Minimal equation
Minimal equation
Simplified equation
| $y^2 + (x^3 + 1)y = -2$ | (homogenize, simplify) |
| $y^2 + (x^3 + z^3)y = -2z^6$ | (dehomogenize, simplify) |
| $y^2 = x^6 + 2x^3 - 7$ | (homogenize, minimize) |
Invariants
| Conductor: | \( N \) | \(=\) | \(142884\) | \(=\) | \( 2^{2} \cdot 3^{6} \cdot 7^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(285768\) | \(=\) | \( 2^{3} \cdot 3^{6} \cdot 7^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(284\) | \(=\) | \( 2^{2} \cdot 71 \) |
| \( I_4 \) | \(=\) | \(1953\) | \(=\) | \( 3^{2} \cdot 7 \cdot 31 \) |
| \( I_6 \) | \(=\) | \(181167\) | \(=\) | \( 3 \cdot 7 \cdot 8627 \) |
| \( I_{10} \) | \(=\) | \(150528\) | \(=\) | \( 2^{10} \cdot 3 \cdot 7^{2} \) |
| \( J_2 \) | \(=\) | \(213\) | \(=\) | \( 3 \cdot 71 \) |
| \( J_4 \) | \(=\) | \(1158\) | \(=\) | \( 2 \cdot 3 \cdot 193 \) |
| \( J_6 \) | \(=\) | \(-2236\) | \(=\) | \( - 2^{2} \cdot 13 \cdot 43 \) |
| \( J_8 \) | \(=\) | \(-454308\) | \(=\) | \( - 2^{2} \cdot 3 \cdot 17^{2} \cdot 131 \) |
| \( J_{10} \) | \(=\) | \(285768\) | \(=\) | \( 2^{3} \cdot 3^{6} \cdot 7^{2} \) |
| \( g_1 \) | \(=\) | \(1804229351/1176\) | ||
| \( g_2 \) | \(=\) | \(69076823/1764\) | ||
| \( g_3 \) | \(=\) | \(-2817919/7938\) |
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: \(\Z/{3}\Z\)
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(3\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(3\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + z^3\) | \(0\) | \(3\) |
2-torsion field: 6.2.18289152.2
BSD invariants
| Hasse-Weil conjecture: | unverified |
| Analytic rank: | \(0\) |
| Mordell-Weil rank: | \(0\) |
| 2-Selmer rank: | \(0\) |
| Regulator: | \( 1 \) |
| Real period: | \( 7.453891 \) |
| Tamagawa product: | \( 3 \) |
| Torsion order: | \( 3 \) |
| Leading coefficient: | \( 2.484630 \) |
| Analytic order of Ш: | \( 1 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(2\) | \(2\) | \(3\) | \(3\) | \(( 1 - T )( 1 + T )\) | |
| \(3\) | \(6\) | \(6\) | \(1\) | \(1\) |  |
| \(7\) | \(2\) | \(2\) | \(1\) | \(1 - 5 T + 7 T^{2}\) |  |
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? |
|---|---|---|
| \(2\) | 2.20.3 | 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 \(\Q (b) \simeq \) 6.0.12252303.1 with defining polynomial:
\(x^{6} + 7\)
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
\(g_4 = -36 b^{5} - 315 b^{2}\)
\(g_6 = -2646 b^{3} - 14175\)
Conductor norm: 5184
\(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
\(g_4 = 36 b^{5} - 315 b^{2}\)
\(g_6 = 2646 b^{3} - 14175\)
Conductor norm: 5184
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 \(x^{12} - 6 x^{11} + 21 x^{10} - 50 x^{9} + 90 x^{8} - 126 x^{7} + 155 x^{6} - 168 x^{5} - 15 x^{4} + 230 x^{3} - 84 x^{2} - 48 x + 64\)
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 \(\frac{77}{50520} a^{11} - \frac{847}{101040} a^{10} + \frac{1337}{101040} a^{9} + \frac{7}{2105} a^{8} - \frac{1039}{10104} a^{7} + \frac{7217}{25260} a^{6} - \frac{700}{1263} a^{5} + \frac{24283}{33680} a^{4} - \frac{9219}{6736} a^{3} + \frac{73759}{50520} a^{2} + \frac{20363}{25260} a - \frac{811}{6315}\) 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 \(F \simeq \) \(\Q(\sqrt{21}) \) with generator \(-\frac{1}{48} a^{10} + \frac{5}{48} a^{9} - \frac{1}{3} a^{8} + \frac{17}{24} a^{7} - \frac{7}{6} a^{6} + \frac{35}{24} a^{5} - \frac{77}{48} a^{4} + \frac{67}{48} a^{3} + \frac{65}{24} a^{2} - \frac{13}{4} a\) with minimal polynomial \(x^{2} - x - 5\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R\) |
