Properties

Label 11664.a.11664.1
Conductor 11664
Discriminant 11664
Sato-Tate group $D_{6,2}$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\mathrm{M}_2(\C)\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

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Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, 0, 0, 0, 0, -1], R![1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, 0, 0, 0, 0, -1]), R([1]))

$y^2 + y = -x^6$

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
\( N \)  =  \( 11664 \)  =  \( 2^{4} \cdot 3^{6} \)
magma: Discriminant(C); Factorization(Integers()!$1);
\( \Delta \)  =  \(11664\)  =  \( 2^{4} \cdot 3^{6} \)

Igusa-Clebsch invariants

magma: IgusaClebschInvariants(C); [Factorization(Integers()!a): a in $1];
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]

Igusa invariants

magma: IgusaInvariants(C); [Factorization(Integers()!a): a in $1];

G2 invariants

magma: G2Invariants(C);

\( I_2 \)  =  \(960\)  =  \( 2^{6} \cdot 3 \cdot 5 \)
\( I_4 \)  =  \(25920\)  =  \( 2^{6} \cdot 3^{4} \cdot 5 \)
\( I_6 \)  =  \(7672320\)  =  \( 2^{9} \cdot 3^{4} \cdot 5 \cdot 37 \)
\( I_{10} \)  =  \(47775744\)  =  \( 2^{16} \cdot 3^{6} \)
\( J_2 \)  =  \(120\)  =  \( 2^{3} \cdot 3 \cdot 5 \)
\( J_4 \)  =  \(330\)  =  \( 2 \cdot 3 \cdot 5 \cdot 11 \)
\( J_6 \)  =  \(-320\)  =  \( -1 \cdot 2^{6} \cdot 5 \)
\( J_8 \)  =  \(-36825\)  =  \( -1 \cdot 3 \cdot 5^{2} \cdot 491 \)
\( J_{10} \)  =  \(11664\)  =  \( 2^{4} \cdot 3^{6} \)
\( g_1 \)  =  \(6400000/3\)
\( g_2 \)  =  \(440000/9\)
\( g_3 \)  =  \(-32000/81\)
Alternative geometric invariants: Igusa-Clebsch, Igusa, G2

Automorphism group

magma: AutomorphismGroup(C); IdentifyGroup($1);
\(\mathrm{Aut}(X)\)\(\simeq\) \(V_4 \) (GAP id : [4,2])
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) \(2D_6 \) (GAP id : [24,8])
magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));

Number of rational Weierstrass points:

\(0\)
magma: f,h:=HyperellipticPolynomials(C); g:=4*f+h^2; HasPointsLocallyEverywhere(g,2) and (#Roots(ChangeRing(g,RealField())) gt 0 or LeadingCoefficient(g) gt 0);

Locally solvable:

yes

Invariants of the Jacobian:

Analytic rank*:

\(1\)
magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1);

2-Selmer rank:

\(1\)
magma: HasSquareSha(Jacobian(C));

Order of Ш*:

square
magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

Torsion:

\(\Z/{3}\Z\)

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $D_{6,2}$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{U}(1)\)

Decomposition

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
  Elliptic curve 432.d1
  Elliptic curve 27.a4

Endomorphisms

of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):
\(\End (J_{})\)\(\simeq\)an order of index \(2\) in \(\Z \times \Z\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R \times \R\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a)\) with defining polynomial \(x^{12} - 3 x^{10} - 8 x^{9} - 6 x^{8} + 12 x^{7} + 47 x^{6} + 78 x^{5} + 78 x^{4} + 50 x^{3} + 21 x^{2} + 6 x + 1\)

not of \(\GL_2\)-type over \(\overline{\Q}\)

Endomorphism ring over \(\overline{\Q}\):
\(\End (J_{\overline{\Q}})\)\(\simeq\)a non-Eichler order of index \(16\) in a maximal order of \(\End (J_{\overline{\Q}}) \otimes \Q\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)\(\simeq\)\(\mathrm{M}_2(\)\(\Q(\sqrt{-3}) \)\()\)
\(\End (J_{\overline{\Q}}) \otimes \R\)\(\simeq\) \(\mathrm{M}_2 (\C)\)

