Minimal equation
Minimal equation
Simplified equation
$y^2 = x^6 - 3x^3 + 2$ | (homogenize, simplify) |
$y^2 = x^6 - 3x^3z^3 + 2z^6$ | (dehomogenize, simplify) |
$y^2 = x^6 - 3x^3 + 2$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(62208\) | \(=\) | \( 2^{8} \cdot 3^{5} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(746496\) | \(=\) | \( 2^{10} \cdot 3^{6} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(142\) | \(=\) | \( 2 \cdot 71 \) |
\( I_4 \) | \(=\) | \(1368\) | \(=\) | \( 2^{3} \cdot 3^{2} \cdot 19 \) |
\( I_6 \) | \(=\) | \(47940\) | \(=\) | \( 2^{2} \cdot 3 \cdot 5 \cdot 17 \cdot 47 \) |
\( I_{10} \) | \(=\) | \(-12\) | \(=\) | \( - 2^{2} \cdot 3 \) |
\( J_2 \) | \(=\) | \(852\) | \(=\) | \( 2^{2} \cdot 3 \cdot 71 \) |
\( J_4 \) | \(=\) | \(-2586\) | \(=\) | \( - 2 \cdot 3 \cdot 431 \) |
\( J_6 \) | \(=\) | \(-2596\) | \(=\) | \( - 2^{2} \cdot 11 \cdot 59 \) |
\( J_8 \) | \(=\) | \(-2224797\) | \(=\) | \( - 3 \cdot 741599 \) |
\( J_{10} \) | \(=\) | \(-746496\) | \(=\) | \( - 2^{10} \cdot 3^{6} \) |
\( g_1 \) | \(=\) | \(-1804229351/3\) | ||
\( g_2 \) | \(=\) | \(154259641/72\) | ||
\( g_3 \) | \(=\) | \(3271609/1296\) |
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: \(1\)
This curve is locally solvable everywhere.
Mordell-Weil group of the Jacobian
Group structure: \(\Z/{6}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((1 : 0 : 1) - (1 : 1 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 + z^3\) | \(0\) | \(6\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((1 : 0 : 1) - (1 : 1 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 + z^3\) | \(0\) | \(6\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((1 : 0 : 1) - (1 : 1/2 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-1/2x^3 + 1/2z^3\) | \(0\) | \(6\) |
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(1\) |
Regulator: | \( 1 \) |
Real period: | \( 8.616344 \) |
Tamagawa product: | \( 6 \) |
Torsion order: | \( 6 \) |
Leading coefficient: | \( 1.436057 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(8\) | \(10\) | \(3\) | \(1\) | |
\(3\) | \(5\) | \(6\) | \(2\) | \(1\) |
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? |
---|---|---|
\(2\) | 2.120.1 | yes |
\(3\) | 3.2880.3 | 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 \(\Q (b) \simeq \) 6.2.1492992.4 with defining polynomial:
\(x^{6} - 2\)
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
\(g_4 = 720 b^{4} - 864 b\)
\(g_6 = 36288 b^{3} - 45792\)
Conductor norm: 81
\(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
\(g_4 = 720 b^{4} + 864 b\)
\(g_6 = -36288 b^{3} - 45792\)
Conductor norm: 81
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 \(x^{12} - 6 x^{10} + 44 x^{6} + 60 x^{4} + 24 x^{2} + 4\)
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 \(-\frac{1}{10} a^{10} + \frac{11}{20} a^{8} + \frac{2}{5} a^{6} - \frac{26}{5} a^{4} - \frac{71}{10} a^{2} - \frac{6}{5}\) 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 \(F \simeq \) \(\Q(\sqrt{2}) \) with generator \(-\frac{12}{31} a^{11} + \frac{145}{62} a^{9} + \frac{3}{62} a^{7} - \frac{561}{31} a^{5} - \frac{654}{31} a^{3} - \frac{129}{31} a\) with minimal polynomial \(x^{2} - 2\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R\) |
Not of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) \(\Q(\sqrt{-6}) \) with generator \(\frac{49}{310} a^{11} - \frac{287}{310} a^{9} - \frac{41}{310} a^{7} + \frac{1064}{155} a^{5} + \frac{1622}{155} a^{3} + \frac{1019}{155} a\) with minimal polynomial \(x^{2} + 6\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R\) |
