Minimal equation
Minimal equation
Simplified equation
$y^2 + y = x^6 - 2x^3 + 1$ | (homogenize, simplify) |
$y^2 + z^3y = x^6 - 2x^3z^3 + z^6$ | (dehomogenize, simplify) |
$y^2 = 4x^6 - 8x^3 + 5$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(72900\) | \(=\) | \( 2^{2} \cdot 3^{6} \cdot 5^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(-291600\) | \(=\) | \( - 2^{4} \cdot 3^{6} \cdot 5^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(184\) | \(=\) | \( 2^{3} \cdot 23 \) |
\( I_4 \) | \(=\) | \(1845\) | \(=\) | \( 3^{2} \cdot 5 \cdot 41 \) |
\( I_6 \) | \(=\) | \(87015\) | \(=\) | \( 3 \cdot 5 \cdot 5801 \) |
\( I_{10} \) | \(=\) | \(150\) | \(=\) | \( 2 \cdot 3 \cdot 5^{2} \) |
\( J_2 \) | \(=\) | \(552\) | \(=\) | \( 2^{3} \cdot 3 \cdot 23 \) |
\( J_4 \) | \(=\) | \(1626\) | \(=\) | \( 2 \cdot 3 \cdot 271 \) |
\( J_6 \) | \(=\) | \(-1616\) | \(=\) | \( - 2^{4} \cdot 101 \) |
\( J_8 \) | \(=\) | \(-883977\) | \(=\) | \( - 3 \cdot 294659 \) |
\( J_{10} \) | \(=\) | \(291600\) | \(=\) | \( 2^{4} \cdot 3^{6} \cdot 5^{2} \) |
\( g_1 \) | \(=\) | \(13181630464/75\) | ||
\( g_2 \) | \(=\) | \(211024448/225\) | ||
\( g_3 \) | \(=\) | \(-3419456/2025\) |
Automorphism group
\(\mathrm{Aut}(X)\) | \(\simeq\) | $C_2$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
|
\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $D_6$ | 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: \(\Z \oplus \Z/{3}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((1 : -1 : 1) - (1 : -1 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - 2z^3\) | \(0.563877\) | \(\infty\) |
\(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3\) | \(0\) | \(3\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((1 : -1 : 1) - (1 : -1 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - 2z^3\) | \(0.563877\) | \(\infty\) |
\(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3\) | \(0\) | \(3\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((1 : -1 : 1) - (1 : -2 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(2x^3 - 3z^3\) | \(0.563877\) | \(\infty\) |
\(D_0 - 2 \cdot(1 : -2 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(2x^3 + z^3\) | \(0\) | \(3\) |
2-torsion field: 6.0.4665600.1
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(1\) |
Mordell-Weil rank: | \(1\) |
2-Selmer rank: | \(1\) |
Regulator: | \( 0.563877 \) |
Real period: | \( 10.31518 \) |
Tamagawa product: | \( 3 \) |
Torsion order: | \( 3 \) |
Leading coefficient: | \( 1.938832 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(2\) | \(4\) | \(3\) | \(1 + 2 T^{2}\) | |
\(3\) | \(6\) | \(6\) | \(1\) | \(1\) | |
\(5\) | \(2\) | \(2\) | \(1\) | \(1 + 5 T^{2}\) |
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? |
---|---|---|
\(2\) | 2.20.3 | no |
\(3\) | 3.2880.1 | yes |
Sato-Tate group
\(\mathrm{ST}\) | \(\simeq\) | $J(E_6)$ |
\(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\) |
Decomposition of the Jacobian
Splits over the number field \(\Q (b) \simeq \) 6.2.450000.1 with defining polynomial:
\(x^{6} - x^{5} - 5 x^{3} - x + 1\)
Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
\(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
\(g_4 = 192 b^{5} - 690 b^{4} + 690 b^{3} - 750 b^{2} + 2280 b - 672\)
\(g_6 = -20160 b^{5} + 10080 b^{4} + 20160 b^{3} + 80640 b^{2} + 40320 b - 39060\)
Conductor norm: 4782969
\(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
\(g_4 = -192 b^{5} - 210 b^{4} + 210 b^{3} + 1650 b^{2} + 1320 b + 672\)