Remainder of the endomorphism lattice by field

Over subfield \(F \simeq \) \(\Q(\sqrt{-3}) \) with generator \(-\frac{384}{59} a^{11} + \frac{296}{59} a^{10} + \frac{914}{59} a^{9} + \frac{2365}{59} a^{8} + \frac{524}{59} a^{7} - \frac{4948}{59} a^{6} - \frac{14170}{59} a^{5} - \frac{19194}{59} a^{4} - \frac{15636}{59} a^{3} - \frac{7870}{59} a^{2} - \frac{2590}{59} a - \frac{546}{59}\) with minimal polynomial \(x^{2} - x + 1\):
\(\End (J_{F})\)\(\simeq\)an order of index \(4\) in \(\Z [\frac{1 + \sqrt{-3}}{2}] \times \Z [\frac{1 + \sqrt{-3}}{2}]\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{-3}) \) \(\times\) \(\Q(\sqrt{-3}) \)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\C \times \C\)
Sato Tate group: $C_6$
not of \(\GL_2\)-type, not simple

Over subfield \(F \simeq \) \(\Q(\sqrt{3}) \) with generator \(-\frac{23}{5} a^{11} + \frac{11}{5} a^{10} + 13 a^{9} + 30 a^{8} + \frac{67}{5} a^{7} - \frac{314}{5} a^{6} - \frac{926}{5} a^{5} - 268 a^{4} - 226 a^{3} - \frac{589}{5} a^{2} - \frac{186}{5} a - \frac{36}{5}\) with minimal polynomial \(x^{2} - 3\):
\(\End (J_{F})\)\(\simeq\)an order of index \(2\) in \(\Z \times \Z\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q\)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\R \times \R\)
Sato Tate group: $D_{3,2}$
of \(\GL_2\)-type, not simple

Over subfield \(F \simeq \) \(\Q(\sqrt{-1}) \) with generator \(-\frac{2499}{295} a^{11} + \frac{327}{59} a^{10} + \frac{6469}{295} a^{9} + \frac{15786}{295} a^{8} + \frac{897}{59} a^{7} - \frac{33294}{295} a^{6} - \frac{19200}{59} a^{5} - \frac{131424}{295} a^{4} - \frac{106596}{295} a^{3} - \frac{51411}{295} a^{2} - \frac{15678}{295} a - \frac{3344}{295}\) with minimal polynomial \(x^{2} + 1\):
\(\End (J_{F})\)\(\simeq\)an order of index \(2\) in \(\Z \times \Z\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q\)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\R \times \R\)
Sato Tate group: $D_{3,2}$
of \(\GL_2\)-type, not simple

Over subfield \(F \simeq \) 3.1.108.1 with generator \(\frac{137}{59} a^{11} - \frac{131}{295} a^{10} - \frac{2147}{295} a^{9} - \frac{4903}{295} a^{8} - \frac{2977}{295} a^{7} + \frac{9336}{295} a^{6} + \frac{30161}{295} a^{5} + \frac{46267}{295} a^{4} + \frac{41313}{295} a^{3} + \frac{22837}{295} a^{2} + \frac{1601}{59} a + \frac{1788}{295}\) with minimal polynomial \(x^{3} - 2\):
\(\End (J_{F})\)\(\simeq\)an order of index \(2\) in \(\Z \times \Z\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q\)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\R \times \R\)
Sato Tate group: $D_{2,1}$
of \(\GL_2\)-type, not simple

