Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x)y = x^6 + 28x^4 + 216x^2 + 546$ | (homogenize, simplify) |
$y^2 + (x^3 + xz^2)y = x^6 + 28x^4z^2 + 216x^2z^4 + 546z^6$ | (dehomogenize, simplify) |
$y^2 = 5x^6 + 114x^4 + 865x^2 + 2184$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(21840\) | \(=\) | \( 2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(-43680\) | \(=\) | \( - 2^{5} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(1049640\) | \(=\) | \( 2^{3} \cdot 3 \cdot 5 \cdot 8747 \) |
\( I_4 \) | \(=\) | \(108888\) | \(=\) | \( 2^{3} \cdot 3 \cdot 13 \cdot 349 \) |
\( I_6 \) | \(=\) | \(38088901380\) | \(=\) | \( 2^{2} \cdot 3 \cdot 5 \cdot 19 \cdot 151 \cdot 173 \cdot 1279 \) |
\( I_{10} \) | \(=\) | \(174720\) | \(=\) | \( 2^{7} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \) |
\( J_2 \) | \(=\) | \(524820\) | \(=\) | \( 2^{2} \cdot 3 \cdot 5 \cdot 8747 \) |
\( J_4 \) | \(=\) | \(11476483202\) | \(=\) | \( 2 \cdot 131 \cdot 1483 \cdot 29537 \) |
\( J_6 \) | \(=\) | \(334614937937280\) | \(=\) | \( 2^{7} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot 281 \cdot 619589 \) |
\( J_8 \) | \(=\) | \(10975736260613779199\) | \(=\) | \( 317 \cdot 1621 \cdot 21359514924607 \) |
\( J_{10} \) | \(=\) | \(43680\) | \(=\) | \( 2^{5} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \) |
\( g_1 \) | \(=\) | \(82948903061048552981340000/91\) | ||
\( g_2 \) | \(=\) | \(3456198828851740672970700/91\) | ||
\( g_3 \) | \(=\) | \(2110004828004044030400\) |
Automorphism group
\(\mathrm{Aut}(X)\) | \(\simeq\) | $C_2^2$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
|
\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $C_2^2$ | 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: \(\Z/{2}\Z \oplus \Z/{2}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(5x^2 + 39z^2\) | \(=\) | \(0,\) | \(5y\) | \(=\) | \(17xz^2\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(x^2 + 7z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(3xz^2\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(5x^2 + 39z^2\) | \(=\) | \(0,\) | \(5y\) | \(=\) | \(17xz^2\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(x^2 + 7z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(3xz^2\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(5x^2 + 39z^2\) | \(=\) | \(0,\) | \(5y\) | \(=\) | \(x^3 + 35xz^2\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(x^2 + 7z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + 7xz^2\) | \(0\) | \(2\) |
2-torsion field: 8.0.14219703912960000.490
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(4\) |
Regulator: | \( 1 \) |
Real period: | \( 4.386039 \) |
Tamagawa product: | \( 2 \) |
Torsion order: | \( 4 \) |
Leading coefficient: | \( 2.193019 \) |
Analytic order of Ш: | \( 4 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(4\) | \(5\) | \(2\) | \(1 - T\) | |
\(3\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 3 T^{2} )\) | |
\(5\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - 2 T + 5 T^{2} )\) | |
\(7\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 4 T + 7 T^{2} )\) | |
\(13\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 2 T + 13 T^{2} )\) |
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? |
---|---|---|
\(2\) | 2.90.6 | 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 40.a
Elliptic curve isogeny class 546.g
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 \(\Q\).