Minimal equation
Minimal equation
Simplified equation
$y^2 + x^3y = x^3 + 3$ | (homogenize, simplify) |
$y^2 + x^3y = x^3z^3 + 3z^6$ | (dehomogenize, simplify) |
$y^2 = x^6 + 4x^3 + 12$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(139968\) | \(=\) | \( 2^{6} \cdot 3^{7} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(-839808\) | \(=\) | \( - 2^{7} \cdot 3^{8} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(116\) | \(=\) | \( 2^{2} \cdot 29 \) |
\( I_4 \) | \(=\) | \(513\) | \(=\) | \( 3^{3} \cdot 19 \) |
\( I_6 \) | \(=\) | \(16623\) | \(=\) | \( 3^{2} \cdot 1847 \) |
\( I_{10} \) | \(=\) | \(432\) | \(=\) | \( 2^{4} \cdot 3^{3} \) |
\( J_2 \) | \(=\) | \(348\) | \(=\) | \( 2^{2} \cdot 3 \cdot 29 \) |
\( J_4 \) | \(=\) | \(1968\) | \(=\) | \( 2^{4} \cdot 3 \cdot 41 \) |
\( J_6 \) | \(=\) | \(-3856\) | \(=\) | \( - 2^{4} \cdot 241 \) |
\( J_8 \) | \(=\) | \(-1303728\) | \(=\) | \( - 2^{4} \cdot 3 \cdot 157 \cdot 173 \) |
\( J_{10} \) | \(=\) | \(839808\) | \(=\) | \( 2^{7} \cdot 3^{8} \) |
\( g_1 \) | \(=\) | \(164089192/27\) | ||
\( g_2 \) | \(=\) | \(7999592/81\) | ||
\( g_3 \) | \(=\) | \(-405362/729\) |
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 | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : -1 : 1) - (1 : -1 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.364885\) | \(\infty\) |
\(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(3\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : -1 : 1) - (1 : -1 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.364885\) | \(\infty\) |
\(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(3\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : -3 : 1) - (1 : -1 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - 2z^3\) | \(0.364885\) | \(\infty\) |
\(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3\) | \(0\) | \(3\) |
2-torsion field: 6.0.1492992.5
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(1\) |
Mordell-Weil rank: | \(1\) |
2-Selmer rank: | \(1\) |
Regulator: | \( 0.364885 \) |
Real period: | \( 10.97281 \) |
Tamagawa product: | \( 6 \) |
Torsion order: | \( 3 \) |
Leading coefficient: | \( 2.669217 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(6\) | \(7\) | \(3\) | \(1\) | |
\(3\) | \(7\) | \(8\) | \(2\) | \(1\) |
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 |
\(3\) | 3.960.1 | 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 \(\Q (b) \simeq \) 6.2.45349632.2 with defining polynomial:
\(x^{6} - 12\)
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
\(g_4 = 72 b^{5} + 540 b^{2}\)
\(g_6 = -9072 b^{3} - 45360\)
Conductor norm: 9
\(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
\(g_4 = -72 b^{5} + 540 b^{2}\)
\(g_6 = 9072 b^{3} - 45360\)
Conductor norm: 9
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 \(x^{12} - 3 x^{10} - 30 x^{8} + 77 x^{6} + 510 x^{4} + 537 x^{2} + 169\)
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 \(-\frac{78}{6137} a^{10} + \frac{225}{6137} a^{8} + \frac{2602}{6137} a^{6} - \frac{7122}{6137} a^{4} - \frac{42018}{6137} a^{2} - \frac{20770}{6137}\) 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 \(F \simeq \) \(\Q(\sqrt{3}) \) with generator \(\frac{411}{9386} a^{11} - \frac{1597}{9386} a^{9} - \frac{10653}{9386} a^{7} + \frac{39369}{9386} a^{5} + \frac{172859}{9386} a^{3} + \frac{91617}{9386} a\) with minimal polynomial \(x^{2} - 3\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R\) |
Not of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) \(\Q(\sqrt{-1}) \) with generator \(-\frac{27}{159562} a^{11} - \frac{829}{159562} a^{9} + \frac{5373}{159562} a^{7} + \frac{19449}{159562} a^{5} - \frac{129483}{159562} a^{3} - \frac{283131}{159562} a\) with minimal polynomial \(x^{2} + 1\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R\) |
Not of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 3.1.972.1 with generator \(\frac{1}{361} a^{10} - \frac{8}{361} a^{8} + \frac{10}{361} a^{6} + \frac{27}{361} a^{4} + \frac{14}{361} a^{2} + \frac{828}{361}\) with minimal polynomial \(x^{3} - 12\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R\) |
