Minimal equation
Minimal equation
Simplified equation
$y^2 + y = 4x^6 - 12x^5 + 3x^4 + 14x^3 - 5x^2 - 4x + 1$ | (homogenize, simplify) |
$y^2 + z^3y = 4x^6 - 12x^5z + 3x^4z^2 + 14x^3z^3 - 5x^2z^4 - 4xz^5 + z^6$ | (dehomogenize, simplify) |
$y^2 = 16x^6 - 48x^5 + 12x^4 + 56x^3 - 20x^2 - 16x + 5$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(1536\) | \(=\) | \( 2^{9} \cdot 3 \) | magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(1536,2),R![1]>*])); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(98304\) | \(=\) | \( 2^{15} \cdot 3 \) | magma: Discriminant(C); Factorization(Integers()!$1);
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Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(1068\) | \(=\) | \( 2^{2} \cdot 3 \cdot 89 \) |
\( I_4 \) | \(=\) | \(38019\) | \(=\) | \( 3 \cdot 19 \cdot 23 \cdot 29 \) |
\( I_6 \) | \(=\) | \(11064156\) | \(=\) | \( 2^{2} \cdot 3 \cdot 19 \cdot 48527 \) |
\( I_{10} \) | \(=\) | \(12\) | \(=\) | \( 2^{2} \cdot 3 \) |
\( J_2 \) | \(=\) | \(4272\) | \(=\) | \( 2^{4} \cdot 3 \cdot 89 \) |
\( J_4 \) | \(=\) | \(354880\) | \(=\) | \( 2^{6} \cdot 5 \cdot 1109 \) |
\( J_6 \) | \(=\) | \(32280576\) | \(=\) | \( 2^{12} \cdot 3 \cdot 37 \cdot 71 \) |
\( J_8 \) | \(=\) | \(2990701568\) | \(=\) | \( 2^{10} \cdot 1091 \cdot 2677 \) |
\( J_{10} \) | \(=\) | \(98304\) | \(=\) | \( 2^{15} \cdot 3 \) |
\( g_1 \) | \(=\) | \(14473882091808\) | ||
\( g_2 \) | \(=\) | \(281451823560\) | ||
\( g_3 \) | \(=\) | \(5992838496\) |
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 everywhere.
Mordell-Weil group of the Jacobian
Group structure: \(\Z/{2}\Z \oplus \Z/{4}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - (1 : -2 : 0) - (1 : 2 : 0)\) | \(2x^2 - z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-z^3\) | \(0\) | \(2\) |
\(D_0 - (1 : -2 : 0) - (1 : 2 : 0)\) | \(x^2 - xz - z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0\) | \(4\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - (1 : -2 : 0) - (1 : 2 : 0)\) | \(2x^2 - z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-z^3\) | \(0\) | \(2\) |
\(D_0 - (1 : -2 : 0) - (1 : 2 : 0)\) | \(x^2 - xz - z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0\) | \(4\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - (1 : -4 : 0) - (1 : 4 : 0)\) | \(2x^2 - z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-z^3\) | \(0\) | \(2\) |
\(D_0 - (1 : -4 : 0) - (1 : 4 : 0)\) | \(x^2 - xz - z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0\) | \(4\) |
2-torsion field: \(\Q(\sqrt{2}, \sqrt{3})\)
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 1 \) |
Real period: | \( 17.68053 \) |
Tamagawa product: | \( 2 \) |
Torsion order: | \( 8 \) |
Leading coefficient: | \( 0.552516 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(9\) | \(15\) | \(2\) | \(1\) | |
\(3\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 3 T^{2} )\) |
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 |
\(3\) | 3.270.2 | no |
Sato-Tate group
\(\mathrm{ST}\) | \(\simeq\) | $N(\mathrm{U}(1)\times\mathrm{SU}(2))$ |
\(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{U}(1)\times\mathrm{SU}(2)\) |
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 48.a
Elliptic curve isogeny class 32.a
Endomorphisms of the Jacobian
Of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
\(\End (J_{})\) | \(\simeq\) | an order of index \(2\) in \(\Z \times \Z\) |
\(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q\) |
\(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\sqrt{-1}) \) with defining polynomial \(x^{2} + 1\)
Not of \(\GL_2\)-type over \(\overline{\Q}\)
Endomorphism ring over \(\overline{\Q}\):
\(\End (J_{\overline{\Q}})\) | \(\simeq\) | an order of index \(8\) in \(\Z \times \Z [\sqrt{-1}]\) |
\(\End (J_{\overline{\Q}}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q(\sqrt{-1}) \) |
\(\End (J_{\overline{\Q}}) \otimes \R\) | \(\simeq\) | \(\R \times \C\) |