Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^2 + 1)y = 7x^6 + 84x^4 + 336x^2 + 446$ | (homogenize, simplify) |
$y^2 + (x^2z + z^3)y = 7x^6 + 84x^4z^2 + 336x^2z^4 + 446z^6$ | (dehomogenize, simplify) |
$y^2 = 28x^6 + 337x^4 + 1346x^2 + 1785$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(799680\) | \(=\) | \( 2^{6} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 17 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-799680\) | \(=\) | \( - 2^{6} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 17 \) | magma: Discriminant(C); Factorization(Integers()!$1);
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Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(2406604\) | \(=\) | \( 2^{2} \cdot 601651 \) |
\( I_4 \) | \(=\) | \(3575875\) | \(=\) | \( 5^{3} \cdot 28607 \) |
\( I_6 \) | \(=\) | \(2866321300076\) | \(=\) | \( 2^{2} \cdot 13 \cdot 163 \cdot 338169101 \) |
\( I_{10} \) | \(=\) | \(99960\) | \(=\) | \( 2^{3} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 17 \) |
\( J_2 \) | \(=\) | \(2406604\) | \(=\) | \( 2^{2} \cdot 601651 \) |
\( J_4 \) | \(=\) | \(241320233284\) | \(=\) | \( 2^{2} \cdot 13 \cdot 4640773717 \) |
\( J_6 \) | \(=\) | \(32263933356762240\) | \(=\) | \( 2^{7} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 17 \cdot 89 \cdot 151 \cdot 1501081 \) |
\( J_8 \) | \(=\) | \(4852764019968313102076\) | \(=\) | \( 2^{2} \cdot 41 \cdot 47 \cdot 127 \cdot 632911 \cdot 7832512601 \) |
\( J_{10} \) | \(=\) | \(799680\) | \(=\) | \( 2^{6} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 17 \) |
\( g_1 \) | \(=\) | \(1261372031256529020641523732016/12495\) | ||
\( g_2 \) | \(=\) | \(52556648780600635811581759084/12495\) | ||
\( g_3 \) | \(=\) | \(233673974755154692792288\) |
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: \(\Z \oplus \Z/{6}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 + 5z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2 + 2z^3\) | \(2.306124\) | \(\infty\) |
\(D_0 - D_\infty\) | \(x^2 + 4z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(2z^3\) | \(0\) | \(6\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 + 5z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2 + 2z^3\) | \(2.306124\) | \(\infty\) |
\(D_0 - D_\infty\) | \(x^2 + 4z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(2z^3\) | \(0\) | \(6\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 + 5z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^2z + 2xz^2 + 5z^3\) | \(2.306124\) | \(\infty\) |
\(D_0 - D_\infty\) | \(x^2 + 4z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^2z + 5z^3\) | \(0\) | \(6\) |
2-torsion field: 8.0.2302157168640000.159
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(1\) |
Mordell-Weil rank: | \(1\) |
2-Selmer rank: | \(6\) |
Regulator: | \( 2.306124 \) |
Real period: | \( 2.035804 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 6 \) |
Leading coefficient: | \( 2.086586 \) |
Analytic order of Ш: | \( 16 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(6\) | \(6\) | \(1\) | \(1 + T\) | |
\(3\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 2 T + 3 T^{2} )\) | |
\(5\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 5 T^{2} )\) | |
\(7\) | \(2\) | \(2\) | \(1\) | \(( 1 - T )^{2}\) | |
\(17\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 6 T + 17 T^{2} )\) |
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? |
---|---|---|
\(2\) | 2.45.1 | 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 57120.bd
Elliptic curve isogeny class 14.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 \(\Q\).