Not of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) \(\Q(\sqrt{-7}) \) with generator \(\frac{189}{8420} a^{11} - \frac{2079}{16840} a^{10} + \frac{7109}{16840} a^{9} - \frac{8199}{8420} a^{8} + \frac{2961}{1684} a^{7} - \frac{5208}{2105} a^{6} + \frac{1386}{421} a^{5} - \frac{62307}{16840} a^{4} + \frac{87}{3368} a^{3} + \frac{12159}{4210} a^{2} + \frac{18}{2105} a - \frac{154}{2105}\) with minimal polynomial \(x^{2} - x + 2\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R\) |
Not of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 3.1.1323.1 with generator \(\frac{7}{240} a^{10} - \frac{7}{48} a^{9} + \frac{29}{60} a^{8} - \frac{127}{120} a^{7} + \frac{28}{15} a^{6} - \frac{301}{120} a^{5} + \frac{273}{80} a^{4} - \frac{287}{80} a^{3} - \frac{169}{120} a^{2} + \frac{35}{12} a + \frac{1}{15}\) with minimal polynomial \(x^{3} - 7\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R\) |
Not of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 3.1.1323.1 with generator \(\frac{1589}{101040} a^{11} - \frac{10213}{101040} a^{10} + \frac{4559}{12630} a^{9} - \frac{44921}{50520} a^{8} + \frac{20839}{12630} a^{7} - \frac{123263}{50520} a^{6} + \frac{106827}{33680} a^{5} - \frac{128401}{33680} a^{4} + \frac{73759}{50520} a^{3} + \frac{72233}{25260} a^{2} - \frac{313}{6315} a - \frac{2413}{2105}\) with minimal polynomial \(x^{3} - 7\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R\) |
Not of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 3.1.1323.1 with generator \(-\frac{1589}{101040} a^{11} + \frac{1211}{16840} a^{10} - \frac{21737}{101040} a^{9} + \frac{20503}{50520} a^{8} - \frac{9963}{16840} a^{7} + \frac{9653}{16840} a^{6} - \frac{67039}{101040} a^{5} + \frac{3367}{8420} a^{4} + \frac{214963}{101040} a^{3} - \frac{24439}{16840} a^{2} - \frac{24141}{8420} a + \frac{6818}{6315}\) with minimal polynomial \(x^{3} - 7\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R\) |
Not of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) \(\Q(\sqrt{-3}, \sqrt{-7})\) with generator \(-\frac{1057}{101040} a^{11} + \frac{3433}{50520} a^{10} - \frac{25921}{101040} a^{9} + \frac{33101}{50520} a^{8} - \frac{4327}{3368} a^{7} + \frac{33061}{16840} a^{6} - \frac{53599}{20208} a^{5} + \frac{76097}{25260} a^{4} - \frac{28193}{20208} a^{3} - \frac{34829}{16840} a^{2} + \frac{51121}{25260} a + \frac{2983}{6315}\) with minimal polynomial \(x^{4} - x^{3} - x^{2} - 2 x + 4\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 6.0.5250987.1 with generator \(\frac{77}{50520} a^{11} - \frac{1897}{50520} a^{10} + \frac{2009}{12630} a^{9} - \frac{2425}{5052} a^{8} + \frac{6034}{6315} a^{7} - \frac{7987}{5052} a^{6} + \frac{32907}{16840} a^{5} - \frac{9065}{3368} a^{4} + \frac{18683}{8420} a^{3} + \frac{36227}{12630} a^{2} - \frac{13328}{6315} a - \frac{1232}{6315}\) with minimal polynomial \(x^{6} - 3 x^{5} + 6 x^{4} + 7 x^{3} - 15 x^{2} - 24 x + 64\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 6.2.36756909.1 with generator \(-\frac{277}{50520} a^{11} + \frac{8149}{151560} a^{10} - \frac{3856}{18945} a^{9} + \frac{10528}{18945} a^{8} - \frac{39113}{37890} a^{7} + \frac{39637}{25260} a^{6} - \frac{269651}{151560} a^{5} + \frac{108037}{50520} a^{4} - \frac{36619}{75780} a^{3} - \frac{298483}{75780} a^{2} + \frac{35144}{18945} a + \frac{12454}{18945}\) with minimal polynomial \(x^{6} - 3 x^{5} + 9 x^{4} - 6 x^{3} - 24 x^{2} + 72 x - 48\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z [\sqrt{3}]\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{3}) \) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 6.2.36756909.1 with generator \(-\frac{2143}{101040} a^{11} + \frac{32623}{303120} a^{10} - \frac{49771}{151560} a^{9} + \frac{98581}{151560} a^{8} - \frac{69803}{75780} a^{7} + \frac{14747}{16840} a^{6} - \frac{219221}{303120} a^{5} + \frac{43031}{101040} a^{4} + \frac{138029}{37890} a^{3} - \frac{89432}{18945} a^{2} - \frac{94129}{37890} a + \frac{52274}{18945}\) with minimal polynomial \(x^{6} - 3 x^{5} + 9 x^{4} - 6 x^{3} - 24 x^{2} + 72 x - 48\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z [\sqrt{3}]\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{3}) \) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 6.2.36756909.1 with generator \(-\frac{1}{360} a^{10} + \frac{1}{72} a^{9} - \frac{1}{45} a^{8} + \frac{1}{180} a^{7} + \frac{1}{10} a^{6} - \frac{47}{180} a^{5} + \frac{61}{120} a^{4} - \frac{217}{360} a^{3} + \frac{367}{180} a^{2} - \frac{16}{9} a - \frac{46}{45}\) with minimal polynomial \(x^{6} - 3 x^{5} + 9 x^{4} - 6 x^{3} - 24 x^{2} + 72 x - 48\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z [\sqrt{3}]\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{3}) \) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 6.0.12252303.1 with generator \(\frac{77}{50520} a^{11} - \frac{847}{101040} a^{10} + \frac{1337}{101040} a^{9} + \frac{7}{2105} a^{8} - \frac{1039}{10104} a^{7} + \frac{7217}{25260} a^{6} - \frac{700}{1263} a^{5} + \frac{24283}{33680} a^{4} - \frac{9219}{6736} a^{3} + \frac{73759}{50520} a^{2} + \frac{45623}{25260} a - \frac{7126}{6315}\) with minimal polynomial \(x^{6} + 7\):
| \(\End (J_{F})\) | \(\simeq\) | 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 \(F \simeq \) 6.0.12252303.1 with generator \(\frac{77}{101040} a^{11} - \frac{1897}{101040} a^{10} + \frac{2009}{25260} a^{9} - \frac{2425}{10104} a^{8} + \frac{3017}{6315} a^{7} - \frac{7987}{10104} a^{6} + \frac{32907}{33680} a^{5} - \frac{9065}{6736} a^{4} + \frac{18683}{16840} a^{3} + \frac{23597}{25260} a^{2} - \frac{349}{6315} a - \frac{6931}{6315}\) with minimal polynomial \(x^{6} + 7\):
| \(\End (J_{F})\) | \(\simeq\) | 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 \(F \simeq \) 6.0.12252303.1 with generator \(\frac{77}{101040} a^{11} + \frac{35}{3368} a^{10} - \frac{2233}{33680} a^{9} + \frac{12293}{50520} a^{8} - \frac{9777}{16840} a^{7} + \frac{18123}{16840} a^{6} - \frac{154721}{101040} a^{5} + \frac{8701}{4210} a^{4} - \frac{83461}{33680} a^{3} + \frac{1771}{3368} a^{2} + \frac{15673}{8420} a - \frac{13}{421}\) with minimal polynomial \(x^{6} + 7\):
| \(\End (J_{F})\) | \(\simeq\) | 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