Not of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 3.1.972.1 with generator \(-\frac{281}{6137} a^{10} + \frac{1146}{6137} a^{8} + \frac{6823}{6137} a^{6} - \frac{26701}{6137} a^{4} - \frac{109954}{6137} a^{2} - \frac{66361}{6137}\) with minimal polynomial \(x^{3} - 12\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R\) |
Not of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 3.1.972.1 with generator \(\frac{264}{6137} a^{10} - \frac{1010}{6137} a^{8} - \frac{6993}{6137} a^{6} + \frac{26242}{6137} a^{4} + \frac{109716}{6137} a^{2} + \frac{52285}{6137}\) with minimal polynomial \(x^{3} - 12\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R\) |
Not of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) \(\Q(\zeta_{12})\) with generator \(\frac{1740}{79781} a^{11} - \frac{13989}{159562} a^{9} - \frac{43932}{79781} a^{7} + \frac{344361}{159562} a^{5} + \frac{702280}{79781} a^{3} + \frac{637179}{159562} a\) with minimal polynomial \(x^{4} - x^{2} + 1\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 6.0.2834352.1 with generator \(-\frac{43}{6137} a^{10} + \frac{99}{12274} a^{8} + \frac{1774}{6137} a^{6} - \frac{5343}{12274} a^{4} - \frac{31686}{6137} a^{2} - \frac{54165}{12274}\) with minimal polynomial \(x^{6} + 12\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 6.0.15116544.4 with generator \(\frac{21}{159562} a^{11} - \frac{1}{722} a^{10} - \frac{844}{79781} a^{9} + \frac{4}{361} a^{8} + \frac{4209}{79781} a^{7} - \frac{5}{361} a^{6} + \frac{11332}{79781} a^{5} - \frac{27}{722} a^{4} - \frac{151231}{159562} a^{3} - \frac{7}{361} a^{2} - \frac{325657}{159562} a - \frac{414}{361}\) with minimal polynomial \(x^{6} - 6 x^{3} + 18\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\sqrt{3}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{3}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 6.2.45349632.2 with generator \(-\frac{3507}{159562} a^{11} + \frac{6580}{79781} a^{9} + \frac{93237}{159562} a^{7} - \frac{162456}{79781} a^{5} - \frac{1534043}{159562} a^{3} - \frac{539936}{79781} a\) with minimal polynomial \(x^{6} - 12\):
\(\End (J_{F})\) | \(\simeq\) | 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 \(F \simeq \) 6.2.45349632.2 with generator \(-\frac{2001}{159562} a^{11} + \frac{6679}{159562} a^{9} + \frac{62279}{159562} a^{7} - \frac{204621}{159562} a^{5} - \frac{962985}{159562} a^{3} - \frac{256889}{159562} a\) with minimal polynomial \(x^{6} - 12\):
\(\End (J_{F})\) | \(\simeq\) | 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 \(F \simeq \) 6.2.45349632.2 with generator \(\frac{162}{4693} a^{11} - \frac{1167}{9386} a^{9} - \frac{4574}{4693} a^{7} + \frac{31149}{9386} a^{5} + \frac{73442}{4693} a^{3} + \frac{78633}{9386} a\) with minimal polynomial \(x^{6} - 12\):
\(\End (J_{F})\) | \(\simeq\) | 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 \(F \simeq \) 6.0.15116544.4 with generator \(\frac{3313}{159562} a^{11} + \frac{281}{12274} a^{10} - \frac{10563}{159562} a^{9} - \frac{573}{6137} a^{8} - \frac{50410}{79781} a^{7} - \frac{6823}{12274} a^{6} + \frac{149748}{79781} a^{5} + \frac{26701}{12274} a^{4} + \frac{830216}{79781} a^{3} + \frac{54977}{6137} a^{2} + \frac{1020453}{159562} a + \frac{66361}{12274}\) with minimal polynomial \(x^{6} - 6 x^{3} + 18\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\sqrt{3}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{3}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 6.0.15116544.4 with generator \(-\frac{1667}{79781} a^{11} - \frac{132}{6137} a^{10} + \frac{12251}{159562} a^{9} + \frac{505}{6137} a^{8} + \frac{46201}{79781} a^{7} + \frac{6993}{12274} a^{6} - \frac{161080}{79781} a^{5} - \frac{13121}{6137} a^{4} - \frac{1509201}{159562} a^{3} - \frac{54858}{6137} a^{2} - \frac{347398}{79781} a - \frac{52285}{12274}\) with minimal polynomial \(x^{6} - 6 x^{3} + 18\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\sqrt{3}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{3}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Of \(\GL_2\)-type, simple