\(g_6 = 20160 b^{5} - 10080 b^{4} - 20160 b^{3} - 80640 b^{2} - 40320 b - 28980\)
Conductor norm: 4782969
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 \(K = \Q (a)\) with defining polynomial \(x^{12} - x^{11} + x^{10} + 10 x^{9} - 5 x^{8} - x^{7} + 26 x^{6} - x^{5} - 5 x^{4} + 10 x^{3} + x^{2} - x + 1\)
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 \(F \simeq \) \(\Q(\sqrt{-3}) \) with generator \(\frac{13}{12} a^{11} - \frac{7}{6} a^{10} + \frac{5}{4} a^{9} + \frac{125}{12} a^{8} - \frac{35}{6} a^{7} - \frac{1}{4} a^{6} + \frac{307}{12} a^{5} - \frac{5}{3} a^{4} - \frac{15}{4} a^{3} + \frac{65}{12} a^{2} + \frac{2}{3} a + \frac{1}{4}\) with minimal polynomial \(x^{2} - x + 1\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) \(\Q(\sqrt{5}) \) with generator \(\frac{1}{36} a^{11} - \frac{1}{2} a^{10} + \frac{7}{36} a^{9} + \frac{5}{36} a^{8} - \frac{31}{6} a^{7} - \frac{19}{36} a^{6} + \frac{103}{36} a^{5} - \frac{35}{3} a^{4} - \frac{221}{36} a^{3} + \frac{101}{36} a^{2} - \frac{7}{3} a - \frac{25}{36}\) with minimal polynomial \(x^{2} - x - 1\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R\) |
Not of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) \(\Q(\sqrt{-15}) \) with generator \(-\frac{1}{3} a^{11} + a^{10} - \frac{4}{3} a^{9} - 2 a^{8} + \frac{23}{3} a^{7} - 6 a^{6} - \frac{13}{3} a^{5} + 16 a^{4} - \frac{22}{3} a^{3} + a^{2} + \frac{17}{3} a - 1\) with minimal polynomial \(x^{2} - x + 4\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R\) |
Not of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 3.1.300.1 with generator \(-\frac{1}{6} a^{11} + \frac{7}{6} a^{10} - a^{9} - \frac{5}{6} a^{8} + \frac{65}{6} a^{7} - 3 a^{6} - \frac{37}{6} a^{5} + \frac{145}{6} a^{4} + 5 a^{3} - \frac{35}{6} a^{2} + \frac{29}{6} a + 2\) with minimal polynomial \(x^{3} - x^{2} - 3 x - 3\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R\) |
Not of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 3.1.300.1 with generator \(\frac{1}{2} a^{11} - \frac{1}{2} a^{10} + \frac{1}{2} a^{9} + \frac{31}{6} a^{8} - \frac{17}{6} a^{7} - \frac{1}{6} a^{6} + \frac{85}{6} a^{5} - \frac{5}{2} a^{4} - \frac{13}{6} a^{3} + \frac{43}{6} a^{2} - \frac{7}{6} a - \frac{1}{6}\) with minimal polynomial \(x^{3} - x^{2} - 3 x - 3\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R\) |
Not of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 3.1.300.1 with generator \(-\frac{1}{3} a^{11} - \frac{2}{3} a^{10} + \frac{1}{2} a^{9} - \frac{13}{3} a^{8} - 8 a^{7} + \frac{19}{6} a^{6} - 8 a^{5} - \frac{65}{3} a^{4} - \frac{17}{6} a^{3} - \frac{4}{3} a^{2} - \frac{11}{3} a - \frac{5}{6}\) with minimal polynomial \(x^{3} - x^{2} - 3 x - 3\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R\) |
Not of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) \(\Q(\sqrt{-3}, \sqrt{5})\) with generator \(-\frac{7}{18} a^{11} + \frac{1}{3} a^{10} - \frac{1}{18} a^{9} - \frac{77}{18} a^{8} + \frac{5}{3} a^{7} + \frac{61}{18} a^{6} - \frac{217}{18} a^{5} - \frac{4}{3} a^{4} + \frac{155}{18} a^{3} - \frac{83}{18} a^{2} - 2 a + \frac{31}{18}\) with minimal polynomial \(x^{4} - x^{3} + 2 x^{2} + x + 1\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 6.0.270000.1 with generator \(\frac{5}{12} a^{11} - \frac{1}{3} a^{10} + \frac{1}{12} a^{9} + \frac{53}{12} a^{8} - \frac{4}{3} a^{7} - \frac{41}{12} a^{6} + \frac{45}{4} a^{5} + \frac{19}{6} a^{4} - \frac{33}{4} a^{3} + \frac{29}{12} a^{2} + \frac{7}{2} a - \frac{11}{12}\) with minimal polynomial \(x^{6} - 3 x^{5} + 4 x^{4} - 3 x^{3} - 2 x^{2} + 3 x + 3\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 6.0.1350000.1 with generator \(\frac{1}{9} a^{11} - \frac{5}{6} a^{10} + \frac{10}{9} a^{9} - \frac{1}{9} a^{8} - \frac{43}{6} a^{7} + \frac{53}{9} a^{6} + \frac{1}{9} a^{5} - \frac{101}{6} a^{4} + \frac{55}{9} a^{3} - \frac{10}{9} a^{2} - \frac{31}{6} a + \frac{8}{9}\) with minimal polynomial \(x^{6} - x^{5} + 5 x^{4} - 5 x^{3} + 10 x^{2} - 6 x + 6\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\sqrt{3}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{3}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 6.2.450000.1 with generator \(-\frac{23}{36} a^{11} + \frac{1}{2} a^{10} - \frac{11}{36} a^{9} - \frac{235}{36} a^{8} + \frac{11}{6} a^{7} + \frac{119}{36} a^{6} - \frac{557}{36} a^{5} - 4 a^{4} + \frac{253}{36} a^{3} - \frac{55}{36} a^{2} - 2 a + \frac{29}{36}\) with minimal polynomial \(x^{6} - x^{5} - 5 x^{3} - x + 1\):
\(\End (J_{F})\) | \(\simeq\) | an order of index \(2\) in \(\Z \times \Z\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q\) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Of \(\GL_2\)-type, not simple
Over subfield \(F \simeq \) 6.2.450000.1 with generator \(\frac{31}{36} a^{11} - \frac{4}{3} a^{10} + \frac{55}{36} a^{9} + \frac{275}{36} a^{8} - \frac{25}{3} a^{7} + \frac{101}{36} a^{6} + \frac{697}{36} a^{5} - \frac{61}{6} a^{4} - \frac{17}{36} a^{3} + \frac{179}{36} a^{2} - \frac{5}{6} a - \frac{1}{36}\) with minimal polynomial \(x^{6} - x^{5} - 5 x^{3} - x + 1\):
\(\End (J_{F})\) | \(\simeq\) | an order of index \(2\) in \(\Z \times \Z\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q\) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Of \(\GL_2\)-type, not simple
Over subfield \(F \simeq \) 6.2.450000.1 with generator \(\frac{1}{12} a^{11} - \frac{1}{6} a^{10} + \frac{5}{12} a^{9} + \frac{5}{12} a^{8} - \frac{5}{6} a^{7} + \frac{31}{12} a^{6} + \frac{7}{12} a^{5} - \frac{5}{3} a^{4} + \frac{65}{12} a^{3} + \frac{5}{12} a^{2} - \frac{1}{3} a + \frac{13}{12}\) with minimal polynomial \(x^{6} - x^{5} - 5 x^{3} - x + 1\):
\(\End (J_{F})\) | \(\simeq\) | an order of index \(2\) in \(\Z \times \Z\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q\) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Of \(\GL_2\)-type, not simple
Over subfield \(F \simeq \) 6.0.1350000.1 with generator \(-\frac{1}{2} a^{11} + \frac{5}{6} a^{10} - a^{9} - \frac{9}{2} a^{8} + \frac{11}{2} a^{7} - \frac{8}{3} a^{6} - \frac{25}{2} a^{5} + \frac{15}{2} a^{4} - a^{3} - \frac{31}{6} a^{2} - \frac{1}{2} a\) with minimal polynomial \(x^{6} - x^{5} + 5 x^{4} - 5 x^{3} + 10 x^{2} - 6 x + 6\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\sqrt{3}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{3}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 6.0.1350000.1 with generator \(-\frac{1}{18} a^{11} + \frac{1}{2} a^{10} - \frac{13}{18} a^{9} + \frac{7}{18} a^{8} + \frac{25}{6} a^{7} - \frac{71}{18} a^{6} + \frac{29}{18} a^{5} + \frac{61}{6} a^{4} - \frac{97}{18} a^{3} + \frac{49}{18} a^{2} + \frac{23}{6} a - \frac{23}{18}\) with minimal polynomial \(x^{6} - x^{5} + 5 x^{4} - 5 x^{3} + 10 x^{2} - 6 x + 6\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\sqrt{3}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{3}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Of \(\GL_2\)-type, simple