Not of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 3.1.108.1 with generator \(-\frac{26}{155} a^{10} + \frac{371}{310} a^{8} - \frac{206}{155} a^{6} - \frac{952}{155} a^{4} - \frac{456}{155} a^{2} + \frac{108}{155}\) with minimal polynomial \(x^{3} - 2\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R\) |
Not of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 3.1.108.1 with generator \(\frac{159}{620} a^{10} - \frac{521}{310} a^{8} + \frac{151}{155} a^{6} + \frac{3209}{310} a^{4} + \frac{1651}{155} a^{2} + \frac{282}{155}\) with minimal polynomial \(x^{3} - 2\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R\) |
Not of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 3.1.108.1 with generator \(-\frac{11}{124} a^{10} + \frac{15}{31} a^{8} + \frac{11}{31} a^{6} - \frac{261}{62} a^{4} - \frac{239}{31} a^{2} - \frac{78}{31}\) with minimal polynomial \(x^{3} - 2\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R\) |
Not of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) \(\Q(\sqrt{2}, \sqrt{-3})\) with generator \(\frac{71}{620} a^{11} - \frac{219}{310} a^{9} + \frac{13}{310} a^{7} + \frac{1741}{310} a^{5} + \frac{824}{155} a^{3} - \frac{187}{155} a\) with minimal polynomial \(x^{4} + 2 x^{2} + 4\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 6.0.34992.1 with generator \(\frac{9}{620} a^{10} - \frac{97}{620} a^{8} + \frac{137}{310} a^{6} + \frac{129}{310} a^{4} - \frac{553}{310} a^{2} - \frac{218}{155}\) with minimal polynomial \(x^{6} - 3 x^{5} + 5 x^{3} - 3 x + 1\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 6.0.4478976.2 with generator \(\frac{23}{310} a^{11} - \frac{203}{620} a^{9} - \frac{247}{310} a^{7} + \frac{588}{155} a^{5} + \frac{2943}{310} a^{3} + \frac{763}{155} a\) with minimal polynomial \(x^{6} + 6 x^{4} + 12 x^{2} + 6\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\sqrt{3}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{3}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 6.2.1492992.4 with generator \(\frac{123}{620} a^{11} - \frac{809}{620} a^{9} + \frac{219}{310} a^{7} + \frac{2693}{310} a^{5} + \frac{1949}{310} a^{3} - \frac{86}{155} a\) with minimal polynomial \(x^{6} - 2\):
\(\End (J_{F})\) | \(\simeq\) | 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 \(F \simeq \) 6.2.1492992.4 with generator \(\frac{15}{62} a^{11} - \frac{189}{124} a^{9} + \frac{33}{62} a^{7} + \frac{308}{31} a^{5} + \frac{771}{62} a^{3} + \frac{131}{31} a\) with minimal polynomial \(x^{6} - 2\):
\(\End (J_{F})\) | \(\simeq\) | 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 \(F \simeq \) 6.2.1492992.4 with generator \(\frac{273}{620} a^{11} - \frac{877}{310} a^{9} + \frac{192}{155} a^{7} + \frac{5773}{310} a^{5} + \frac{2902}{155} a^{3} + \frac{569}{155} a\) with minimal polynomial \(x^{6} - 2\):
\(\End (J_{F})\) | \(\simeq\) | 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 \(F \simeq \) 6.0.4478976.2 with generator \(\frac{61}{620} a^{11} - \frac{117}{155} a^{9} + \frac{343}{310} a^{7} + \frac{1081}{310} a^{5} - \frac{126}{155} a^{3} - \frac{272}{155} a\) with minimal polynomial \(x^{6} + 6 x^{4} + 12 x^{2} + 6\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\sqrt{3}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{3}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 6.0.4478976.2 with generator \(\frac{9}{620} a^{11} - \frac{97}{620} a^{9} + \frac{137}{310} a^{7} + \frac{129}{310} a^{5} - \frac{553}{310} a^{3} - \frac{528}{155} a\) with minimal polynomial \(x^{6} + 6 x^{4} + 12 x^{2} + 6\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\sqrt{3}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{3}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Of \(\GL_2\)-type, simple