Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + 1)y = -3x^6 + 5x^5 - 11x^4 + 10x^3 - 11x^2 + 5x - 3$ | (homogenize, simplify) |
$y^2 + (x^3 + z^3)y = -3x^6 + 5x^5z - 11x^4z^2 + 10x^3z^3 - 11x^2z^4 + 5xz^5 - 3z^6$ | (dehomogenize, simplify) |
$y^2 = -11x^6 + 20x^5 - 44x^4 + 42x^3 - 44x^2 + 20x - 11$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(2058\) | \(=\) | \( 2 \cdot 3 \cdot 7^{3} \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-16464\) | \(=\) | \( - 2^{4} \cdot 3 \cdot 7^{3} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(16716\) | \(=\) | \( 2^{2} \cdot 3 \cdot 7 \cdot 199 \) |
\( I_4 \) | \(=\) | \(21945\) | \(=\) | \( 3 \cdot 5 \cdot 7 \cdot 11 \cdot 19 \) |
\( I_6 \) | \(=\) | \(119839251\) | \(=\) | \( 3 \cdot 7^{2} \cdot 19 \cdot 107 \cdot 401 \) |
\( I_{10} \) | \(=\) | \(2107392\) | \(=\) | \( 2^{11} \cdot 3 \cdot 7^{3} \) |
\( J_2 \) | \(=\) | \(4179\) | \(=\) | \( 3 \cdot 7 \cdot 199 \) |
\( J_4 \) | \(=\) | \(726754\) | \(=\) | \( 2 \cdot 7 \cdot 23 \cdot 37 \cdot 61 \) |
\( J_6 \) | \(=\) | \(168337344\) | \(=\) | \( 2^{6} \cdot 3 \cdot 7^{2} \cdot 29 \cdot 617 \) |
\( J_8 \) | \(=\) | \(43827596015\) | \(=\) | \( 5 \cdot 7^{2} \cdot 223 \cdot 802189 \) |
\( J_{10} \) | \(=\) | \(16464\) | \(=\) | \( 2^{4} \cdot 3 \cdot 7^{3} \) |
\( g_1 \) | \(=\) | \(1238643936365031/16\) | ||
\( g_2 \) | \(=\) | \(25772655805407/8\) | ||
\( g_3 \) | \(=\) | \(178562334636\) |
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 $\R$.
Mordell-Weil group of the Jacobian
Group structure: \(\Z/{8}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2 - z^3\) | \(0\) | \(8\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2 - z^3\) | \(0\) | \(8\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + 2xz^2 - z^3\) | \(0\) | \(8\) |
2-torsion field: 8.0.152473104.1
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 1 \) |
Real period: | \( 6.719123 \) |
Tamagawa product: | \( 4 \) |
Torsion order: | \( 8 \) |
Leading coefficient: | \( 0.839890 \) |
Analytic order of Ш: | \( 2 \) (rounded) |
Order of Ш: | twice a square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(1\) | \(4\) | \(4\) | \(( 1 - T )( 1 - T + 2 T^{2} )\) | |
\(3\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 3 T^{2} )\) | |
\(7\) | \(3\) | \(3\) | \(1\) | \(1 + T\) |
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 |
\(3\) | 3.270.2 | 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 49.a
Elliptic curve isogeny class 42.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 \(K = \Q (a) \simeq \) \(\Q(\sqrt{-7}) \) with defining polynomial \(x^{2} - x + 2\)
Not of \(\GL_2\)-type over \(\overline{\Q}\)
Endomorphism ring over \(\overline{\Q}\):
\(\End (J_{\overline{\Q}})\) | \(\simeq\) | an order of index \(4\) in \(\Z \times \Z [\frac{1 + \sqrt{-7}}{2}]\) |
\(\End (J_{\overline{\Q}}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q(\sqrt{-7}) \) |
\(\End (J_{\overline{\Q}}) \otimes \R\) | \(\simeq\) | \(\R \times \C\) |