Minimal equation
Minimal equation
Simplified equation
$y^2 + x^2y = -7x^6 + 108x^4 - 446x^2 + 209$ | (homogenize, simplify) |
$y^2 + x^2zy = -7x^6 + 108x^4z^2 - 446x^2z^4 + 209z^6$ | (dehomogenize, simplify) |
$y^2 = -28x^6 + 433x^4 - 1784x^2 + 836$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(421344\) | \(=\) | \( 2^{5} \cdot 3^{2} \cdot 7 \cdot 11 \cdot 19 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(842688\) | \(=\) | \( 2^{6} \cdot 3^{2} \cdot 7 \cdot 11 \cdot 19 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(2247184\) | \(=\) | \( 2^{4} \cdot 140449 \) |
\( I_4 \) | \(=\) | \(78904809991\) | \(=\) | \( 7 \cdot 40129 \cdot 280897 \) |
\( I_6 \) | \(=\) | \(51716421805730562\) | \(=\) | \( 2 \cdot 3^{2} \cdot 29 \cdot 99073604991821 \) |
\( I_{10} \) | \(=\) | \(105336\) | \(=\) | \( 2^{3} \cdot 3^{2} \cdot 7 \cdot 11 \cdot 19 \) |
\( J_2 \) | \(=\) | \(2247184\) | \(=\) | \( 2^{4} \cdot 140449 \) |
\( J_4 \) | \(=\) | \(157806623750\) | \(=\) | \( 2 \cdot 5^{4} \cdot 31 \cdot 179 \cdot 22751 \) |
\( J_6 \) | \(=\) | \(13134015401810688\) | \(=\) | \( 2^{8} \cdot 3^{2} \cdot 7 \cdot 11 \cdot 19 \cdot 401 \cdot 467 \cdot 20807 \) |
\( J_8 \) | \(=\) | \(1152904691832121260023\) | \(=\) | \( 38014807 \cdot 30327779694689 \) |
\( J_{10} \) | \(=\) | \(842688\) | \(=\) | \( 2^{6} \cdot 3^{2} \cdot 7 \cdot 11 \cdot 19 \) |
\( g_1 \) | \(=\) | \(895391971763859078040408047616/13167\) | ||
\( g_2 \) | \(=\) | \(27980866301321432857668080000/13167\) | ||
\( g_3 \) | \(=\) | \(78706025099853921145856\) |
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
Number of rational Weierstrass points: \(0\)
This curve is locally solvable except over $\Q_{3}$.
Mordell-Weil group of the Jacobian
Group structure: \(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(9533214251x^2 - 5011014400z^2\) | \(=\) | \(0,\) | \(\)\(15\!\cdots\!40\)\(y\) | \(=\) | \(\)\(39\!\cdots\!19\)\(xz^2 - \)\(40\!\cdots\!00\)\(z^3\) | \(48.20794\) | \(\infty\) |
\(D_0 - D_\infty\) | \(28x^2 - 209z^2\) | \(=\) | \(0,\) | \(56y\) | \(=\) | \(-209z^3\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(x^2 - 2xz - 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2 - z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(9533214251x^2 - 5011014400z^2\) | \(=\) | \(0,\) | \(\)\(15\!\cdots\!40\)\(y\) | \(=\) | \(\)\(39\!\cdots\!19\)\(xz^2 - \)\(40\!\cdots\!00\)\(z^3\) | \(48.20794\) | \(\infty\) |
\(D_0 - D_\infty\) | \(28x^2 - 209z^2\) | \(=\) | \(0,\) | \(56y\) | \(=\) | \(-209z^3\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(x^2 - 2xz - 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2 - z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(9533214251x^2 - 5011014400z^2\) | \(=\) | \(0,\) | \(\)\(15\!\cdots\!40\)\(y\) | \(=\) | \(x^2z + \)\(79\!\cdots\!38\)\(xz^2 - \)\(81\!\cdots\!00\)\(z^3\) | \(48.20794\) | \(\infty\) |
\(D_0 - D_\infty\) | \(28x^2 - 209z^2\) | \(=\) | \(0,\) | \(56y\) | \(=\) | \(x^2z - 418z^3\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(x^2 - 2xz - 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^2z - 2xz^2 - 2z^3\) | \(0\) | \(2\) |
2-torsion field: \(\Q(\sqrt{3}, \sqrt{1463})\)
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(1\) |
Mordell-Weil rank: | \(1\) |
2-Selmer rank: | \(5\) |
Regulator: | \( 48.20794 \) |
Real period: | \( 0.148975 \) |
Tamagawa product: | \( 2 \) |
Torsion order: | \( 4 \) |
Leading coefficient: | \( 3.590893 \) |
Analytic order of Ш: | \( 4 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(5\) | \(6\) | \(2\) | \(1 + T + 2 T^{2}\) | |
\(3\) | \(2\) | \(2\) | \(1\) | \(( 1 - T )^{2}\) | |
\(7\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 4 T + 7 T^{2} )\) | |
\(11\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - 4 T + 11 T^{2} )\) | |
\(19\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 4 T + 19 T^{2} )\) |
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? |
---|---|---|
\(2\) | 2.180.7 | yes |
\(3\) | 3.90.1 | 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
Splits over \(\Q\)
Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
Elliptic curve isogeny class 20064.s
Elliptic curve isogeny class 21.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 \(\Q\).