Minimal equation
Minimal equation
Simplified equation
$y^2 = x^6 - 2x^5 - 3x^4 + 5x^3 + 3x^2 - 2x - 1$ | (homogenize, simplify) |
$y^2 = x^6 - 2x^5z - 3x^4z^2 + 5x^3z^3 + 3x^2z^4 - 2xz^5 - z^6$ | (dehomogenize, simplify) |
$y^2 = x^6 - 2x^5 - 3x^4 + 5x^3 + 3x^2 - 2x - 1$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(46400\) | \(=\) | \( 2^{6} \cdot 5^{2} \cdot 29 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(928000\) | \(=\) | \( 2^{8} \cdot 5^{3} \cdot 29 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(694\) | \(=\) | \( 2 \cdot 347 \) |
\( I_4 \) | \(=\) | \(1996\) | \(=\) | \( 2^{2} \cdot 499 \) |
\( I_6 \) | \(=\) | \(478406\) | \(=\) | \( 2 \cdot 251 \cdot 953 \) |
\( I_{10} \) | \(=\) | \(3625\) | \(=\) | \( 5^{3} \cdot 29 \) |
\( J_2 \) | \(=\) | \(1388\) | \(=\) | \( 2^{2} \cdot 347 \) |
\( J_4 \) | \(=\) | \(74950\) | \(=\) | \( 2 \cdot 5^{2} \cdot 1499 \) |
\( J_6 \) | \(=\) | \(4840100\) | \(=\) | \( 2^{2} \cdot 5^{2} \cdot 29 \cdot 1669 \) |
\( J_8 \) | \(=\) | \(275139075\) | \(=\) | \( 3 \cdot 5^{2} \cdot 541 \cdot 6781 \) |
\( J_{10} \) | \(=\) | \(928000\) | \(=\) | \( 2^{8} \cdot 5^{3} \cdot 29 \) |
\( g_1 \) | \(=\) | \(20123678266028/3625\) | ||
\( g_2 \) | \(=\) | \(62631102577/290\) | ||
\( g_3 \) | \(=\) | \(200962621/20\) |
Automorphism group
\(\mathrm{Aut}(X)\) | \(\simeq\) | $C_2$ | 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: \(\Z \oplus \Z/{6}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((1 : -1 : 1) - (1 : -1 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - 2z^3\) | \(0.690071\) | \(\infty\) |
\((-1 : -1 : 2) + (1 : 1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \((x - z) (2x + z)\) | \(=\) | \(0,\) | \(4y\) | \(=\) | \(3xz^2 + z^3\) | \(0\) | \(6\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((1 : -1 : 1) - (1 : -1 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - 2z^3\) | \(0.690071\) | \(\infty\) |
\((-1 : -1 : 2) + (1 : 1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \((x - z) (2x + z)\) | \(=\) | \(0,\) | \(4y\) | \(=\) | \(3xz^2 + z^3\) | \(0\) | \(6\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((1 : -1/2 : 1) - (1 : -1/2 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(1/2x^3 - z^3\) | \(0.690071\) | \(\infty\) |
\((-1 : -1/2 : 2) + (1 : 1/2 : 1) - (1 : -1/2 : 0) - (1 : 1/2 : 0)\) | \((x - z) (2x + z)\) | \(=\) | \(0,\) | \(4y\) | \(=\) | \(3/2xz^2 + 1/2z^3\) | \(0\) | \(6\) |
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(1\) |
Mordell-Weil rank: | \(1\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 0.690071 \) |
Real period: | \( 11.11146 \) |
Tamagawa product: | \( 9 \) |
Torsion order: | \( 6 \) |
Leading coefficient: | \( 1.916926 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(6\) | \(8\) | \(3\) | \(1\) | |
\(5\) | \(2\) | \(3\) | \(3\) | \(( 1 - T )( 1 + T )\) | |
\(29\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 6 T + 29 T^{2} )\) |
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? |
---|---|---|
\(2\) | 2.90.1 | yes |
\(3\) | 3.2880.5 | yes |
Sato-Tate group
\(\mathrm{ST}\) | \(\simeq\) | $N(\mathrm{SU}(2)\times\mathrm{SU}(2))$ |
\(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\times\mathrm{SU}(2)\) |
Decomposition of the Jacobian
Splits over the number field \(\Q (b) \simeq \) \(\Q(\sqrt{-1}) \) with defining polynomial:
\(x^{2} + 1\)
Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
Elliptic curve isogeny class 2.0.4.1-2900.4-a
Elliptic curve isogeny class 2.0.4.1-2900.3-a
Endomorphisms of the Jacobian
Not of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
\(\End (J_{})\) | \(\simeq\) | \(\Z\) |
\(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
\(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\R\) |
Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\sqrt{-1}) \) with defining polynomial \(x^{2} + 1\)
Of \(\GL_2\)-type over \(\overline{\Q}\)
Endomorphism ring over \(\overline{\Q}\):
\(\End (J_{\overline{\Q}})\) | \(\simeq\) | an order of index \(2\) in \(\Z \times \Z\) |
\(\End (J_{\overline{\Q}}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q\) |
\(\End (J_{\overline{\Q}}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |