tamagawa product: | \( 9 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 1.170768 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(4\) | \(8\) | \(9\) | \(1\) | |
\(47\) | \(2\) | \(2\) | \(1\) | \(( 1 - T )^{2}\) |
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? |
---|---|---|
\(2\) | 2.72.1 | 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 Jacobian of this curve is an optimal quotient of $J_0(188)$. 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.