Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x^2 + x + 1)y = x^4 - x^3 + x^2 - x$ | (homogenize, simplify) |
$y^2 + (x^3 + x^2z + xz^2 + z^3)y = x^4z^2 - x^3z^3 + x^2z^4 - xz^5$ | (dehomogenize, simplify) |
$y^2 = x^6 + 2x^5 + 7x^4 + 7x^2 - 2x + 1$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(12544\) | \(=\) | \( 2^{8} \cdot 7^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(-25088\) | \(=\) | \( - 2^{9} \cdot 7^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(74\) | \(=\) | \( 2 \cdot 37 \) |
\( I_4 \) | \(=\) | \(142\) | \(=\) | \( 2 \cdot 71 \) |
\( I_6 \) | \(=\) | \(3272\) | \(=\) | \( 2^{3} \cdot 409 \) |
\( I_{10} \) | \(=\) | \(98\) | \(=\) | \( 2 \cdot 7^{2} \) |
\( J_2 \) | \(=\) | \(148\) | \(=\) | \( 2^{2} \cdot 37 \) |
\( J_4 \) | \(=\) | \(534\) | \(=\) | \( 2 \cdot 3 \cdot 89 \) |
\( J_6 \) | \(=\) | \(-196\) | \(=\) | \( - 2^{2} \cdot 7^{2} \) |
\( J_8 \) | \(=\) | \(-78541\) | \(=\) | \( -78541 \) |
\( J_{10} \) | \(=\) | \(25088\) | \(=\) | \( 2^{9} \cdot 7^{2} \) |
\( g_1 \) | \(=\) | \(138687914/49\) | ||
\( g_2 \) | \(=\) | \(13524351/196\) | ||
\( g_3 \) | \(=\) | \(-1369/8\) |
Automorphism group
\(\mathrm{Aut}(X)\) | \(\simeq\) | $C_4$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
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\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $D_4$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
|
Rational points
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((1 : 0 : 1)\) | \((-1 : -2 : 1)\) |
\((-1 : 2 : 1)\) | \((1 : -4 : 1)\) | \((4 : 5 : 3)\) | \((-3 : 80 : 4)\) | \((-3 : -105 : 4)\) | \((4 : -180 : 3)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((1 : 0 : 1)\) | \((-1 : -2 : 1)\) |
\((-1 : 2 : 1)\) | \((1 : -4 : 1)\) | \((4 : 5 : 3)\) | \((-3 : 80 : 4)\) | \((-3 : -105 : 4)\) | \((4 : -180 : 3)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((-1 : -4 : 1)\) | \((-1 : 4 : 1)\) |
\((1 : -4 : 1)\) | \((1 : 4 : 1)\) | \((-3 : -185 : 4)\) | \((-3 : 185 : 4)\) | \((4 : -185 : 3)\) | \((4 : 185 : 3)\) |
Number of rational Weierstrass points: \(0\)
This curve is locally solvable everywhere.
Mordell-Weil group of the Jacobian
Group structure: \(\Z \oplus \Z \oplus \Z/{2}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : 0 : 1) - (1 : 0 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3\) | \(0.240990\) | \(\infty\) |
\((1 : -4 : 1) - (1 : 0 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - 3z^3\) | \(0.240990\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2 + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : 0 : 1) - (1 : 0 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3\) | \(0.240990\) | \(\infty\) |
\((1 : -4 : 1) - (1 : 0 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - 3z^3\) | \(0.240990\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2 + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : 1 : 1) - (1 : 1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 + x^2z + xz^2 + z^3\) | \(0.240990\) | \(\infty\) |
\((1 : -4 : 1) - (1 : 1 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 + x^2z + xz^2 - 5z^3\) | \(0.240990\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2 + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + x^2z + xz^2 + z^3\) | \(0\) | \(2\) |
2-torsion field: 8.0.3211264.1
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(2\) |
Mordell-Weil rank: | \(2\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 0.058076 \) |
Real period: | \( 15.06127 \) |
Tamagawa product: | \( 2 \) |
Torsion order: | \( 2 \) |
Leading coefficient: | \( 0.437353 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(8\) | \(9\) | \(2\) | \(1 + 2 T + 2 T^{2}\) | |
\(7\) | \(2\) | \(2\) | \(1\) | \(1 + T^{2}\) |
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 |
\(3\) | 3.540.6 | no |
Sato-Tate group
\(\mathrm{ST}\) | \(\simeq\) | $E_4$ |
\(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\) |
Decomposition of the Jacobian
Splits over the number field \(\Q (b) \simeq \) \(\Q(\zeta_{16})^+\) with defining polynomial:
\(x^{4} - 4 x^{2} + 2\)
Decomposes up to isogeny as the square of the elliptic curve isogeny class:
\(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
\(g_4 = -7232 b^{3} - 5568 b^{2} + 24704 b + 18976\)
\(g_6 = 1236992 b^{3} + 945664 b^{2} - 4222208 b - 3231232\)
Conductor norm: 2401
Endomorphisms of the Jacobian
Of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
\(\End (J_{})\) | \(\simeq\) | \(\Z [\sqrt{-1}]\) |
\(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-1}) \) |
\(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\C\) |
Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\zeta_{16})^+\) with defining polynomial \(x^{4} - 4 x^{2} + 2\)
Not of \(\GL_2\)-type over \(\overline{\Q}\)
Endomorphism ring over \(\overline{\Q}\):
\(\End (J_{\overline{\Q}})\) | \(\simeq\) | a non-Eichler order of index \(4\) in a maximal order of \(\End (J_{\overline{\Q}}) \otimes \Q\) |
\(\End (J_{\overline{\Q}}) \otimes \Q \) | \(\simeq\) | \(\mathrm{M}_2(\)\(\Q\)\()\) |
\(\End (J_{\overline{\Q}}) \otimes \R\) | \(\simeq\) | \(\mathrm{M}_2 (\R)\) |
Remainder of the endomorphism lattice by field
Over subfield \(F \simeq \) \(\Q(\sqrt{2}) \) with generator \(a^{2} - 2\) with minimal polynomial \(x^{2} - 2\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\sqrt{-1}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-1}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple