Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + 1)y = x^3 + 1$ | (homogenize, simplify) |
$y^2 + (x^3 + z^3)y = x^3z^3 + z^6$ | (dehomogenize, simplify) |
$y^2 = x^6 + 6x^3 + 5$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(6075\) | \(=\) | \( 3^{5} \cdot 5^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(18225\) | \(=\) | \( 3^{6} \cdot 5^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(164\) | \(=\) | \( 2^{2} \cdot 41 \) |
\( I_4 \) | \(=\) | \(2745\) | \(=\) | \( 3^{2} \cdot 5 \cdot 61 \) |
\( I_6 \) | \(=\) | \(106365\) | \(=\) | \( 3 \cdot 5 \cdot 7 \cdot 1013 \) |
\( I_{10} \) | \(=\) | \(-9600\) | \(=\) | \( - 2^{7} \cdot 3 \cdot 5^{2} \) |
\( J_2 \) | \(=\) | \(123\) | \(=\) | \( 3 \cdot 41 \) |
\( J_4 \) | \(=\) | \(-399\) | \(=\) | \( - 3 \cdot 7 \cdot 19 \) |
\( J_6 \) | \(=\) | \(-409\) | \(=\) | \( -409 \) |
\( J_8 \) | \(=\) | \(-52377\) | \(=\) | \( - 3 \cdot 13 \cdot 17 \cdot 79 \) |
\( J_{10} \) | \(=\) | \(-18225\) | \(=\) | \( - 3^{6} \cdot 5^{2} \) |
\( g_1 \) | \(=\) | \(-115856201/75\) | ||
\( g_2 \) | \(=\) | \(9166493/225\) | ||
\( g_3 \) | \(=\) | \(687529/2025\) |
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: \(1\)
This curve is locally solvable everywhere.
Mordell-Weil group of the Jacobian
Group structure: \(\Z/{6}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 0 : 1) - (1 : 0 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0\) | \(6\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 0 : 1) - (1 : 0 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0\) | \(6\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 0 : 1) - (1 : 1 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0\) | \(6\) |
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(1\) |
Regulator: | \( 1 \) |
Real period: | \( 15.57466 \) |
Tamagawa product: | \( 2 \) |
Torsion order: | \( 6 \) |
Leading coefficient: | \( 0.865259 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(3\) | \(5\) | \(6\) | \(2\) | \(1\) | |
\(5\) | \(2\) | \(2\) | \(1\) | \(1 + 5 T^{2}\) |
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? |
---|---|---|
\(2\) | 2.120.1 | yes |
\(3\) | 3.2880.4 | 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 \(\Q (b) \simeq \) 6.2.2278125.1 with defining polynomial:
\(x^{6} - 5\)
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
\(g_4 = 108 b^{5} + 225 b^{2}\)
\(g_6 = -5670 b^{3} - 9855\)
Conductor norm: 81
\(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
\(g_4 = -108 b^{5} + 225 b^{2}\)
\(g_6 = 5670 b^{3} - 9855\)
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 \(x^{12} - 6 x^{11} + 27 x^{10} - 80 x^{9} + 195 x^{8} - 366 x^{7} + 551 x^{6} - 642 x^{5} + 585 x^{4} - 400 x^{3} + 177 x^{2} - 42 x + 4\)
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 \(\frac{433}{184} a^{11} - \frac{4763}{368} a^{10} + \frac{21035}{368} a^{9} - \frac{58935}{368} a^{8} + \frac{70075}{184} a^{7} - \frac{62209}{92} a^{6} + \frac{178651}{184} a^{5} - \frac{384865}{368} a^{4} + \frac{324585}{368} a^{3} - \frac{194285}{368} a^{2} + \frac{32181}{184} a - \frac{2031}{92}\) 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 \(\frac{93}{92} a^{11} - \frac{1023}{184} a^{10} + \frac{4499}{184} a^{9} - \frac{12573}{184} a^{8} + \frac{14871}{92} a^{7} - \frac{6573}{23} a^{6} + \frac{37407}{92} a^{5} - \frac{79797}{184} a^{4} + \frac{66117}{184} a^{3} - \frac{38823}{184} a^{2} + \frac{5841}{92} a - \frac{257}{46}\) 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 \(-\frac{19}{16} a^{10} + \frac{95}{16} a^{9} - \frac{413}{16} a^{8} + \frac{541}{8} a^{7} - \frac{629}{4} a^{6} + 260 a^{5} - \frac{5673}{16} a^{4} + \frac{5485}{16} a^{3} - \frac{4255}{16} a^{2} + \frac{1027}{8} a - \frac{85}{4}\) 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 \(F \simeq \) 3.1.675.1 with generator \(\frac{291}{92} a^{11} - \frac{6241}{368} a^{10} + \frac{27445}{368} a^{9} - \frac{75591}{368} a^{8} + \frac{89331}{184} a^{7} - \frac{77923}{92} a^{6} + \frac{110397}{92} a^{5} - \frac{463635}{368} a^{4} + \frac{381951}{368} a^{3} - \frac{218125}{368} a^{2} + \frac{31917}{184} a - \frac{1615}{92}\) with minimal polynomial \(x^{3} - 5\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R\) |
Not of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 3.1.675.1 with generator \(-\frac{291}{92} a^{11} + \frac{6563}{368} a^{10} - \frac{29055}{368} a^{9} + \frac{82629}{368} a^{8} - \frac{98577}{184} a^{7} + \frac{88733}{92} a^{6} - \frac{128337}{92} a^{5} + \frac{562305}{368} a^{4} - \frac{478045}{368} a^{3} + \frac{292967}{368} a^{2} - \frac{49995}{184} a + \frac{3225}{92}\) with minimal polynomial \(x^{3} - 5\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R\) |
Not of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 3.1.675.1 with generator \(-\frac{7}{8} a^{10} + \frac{35}{8} a^{9} - \frac{153}{8} a^{8} + \frac{201}{4} a^{7} - \frac{235}{2} a^{6} + 195 a^{5} - \frac{2145}{8} a^{4} + \frac{2089}{8} a^{3} - \frac{1627}{8} a^{2} + \frac{393}{4} a - \frac{35}{2}\) with minimal polynomial \(x^{3} - 5\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R\) |
Not of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) \(\Q(\sqrt{-3}, \sqrt{5})\) with generator \(\frac{247}{368} a^{11} - \frac{285}{92} a^{10} + \frac{2463}{184} a^{9} - \frac{12145}{368} a^{8} + \frac{13945}{184} a^{7} - \frac{10725}{92} a^{6} + \frac{55997}{368} a^{5} - \frac{11849}{92} a^{4} + \frac{16549}{184} a^{3} - \frac{9387}{368} a^{2} - \frac{1561}{184} a + \frac{265}{92}\) 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 \(F \simeq \) 6.0.1366875.1 with generator \(-\frac{149}{184} a^{11} + \frac{739}{184} a^{10} - \frac{3205}{184} a^{9} + \frac{1041}{23} a^{8} - \frac{2407}{23} a^{7} + \frac{7857}{46} a^{6} - \frac{42143}{184} a^{5} + \frac{39385}{184} a^{4} - \frac{28683}{184} a^{3} + \frac{1490}{23} a^{2} + \frac{33}{23} a - \frac{104}{23}\) with minimal polynomial \(x^{6} - 3 x^{5} + 6 x^{4} + 3 x^{3} - 9 x^{2} - 18 x + 36\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 6.0.6834375.1 with generator \(\frac{591}{368} a^{11} - \frac{201}{23} a^{10} + \frac{7071}{184} a^{9} - \frac{39289}{368} a^{8} + \frac{46449}{184} a^{7} - \frac{40837}{92} a^{6} + \frac{231869}{368} a^{5} - \frac{15346}{23} a^{4} + \frac{101237}{184} a^{3} - \frac{116763}{368} a^{2} + \frac{17035}{184} a - \frac{699}{92}\) with minimal polynomial \(x^{6} - 3 x^{5} - 5 x^{3} + 15 x^{2} + 12 x + 4\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\sqrt{3}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{3}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 6.2.2278125.1 with generator \(\frac{167}{92} a^{11} - \frac{3789}{368} a^{10} + \frac{16797}{368} a^{9} - \frac{47951}{368} a^{8} + \frac{57311}{184} a^{7} - \frac{51791}{92} a^{6} + \frac{75195}{92} a^{5} - \frac{331527}{368} a^{4} + \frac{284023}{368} a^{3} - \frac{176605}{368} a^{2} + \frac{31153}{184} a - \frac{2187}{92}\) with minimal polynomial \(x^{6} - 5\):
\(\End (J_{F})\) | \(\simeq\) | 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 \(F \simeq \) 6.2.2278125.1 with generator \(\frac{167}{46} a^{11} - \frac{1837}{92} a^{10} + \frac{8111}{92} a^{9} - \frac{11361}{46} a^{8} + \frac{27011}{46} a^{7} - \frac{23975}{23} a^{6} + \frac{34412}{23} a^{5} - \frac{148203}{92} a^{4} + \frac{124865}{92} a^{3} - \frac{18663}{23} a^{2} + \frac{12207}{46} a - \frac{760}{23}\) with minimal polynomial \(x^{6} - 5\):
\(\End (J_{F})\) | \(\simeq\) | 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 \(F \simeq \) 6.2.2278125.1 with generator \(\frac{167}{92} a^{11} - \frac{3559}{368} a^{10} + \frac{15647}{368} a^{9} - \frac{42937}{368} a^{8} + \frac{50733}{184} a^{7} - \frac{44109}{92} a^{6} + \frac{62453}{92} a^{5} - \frac{261285}{368} a^{4} + \frac{215437}{368} a^{3} - \frac{122003}{368} a^{2} + \frac{17675}{184} a - \frac{853}{92}\) with minimal polynomial \(x^{6} - 5\):
\(\End (J_{F})\) | \(\simeq\) | 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 \(F \simeq \) 6.0.6834375.1 with generator \(-\frac{573}{368} a^{11} + \frac{739}{92} a^{10} - \frac{6479}{184} a^{9} + \frac{34807}{368} a^{8} - \frac{40927}{184} a^{7} + \frac{34809}{92} a^{6} - \frac{194631}{368} a^{5} + \frac{49321}{92} a^{4} - \frac{79561}{184} a^{3} + \frac{84549}{368} a^{2} - \frac{10581}{184} a + \frac{433}{92}\) with minimal polynomial \(x^{6} - 3 x^{5} - 5 x^{3} + 15 x^{2} + 12 x + 4\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\sqrt{3}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{3}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 6.0.6834375.1 with generator \(-\frac{1}{8} a^{10} + \frac{5}{8} a^{9} - \frac{21}{8} a^{8} + \frac{27}{4} a^{7} - 15 a^{6} + 24 a^{5} - \frac{239}{8} a^{4} + \frac{211}{8} a^{3} - \frac{135}{8} a^{2} + \frac{27}{4} a + \frac{3}{2}\) with minimal polynomial \(x^{6} - 3 x^{5} - 5 x^{3} + 15 x^{2} + 12 x + 4\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\sqrt{3}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{3}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Of \(\GL_2\)-type, simple