Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x + 1)y = x^4 + 5x^3 + 5x^2 + x$ | (homogenize, simplify) |
$y^2 + (x^3 + xz^2 + z^3)y = x^4z^2 + 5x^3z^3 + 5x^2z^4 + xz^5$ | (dehomogenize, simplify) |
$y^2 = x^6 + 6x^4 + 22x^3 + 21x^2 + 6x + 1$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(123201\) | \(=\) | \( 3^{6} \cdot 13^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(-369603\) | \(=\) | \( - 3^{7} \cdot 13^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(108\) | \(=\) | \( 2^{2} \cdot 3^{3} \) |
\( I_4 \) | \(=\) | \(3393\) | \(=\) | \( 3^{2} \cdot 13 \cdot 29 \) |
\( I_6 \) | \(=\) | \(88335\) | \(=\) | \( 3^{2} \cdot 5 \cdot 13 \cdot 151 \) |
\( I_{10} \) | \(=\) | \(-194688\) | \(=\) | \( - 2^{7} \cdot 3^{2} \cdot 13^{2} \) |
\( J_2 \) | \(=\) | \(81\) | \(=\) | \( 3^{4} \) |
\( J_4 \) | \(=\) | \(-999\) | \(=\) | \( - 3^{3} \cdot 37 \) |
\( J_6 \) | \(=\) | \(-3267\) | \(=\) | \( - 3^{3} \cdot 11^{2} \) |
\( J_8 \) | \(=\) | \(-315657\) | \(=\) | \( - 3^{6} \cdot 433 \) |
\( J_{10} \) | \(=\) | \(-369603\) | \(=\) | \( - 3^{7} \cdot 13^{2} \) |
\( g_1 \) | \(=\) | \(-1594323/169\) | ||
\( g_2 \) | \(=\) | \(242757/169\) | ||
\( g_3 \) | \(=\) | \(9801/169\) |
Automorphism group
\(\mathrm{Aut}(X)\) | \(\simeq\) | $C_6$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
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\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $D_6$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
|
Rational points
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((-1 : 0 : 1)\) | \((0 : -1 : 1)\) | \((-1 : 1 : 1)\) |
\((-5 : 5 : 2)\) | \((2 : 37 : 3)\) | \((-3 : 52 : 5)\) | \((-3 : -75 : 5)\) | \((2 : -90 : 3)\) | \((-5 : 132 : 2)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((-1 : 0 : 1)\) | \((0 : -1 : 1)\) | \((-1 : 1 : 1)\) |
\((-5 : 5 : 2)\) | \((2 : 37 : 3)\) | \((-3 : 52 : 5)\) | \((-3 : -75 : 5)\) | \((2 : -90 : 3)\) | \((-5 : 132 : 2)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((-1 : -1 : 1)\) | \((-1 : 1 : 1)\) |
\((2 : -127 : 3)\) | \((2 : 127 : 3)\) | \((-5 : -127 : 2)\) | \((-5 : 127 : 2)\) | \((-3 : -127 : 5)\) | \((-3 : 127 : 5)\) |
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 | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 1 : 1) - (1 : -1 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(z^3\) | \(0.170953\) | \(\infty\) |
\((0 : 0 : 1) - (1 : 0 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3\) | \(0.170953\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 1 : 1) - (1 : -1 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(z^3\) | \(0.170953\) | \(\infty\) |
\((0 : 0 : 1) - (1 : 0 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3\) | \(0.170953\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 1 : 1) - (1 : -1 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + xz^2 + 3z^3\) | \(0.170953\) | \(\infty\) |
\((0 : 1 : 1) - (1 : 1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 + xz^2 + z^3\) | \(0.170953\) | \(\infty\) |
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(2\) |
Mordell-Weil rank: | \(2\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 0.021918 \) |
Real period: | \( 16.98607 \) |
Tamagawa product: | \( 3 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 1.116945 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(3\) | \(6\) | \(7\) | \(3\) | \(1\) | |
\(13\) | \(2\) | \(2\) | \(1\) | \(1 + 2 T + 13 T^{2}\) |
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? |
---|---|---|
\(2\) | 2.40.3 | 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 \(\Q (b) \simeq \) \(\Q(\zeta_{13})^+\) with defining polynomial:
\(x^{6} - x^{5} - 5 x^{4} + 4 x^{3} + 6 x^{2} - 3 x - 1\)
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
\(g_4 = -\frac{67149}{16} b^{5} + \frac{18351}{2} b^{4} + \frac{165915}{16} b^{3} - \frac{235161}{8} b^{2} + \frac{33759}{4} b + \frac{15633}{4}\)
\(g_6 = -\frac{94866525}{64} b^{5} + \frac{50567517}{16} b^{4} + \frac{245069253}{64} b^{3} - \frac{164210085}{16} b^{2} + \frac{88140663}{32} b + \frac{83317221}{64}\)
Conductor norm: 387420489
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 \(K = \Q (a) \simeq \) \(\Q(\zeta_{13})^+\) with defining polynomial \(x^{6} - x^{5} - 5 x^{4} + 4 x^{3} + 6 x^{2} - 3 x - 1\)
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 \(F \simeq \) \(\Q(\sqrt{13}) \) with generator \(-a^{4} + a^{3} + 4 a^{2} - 2 a - 2\) with minimal polynomial \(x^{2} - x - 3\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 3.3.169.1 with generator \(a^{3} - a^{2} - 3 a + 2\) with minimal polynomial \(x^{3} - x^{2} - 4 x - 1\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple