Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + 1)y = -5x^4 + 10x^3 - 5x^2$ | (homogenize, simplify) |
$y^2 + (x^3 + z^3)y = -5x^4z^2 + 10x^3z^3 - 5x^2z^4$ | (dehomogenize, simplify) |
$y^2 = x^6 - 20x^4 + 42x^3 - 20x^2 + 1$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(8450\) | \(=\) | \( 2 \cdot 5^{2} \cdot 13^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(84500\) | \(=\) | \( 2^{2} \cdot 5^{3} \cdot 13^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(1972\) | \(=\) | \( 2^{2} \cdot 17 \cdot 29 \) |
\( I_4 \) | \(=\) | \(60889\) | \(=\) | \( 60889 \) |
\( I_6 \) | \(=\) | \(35769757\) | \(=\) | \( 35769757 \) |
\( I_{10} \) | \(=\) | \(10816000\) | \(=\) | \( 2^{9} \cdot 5^{3} \cdot 13^{2} \) |
\( J_2 \) | \(=\) | \(493\) | \(=\) | \( 17 \cdot 29 \) |
\( J_4 \) | \(=\) | \(7590\) | \(=\) | \( 2 \cdot 3 \cdot 5 \cdot 11 \cdot 23 \) |
\( J_6 \) | \(=\) | \(128000\) | \(=\) | \( 2^{10} \cdot 5^{3} \) |
\( J_8 \) | \(=\) | \(1373975\) | \(=\) | \( 5^{2} \cdot 54959 \) |
\( J_{10} \) | \(=\) | \(84500\) | \(=\) | \( 2^{2} \cdot 5^{3} \cdot 13^{2} \) |
\( g_1 \) | \(=\) | \(29122898485693/84500\) | ||
\( g_2 \) | \(=\) | \(90945776163/8450\) | ||
\( g_3 \) | \(=\) | \(62220544/169\) |
Automorphism group
\(\mathrm{Aut}(X)\) | \(\simeq\) | $C_2^2$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
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\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $C_2^2$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
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Rational points
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((1 : 0 : 1)\) | \((1 : -2 : 1)\) |
\((1 : -4 : 2)\) | \((2 : -4 : 1)\) | \((1 : -5 : 2)\) | \((2 : -5 : 1)\) | \((1 : -10 : 3)\) | \((3 : -10 : 1)\) |
\((1 : -18 : 3)\) | \((3 : -18 : 1)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((1 : 0 : 1)\) | \((1 : -2 : 1)\) |
\((1 : -4 : 2)\) | \((2 : -4 : 1)\) | \((1 : -5 : 2)\) | \((2 : -5 : 1)\) | \((1 : -10 : 3)\) | \((3 : -10 : 1)\) |
\((1 : -18 : 3)\) | \((3 : -18 : 1)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((1 : -1 : 2)\) | \((1 : 1 : 2)\) |
\((1 : -2 : 1)\) | \((1 : 2 : 1)\) | \((2 : -1 : 1)\) | \((2 : 1 : 1)\) | \((1 : -8 : 3)\) | \((1 : 8 : 3)\) |
\((3 : -8 : 1)\) | \((3 : 8 : 1)\) |
Number of rational Weierstrass points: \(0\)
This curve is locally solvable everywhere.
Mordell-Weil group of the Jacobian
Group structure: \(\Z \oplus \Z \oplus \Z/{6}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((2 : -5 : 1) - (1 : -1 : 0)\) | \(z (x - 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-5z^3\) | \(0.585232\) | \(\infty\) |
\((1 : -2 : 1) - (1 : -1 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2z^3\) | \(0.187757\) | \(\infty\) |
\((0 : 0 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(6\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((2 : -5 : 1) - (1 : -1 : 0)\) | \(z (x - 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-5z^3\) | \(0.585232\) | \(\infty\) |
\((1 : -2 : 1) - (1 : -1 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2z^3\) | \(0.187757\) | \(\infty\) |
\((0 : 0 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(6\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((2 : -1 : 1) - (1 : -1 : 0)\) | \(z (x - 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - 9z^3\) | \(0.585232\) | \(\infty\) |
\((1 : -2 : 1) - (1 : -1 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - 3z^3\) | \(0.187757\) | \(\infty\) |
\((0 : 1 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + z^3\) | \(0\) | \(6\) |
2-torsion field: \(\Q(\sqrt{5}, \sqrt{13})\)
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(2\) |
Mordell-Weil rank: | \(2\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 0.109881 \) |
Real period: | \( 20.36995 \) |
Tamagawa product: | \( 6 \) |
Torsion order: | \( 6 \) |
Leading coefficient: | \( 0.373046 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(1\) | \(2\) | \(2\) | \(( 1 + T )( 1 + T + 2 T^{2} )\) | |
\(5\) | \(2\) | \(3\) | \(3\) | \(( 1 - T )( 1 + T )\) | |
\(13\) | \(2\) | \(2\) | \(1\) | \(( 1 - T )( 1 + T )\) |
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? |
---|---|---|
\(2\) | 2.180.4 | yes |
\(3\) | 3.720.4 | yes |
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
Splits over \(\Q\)
Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
Elliptic curve isogeny class 65.a
Elliptic curve isogeny class 130.a
Endomorphisms of the Jacobian
Of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
\(\End (J_{})\) | \(\simeq\) | an order of index \(2\) in \(\Z \times \Z\) |
\(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q\) |
\(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).