Minimal equation
Minimal equation
Simplified equation
$y^2 + (x + 1)y = -x^6 + 30x^5 - 278x^4 + 805x^3 - 228x^2 + 22x - 1$ | (homogenize, simplify) |
$y^2 + (xz^2 + z^3)y = -x^6 + 30x^5z - 278x^4z^2 + 805x^3z^3 - 228x^2z^4 + 22xz^5 - z^6$ | (dehomogenize, simplify) |
$y^2 = -4x^6 + 120x^5 - 1112x^4 + 3220x^3 - 911x^2 + 90x - 3$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(483552\) | \(=\) | \( 2^{5} \cdot 3^{2} \cdot 23 \cdot 73 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(483552\) | \(=\) | \( 2^{5} \cdot 3^{2} \cdot 23 \cdot 73 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(11607752\) | \(=\) | \( 2^{3} \cdot 13 \cdot 239 \cdot 467 \) |
\( I_4 \) | \(=\) | \(391864\) | \(=\) | \( 2^{3} \cdot 11 \cdot 61 \cdot 73 \) |
\( I_6 \) | \(=\) | \(1516173091284\) | \(=\) | \( 2^{2} \cdot 3 \cdot 173^{2} \cdot 4221583 \) |
\( I_{10} \) | \(=\) | \(1934208\) | \(=\) | \( 2^{7} \cdot 3^{2} \cdot 23 \cdot 73 \) |
\( J_2 \) | \(=\) | \(5803876\) | \(=\) | \( 2^{2} \cdot 13 \cdot 239 \cdot 467 \) |
\( J_4 \) | \(=\) | \(1403540627330\) | \(=\) | \( 2 \cdot 5 \cdot 195739 \cdot 717047 \) |
\( J_6 \) | \(=\) | \(452554166887189632\) | \(=\) | \( 2^{7} \cdot 3^{2} \cdot 23 \cdot 73 \cdot 233973888479 \) |
\( J_8 \) | \(=\) | \(164160493832666167421183\) | \(=\) | \( 1854576793 \cdot 88516417574231 \) |
\( J_{10} \) | \(=\) | \(483552\) | \(=\) | \( 2^{5} \cdot 3^{2} \cdot 23 \cdot 73 \) |
\( g_1 \) | \(=\) | \(205797761914146968633542599227168/15111\) | ||
\( g_2 \) | \(=\) | \(8574906347408441043142078203940/15111\) | ||
\( g_3 \) | \(=\) | \(31525619855581992833940416\) |
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 $\Q_{2}$ and $\Q_{3}$.
Mordell-Weil group of the Jacobian
Group structure: \(\Z/{2}\Z \oplus \Z/{2}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(2x^2 - 30xz + 3z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2 - z^3\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(x^2 - 10xz + z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2 - z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(2x^2 - 30xz + 3z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2 - z^3\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(x^2 - 10xz + z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2 - z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(2x^2 - 30xz + 3z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2 - z^3\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(x^2 - 10xz + z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2 - z^3\) | \(0\) | \(2\) |
2-torsion field: 8.8.42185940332325175296.1
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(4\) |
Regulator: | \( 1 \) |
Real period: | \( 4.625149 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 4 \) |
Leading coefficient: | \( 4.625149 \) |
Analytic order of Ш: | \( 16 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(5\) | \(5\) | \(1\) | \(1 - T\) | |
\(3\) | \(2\) | \(2\) | \(1\) | \(( 1 - T )^{2}\) | |
\(23\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 8 T + 23 T^{2} )\) | |
\(73\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - 10 T + 73 T^{2} )\) |
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 10074.r
Elliptic curve isogeny class 48.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 \(\Q\).