Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x + 1)y = -2x^5 + 3x^4 - 3x^2$ | (homogenize, simplify) |
$y^2 + (x^3 + xz^2 + z^3)y = -2x^5z + 3x^4z^2 - 3x^2z^4$ | (dehomogenize, simplify) |
$y^2 = x^6 - 8x^5 + 14x^4 + 2x^3 - 11x^2 + 2x + 1$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(3969\) | \(=\) | \( 3^{4} \cdot 7^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(35721\) | \(=\) | \( 3^{6} \cdot 7^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(268\) | \(=\) | \( 2^{2} \cdot 67 \) |
\( I_4 \) | \(=\) | \(2961\) | \(=\) | \( 3^{2} \cdot 7 \cdot 47 \) |
\( I_6 \) | \(=\) | \(216951\) | \(=\) | \( 3 \cdot 7 \cdot 10331 \) |
\( I_{10} \) | \(=\) | \(18816\) | \(=\) | \( 2^{7} \cdot 3 \cdot 7^{2} \) |
\( J_2 \) | \(=\) | \(201\) | \(=\) | \( 3 \cdot 67 \) |
\( J_4 \) | \(=\) | \(573\) | \(=\) | \( 3 \cdot 191 \) |
\( J_6 \) | \(=\) | \(-563\) | \(=\) | \( -563 \) |
\( J_8 \) | \(=\) | \(-110373\) | \(=\) | \( - 3 \cdot 36791 \) |
\( J_{10} \) | \(=\) | \(35721\) | \(=\) | \( 3^{6} \cdot 7^{2} \) |
\( g_1 \) | \(=\) | \(1350125107/147\) | ||
\( g_2 \) | \(=\) | \(57445733/441\) | ||
\( g_3 \) | \(=\) | \(-2527307/3969\) |
Automorphism group
\(\mathrm{Aut}(X)\) | \(\simeq\) | $C_6$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
|
\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $D_6$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
|
Rational points
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((-1 : -1 : 1)\) | \((1 : -1 : 1)\) |
\((-1 : 2 : 1)\) | \((1 : -2 : 1)\) | \((2 : -4 : 1)\) | \((1 : -5 : 2)\) | \((2 : -7 : 1)\) | \((1 : -8 : 2)\) |
\((1 : -16 : 6)\) | \((-5 : -46 : 1)\) | \((6 : -135 : 5)\) | \((-5 : 175 : 1)\) | \((1 : -237 : 6)\) | \((6 : -356 : 5)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((-1 : -1 : 1)\) | \((1 : -1 : 1)\) |
\((-1 : 2 : 1)\) | \((1 : -2 : 1)\) | \((2 : -4 : 1)\) | \((1 : -5 : 2)\) | \((2 : -7 : 1)\) | \((1 : -8 : 2)\) |
\((1 : -16 : 6)\) | \((-5 : -46 : 1)\) | \((6 : -135 : 5)\) | \((-5 : 175 : 1)\) | \((1 : -237 : 6)\) | \((6 : -356 : 5)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((1 : -1 : 1)\) | \((1 : 1 : 1)\) |
\((-1 : -3 : 1)\) | \((-1 : 3 : 1)\) | \((1 : -3 : 2)\) | \((1 : 3 : 2)\) | \((2 : -3 : 1)\) | \((2 : 3 : 1)\) |
\((-5 : -221 : 1)\) | \((-5 : 221 : 1)\) | \((1 : -221 : 6)\) | \((1 : 221 : 6)\) | \((6 : -221 : 5)\) | \((6 : 221 : 5)\) |
Number of rational Weierstrass points: \(0\)
This curve is locally solvable everywhere.
Mordell-Weil group of the Jacobian
Group structure: \(\Z \oplus \Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 2 : 1) + (1 : -2 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2xz^2\) | \(0.064863\) | \(\infty\) |
\((0 : -1 : 1) + (2 : -7 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x - 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-3xz^2 - z^3\) | \(0.064863\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 2 : 1) + (1 : -2 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2xz^2\) | \(0.064863\) | \(\infty\) |
\((0 : -1 : 1) + (2 : -7 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x - 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-3xz^2 - z^3\) | \(0.064863\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 3 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - 3xz^2 + z^3\) | \(0.064863\) | \(\infty\) |
\((0 : -1 : 1) + (2 : -3 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x (x - 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - 5xz^2 - z^3\) | \(0.064863\) | \(\infty\) |
2-torsion field: 9.9.62523502209.1
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(2\) |
Mordell-Weil rank: | \(2\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 0.003155 \) |
Real period: | \( 23.23416 \) |
Tamagawa product: | \( 3 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 0.219944 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(3\) | \(4\) | \(6\) | \(3\) | \(1 + 3 T + 3 T^{2}\) | |
\(7\) | \(2\) | \(2\) | \(1\) | \(1 + 5 T + 7 T^{2}\) |
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? |
---|---|---|
\(2\) | 2.80.1 | no |
\(3\) | 3.480.12 | no |
Sato-Tate group
\(\mathrm{ST}\) | \(\simeq\) | $E_6$ |
\(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\) |
Decomposition of the Jacobian
Splits over the number field \(\Q (b) \simeq \) 6.6.330812181.1 with defining polynomial:
\(x^{6} - 21 x^{4} - 14 x^{3} + 63 x^{2} + 21 x - 35\)
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
\(g_4 = \frac{29108241}{62500} b^{5} + \frac{81976293}{250000} b^{4} - \frac{1198505889}{125000} b^{3} - \frac{3288044907}{250000} b^{2} + \frac{5106100923}{250000} b + \frac{1196447679}{50000}\)
\(g_6 = -\frac{248712019479}{3125000} b^{5} - \frac{1355533889409}{25000000} b^{4} + \frac{10205353689741}{6250000} b^{3} + \frac{55772501101191}{25000000} b^{2} - \frac{86931059013099}{25000000} b - \frac{2533145646969}{625000}\)
Conductor norm: 1
Endomorphisms of the Jacobian
Of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
\(\End (J_{})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
\(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
\(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\C\) |
Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) 6.6.330812181.1 with defining polynomial \(x^{6} - 21 x^{4} - 14 x^{3} + 63 x^{2} + 21 x - 35\)
Not of \(\GL_2\)-type over \(\overline{\Q}\)
Endomorphism ring over \(\overline{\Q}\):
\(\End (J_{\overline{\Q}})\) | \(\simeq\) | an Eichler order of index \(3\) 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 \(F \simeq \) \(\Q(\sqrt{21}) \) with generator \(\frac{9}{125} a^{5} - \frac{12}{125} a^{4} - \frac{173}{125} a^{3} + \frac{63}{125} a^{2} + \frac{483}{125} a + \frac{9}{25}\) with minimal polynomial \(x^{2} - x - 5\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 3.3.3969.2 with generator \(-\frac{12}{125} a^{5} - \frac{9}{125} a^{4} + \frac{214}{125} a^{3} + \frac{391}{125} a^{2} - \frac{119}{125} a - \frac{112}{25}\) with minimal polynomial \(x^{3} - 21 x - 35\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple