Minimal equation
Minimal equation
Simplified equation
| $y^2 + (x^3 + 1)y = -x^6 + 2x^3 - 2$ | (homogenize, simplify) |
| $y^2 + (x^3 + z^3)y = -x^6 + 2x^3z^3 - 2z^6$ | (dehomogenize, simplify) |
| $y^2 = -3x^6 + 10x^3 - 7$ | (homogenize, minimize) |
Invariants
| Conductor: | \( N \) | \(=\) | \(107163\) | \(=\) | \( 3^{7} \cdot 7^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(321489\) | \(=\) | \( 3^{8} \cdot 7^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(740\) | \(=\) | \( 2^{2} \cdot 5 \cdot 37 \) |
| \( I_4 \) | \(=\) | \(38745\) | \(=\) | \( 3^{3} \cdot 5 \cdot 7 \cdot 41 \) |
| \( I_6 \) | \(=\) | \(7033005\) | \(=\) | \( 3^{2} \cdot 5 \cdot 7 \cdot 83 \cdot 269 \) |
| \( I_{10} \) | \(=\) | \(-169344\) | \(=\) | \( - 2^{7} \cdot 3^{3} \cdot 7^{2} \) |
| \( J_2 \) | \(=\) | \(555\) | \(=\) | \( 3 \cdot 5 \cdot 37 \) |
| \( J_4 \) | \(=\) | \(-1695\) | \(=\) | \( - 3 \cdot 5 \cdot 113 \) |
| \( J_6 \) | \(=\) | \(-1705\) | \(=\) | \( - 5 \cdot 11 \cdot 31 \) |
| \( J_8 \) | \(=\) | \(-954825\) | \(=\) | \( - 3 \cdot 5^{2} \cdot 29 \cdot 439 \) |
| \( J_{10} \) | \(=\) | \(-321489\) | \(=\) | \( - 3^{8} \cdot 7^{2} \) |
| \( g_1 \) | \(=\) | \(-216699865625/1323\) | ||
| \( g_2 \) | \(=\) | \(3577368125/3969\) | ||
| \( g_3 \) | \(=\) | \(58353625/35721\) |
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: \(\Z \oplus \Z/{2}\Z\)
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - D_\infty\) | \(x^2 + xz + 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2\) | \(1.321347\) | \(\infty\) |
| \(D_0 - D_\infty\) | \(x^2 + xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0\) | \(2\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - D_\infty\) | \(x^2 + xz + 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2\) | \(1.321347\) | \(\infty\) |
| \(D_0 - D_\infty\) | \(x^2 + xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0\) | \(2\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - D_\infty\) | \(x^2 + xz + 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + 2xz^2 + z^3\) | \(1.321347\) | \(\infty\) |
| \(D_0 - D_\infty\) | \(x^2 + xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - z^3\) | \(0\) | \(2\) |
BSD invariants
| Hasse-Weil conjecture: | unverified |
| Analytic rank: | \(1\) |
| Mordell-Weil rank: | \(1\) |
| 2-Selmer rank: | \(2\) |
| Regulator: | \( 1.321347 \) |
| Real period: | \( 3.316199 \) |
| Tamagawa product: | \( 2 \) |
| Torsion order: | \( 2 \) |
| Leading coefficient: | \( 2.190926 \) |
| Analytic order of Ш: | \( 1 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(3\) | \(7\) | \(8\) | \(2\) | \(1\) |  |
| \(7\) | \(2\) | \(2\) | \(1\) | \(1 + 4 T + 7 T^{2}\) |  |
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? |
|---|---|---|
| \(2\) | 2.120.1 | yes |
| \(3\) | 3.960.3 | no |
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 \(\Q (b) \simeq \) 6.2.2977309629.4 with defining polynomial:
\(x^{6} - 84 x^{3} + 63\)
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
\(g_4 = \frac{5}{2} b^{4} + \frac{195}{2} b\)
\(g_6 = 630 b^{3} - 3699\)
Conductor norm: 9
\(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
\(g_4 = -\frac{35}{2} b^{4} + \frac{2955}{2} b\)
\(g_6 = -630 b^{3} + 49221\)
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 \(x^{12} - 6 x^{11} + 21 x^{10} - 50 x^{9} + 90 x^{8} - 126 x^{7} + 267 x^{6} - 504 x^{5} - 855 x^{4} + 2470 x^{3} - 924 x^{2} - 384 x + 4096\)
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 \(-\frac{427}{768960} a^{11} + \frac{4697}{1537920} a^{10} - \frac{17143}{1537920} a^{9} + \frac{3493}{128160} a^{8} - \frac{14657}{256320} a^{7} + \frac{12089}{128160} a^{6} - \frac{7133}{32040} a^{5} + \frac{189203}{512640} a^{4} + \frac{102809}{512640} a^{3} - \frac{489307}{768960} a^{2} + \frac{166061}{384480} a + \frac{4816}{12015}\) 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 \(-\frac{1}{1152} a^{10} + \frac{5}{1152} a^{9} - \frac{1}{72} a^{8} + \frac{17}{576} a^{7} - \frac{7}{144} a^{6} + \frac{35}{576} a^{5} - \frac{133}{1152} a^{4} + \frac{179}{1152} a^{3} + \frac{709}{576} a^{2} - \frac{125}{96} a + \frac{1}{3}\) 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 \(-\frac{7}{16020} a^{11} + \frac{77}{32040} a^{10} - \frac{989}{96120} a^{9} + \frac{151}{5340} a^{8} - \frac{119}{1780} a^{7} + \frac{476}{4005} a^{6} - \frac{112}{445} a^{5} + \frac{4123}{10680} a^{4} - \frac{7433}{32040} a^{3} + \frac{49}{8010} a^{2} + \frac{5968}{4005} a - \frac{2827}{12015}\) 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 \(F \simeq \) 3.1.11907.1 with generator \(-\frac{7}{3456} a^{10} + \frac{35}{3456} a^{9} - \frac{1}{24} a^{8} + \frac{61}{576} a^{7} - \frac{35}{144} a^{6} + \frac{77}{192} a^{5} - \frac{1057}{1152} a^{4} + \frac{1463}{1152} a^{3} - \frac{101}{192} a^{2} - \frac{49}{864} a + \frac{139}{27}\) with minimal polynomial \(x^{3} - 63\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R\) |
Not of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 3.1.11907.1 with generator \(-\frac{917}{1537920} a^{11} + \frac{6601}{1537920} a^{10} - \frac{643}{32040} a^{9} + \frac{16343}{256320} a^{8} - \frac{10129}{64080} a^{7} + \frac{26663}{85440} a^{6} - \frac{303779}{512640} a^{5} + \frac{536389}{512640} a^{4} - \frac{24661}{28480} a^{3} + \frac{35917}{384480} a^{2} + \frac{45569}{12015} a - \frac{17669}{4005}\) with minimal polynomial \(x^{3} - 63\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R\) |
Not of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 3.1.11907.1 with generator \(\frac{917}{1537920} a^{11} - \frac{581}{256320} a^{10} + \frac{15289}{1537920} a^{9} - \frac{5663}{256320} a^{8} + \frac{4457}{85440} a^{7} - \frac{17689}{256320} a^{6} + \frac{98189}{512640} a^{5} - \frac{917}{7120} a^{4} - \frac{207137}{512640} a^{3} + \frac{332671}{768960} a^{2} - \frac{478801}{128160} a - \frac{8848}{12015}\) with minimal polynomial \(x^{3} - 63\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R\) |
Not of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) \(\Q(\sqrt{-3}, \sqrt{-7})\) with generator \(\frac{91}{1537920} a^{11} - \frac{73}{96120} a^{10} + \frac{3997}{1537920} a^{9} - \frac{1649}{256320} a^{8} + \frac{2543}{256320} a^{7} - \frac{343}{28480} a^{6} + \frac{903}{56960} a^{5} - \frac{4207}{85440} a^{4} - \frac{71041}{512640} a^{3} + \frac{720263}{768960} a^{2} - \frac{46879}{384480} a - \frac{1819}{12015}\) 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 \(F \simeq \) 6.0.425329947.3 with generator \(-\frac{917}{4613760} a^{11} + \frac{2429}{1153440} a^{10} - \frac{46439}{4613760} a^{9} + \frac{27023}{768960} a^{8} - \frac{67661}{768960} a^{7} + \frac{142289}{768960} a^{6} - \frac{509369}{1537920} a^{5} + \frac{503377}{768960} a^{4} - \frac{1094933}{1537920} a^{3} + \frac{476339}{2306880} a^{2} + \frac{1480013}{1153440} a - \frac{114862}{36045}\) with minimal polynomial \(x^{6} + 147\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 6.2.2977309629.4 with generator \(\frac{67}{128160} a^{11} - \frac{1919}{512640} a^{10} + \frac{6671}{512640} a^{9} - \frac{2009}{64080} a^{8} + \frac{4609}{85440} a^{7} - \frac{259}{3560} a^{6} + \frac{28679}{256320} a^{5} - \frac{16667}{56960} a^{4} - \frac{98933}{170880} a^{3} + \frac{185533}{85440} a^{2} + \frac{53323}{128160} a - \frac{5582}{4005}\) with minimal polynomial \(x^{6} - 84 x^{3} + 63\):
| \(\End (J_{F})\) | \(\simeq\) | 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 \(F \simeq \) 6.2.2977309629.4 with generator \(-\frac{67}{128160} a^{11} + \frac{343}{170880} a^{10} - \frac{2221}{512640} a^{9} + \frac{229}{64080} a^{8} + \frac{1303}{256320} a^{7} - \frac{98}{4005} a^{6} + \frac{2471}{256320} a^{5} - \frac{53807}{512640} a^{4} + \frac{626989}{512640} a^{3} - \frac{181909}{256320} a^{2} - \frac{280273}{128160} a + \frac{1577}{4005}\) with minimal polynomial \(x^{6} - 84 x^{3} + 63\):
| \(\End (J_{F})\) | \(\simeq\) | 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 \(F \simeq \) 6.2.2977309629.4 with generator \(\frac{1}{576} a^{10} - \frac{5}{576} a^{9} + \frac{1}{36} a^{8} - \frac{17}{288} a^{7} + \frac{7}{72} a^{6} - \frac{35}{288} a^{5} + \frac{229}{576} a^{4} - \frac{371}{576} a^{3} - \frac{421}{288} a^{2} + \frac{85}{48} a + 1\) with minimal polynomial \(x^{6} - 84 x^{3} + 63\):
| \(\End (J_{F})\) | \(\simeq\) | 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 \(F \simeq \) 6.0.992436543.1 with generator \(\frac{427}{768960} a^{11} - \frac{4697}{1537920} a^{10} + \frac{17143}{1537920} a^{9} - \frac{3493}{128160} a^{8} + \frac{14657}{256320} a^{7} - \frac{12089}{128160} a^{6} + \frac{7133}{32040} a^{5} - \frac{189203}{512640} a^{4} - \frac{102809}{512640} a^{3} + \frac{489307}{768960} a^{2} + \frac{218419}{384480} a - \frac{4816}{12015}\) with minimal polynomial \(x^{6} + 63\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z [\sqrt{3}]\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{3}) \) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 6.0.992436543.1 with generator \(\frac{427}{1537920} a^{11} - \frac{791}{1537920} a^{10} + \frac{49}{96120} a^{9} + \frac{1847}{256320} a^{8} - \frac{1561}{64080} a^{7} + \frac{19061}{256320} a^{6} - \frac{45731}{512640} a^{5} + \frac{140581}{512640} a^{4} - \frac{188461}{256320} a^{3} + \frac{31213}{384480} a^{2} + \frac{9761}{12015} a - \frac{39343}{12015}\) with minimal polynomial \(x^{6} + 63\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z [\sqrt{3}]\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{3}) \) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 6.0.992436543.1 with generator \(-\frac{427}{1537920} a^{11} + \frac{217}{85440} a^{10} - \frac{5453}{512640} a^{9} + \frac{8833}{256320} a^{8} - \frac{6967}{85440} a^{7} + \frac{14413}{85440} a^{6} - \frac{159859}{512640} a^{5} + \frac{13741}{21360} a^{4} - \frac{30457}{56960} a^{3} - \frac{426881}{768960} a^{2} + \frac{3479}{14240} a - \frac{11509}{4005}\) with minimal polynomial \(x^{6} + 63\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z [\sqrt{3}]\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{3}) \) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Of \(\GL_2\)-type, simple