Over subfield \(F \simeq \) 3.1.108.1 with generator \(-\frac{49}{5} a^{11} + \frac{28}{5} a^{10} + 26 a^{9} + 64 a^{8} + \frac{111}{5} a^{7} - \frac{647}{5} a^{6} - \frac{1938}{5} a^{5} - 545 a^{4} - 457 a^{3} - \frac{1157}{5} a^{2} - \frac{368}{5} a - \frac{78}{5}\) with minimal polynomial \(x^{3} - 2\):
\(\End (J_{F})\)\(\simeq\)an order of index \(2\) in \(\Z \times \Z\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q\)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\R \times \R\)
Sato Tate group: $D_{2,1}$
of \(\GL_2\)-type, not simple

Over subfield \(F \simeq \) 3.1.108.1 with generator \(\frac{2206}{295} a^{11} - \frac{1521}{295} a^{10} - \frac{5523}{295} a^{9} - \frac{13977}{295} a^{8} - \frac{3572}{295} a^{7} + \frac{28837}{295} a^{6} + \frac{84181}{295} a^{5} + \frac{114508}{295} a^{4} + \frac{93502}{295} a^{3} + \frac{45426}{295} a^{2} + \frac{13707}{295} a + \frac{2814}{295}\) with minimal polynomial \(x^{3} - 2\):
\(\End (J_{F})\)\(\simeq\)an order of index \(2\) in \(\Z \times \Z\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q\)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\R \times \R\)
Sato Tate group: $D_{2,1}$
of \(\GL_2\)-type, not simple

Over subfield \(F \simeq \) \(\Q(\zeta_{12})\) with generator \(-\frac{571}{295} a^{11} + \frac{493}{295} a^{10} + \frac{1317}{295} a^{9} + \frac{3468}{295} a^{8} + \frac{266}{295} a^{7} - \frac{7384}{295} a^{6} - \frac{20683}{295} a^{5} - \frac{26182}{295} a^{4} - \frac{19963}{295} a^{3} - \frac{1666}{59} a^{2} - \frac{2352}{295} a - \frac{122}{59}\) with minimal polynomial \(x^{4} - x^{2} + 1\):
\(\End (J_{F})\)\(\simeq\)an order of index \(4\) in \(\Z [\frac{1 + \sqrt{-3}}{2}] \times \Z [\frac{1 + \sqrt{-3}}{2}]\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{-3}) \) \(\times\) \(\Q(\sqrt{-3}) \)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\C \times \C\)
Sato Tate group: $C_3$
not of \(\GL_2\)-type, not simple

Over subfield \(F \simeq \) 6.0.34992.1 with generator \(\frac{28}{59} a^{11} - \frac{457}{295} a^{10} + \frac{81}{295} a^{9} - \frac{161}{295} a^{8} + \frac{1606}{295} a^{7} + \frac{1477}{295} a^{6} + \frac{617}{295} a^{5} - \frac{4061}{295} a^{4} - \frac{6564}{295} a^{3} - \frac{5006}{295} a^{2} - \frac{277}{59} a + \frac{11}{295}\) with minimal polynomial \(x^{6} - 3 x^{5} + 5 x^{3} - 3 x + 1\):
\(\End (J_{F})\)\(\simeq\)an order of index \(4\) in \(\Z [\frac{1 + \sqrt{-3}}{2}] \times \Z [\frac{1 + \sqrt{-3}}{2}]\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{-3}) \) \(\times\) \(\Q(\sqrt{-3}) \)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\C \times \C\)
Sato Tate group: $C_2$
not of \(\GL_2\)-type, not simple

Over subfield \(F \simeq \) 6.2.559872.1 with generator \(\frac{2132}{295} a^{11} - \frac{1786}{295} a^{10} - \frac{5184}{295} a^{9} - \frac{12411}{295} a^{8} - \frac{1717}{295} a^{7} + \frac{28428}{295} a^{6} + \frac{76211}{295} a^{5} + \frac{98539}{295} a^{4} + \frac{74651}{295} a^{3} + \frac{6717}{59} a^{2} + \frac{10229}{295} a + \frac{461}{59}\) with minimal polynomial \(x^{6} - 3 x^{4} - 2 x^{3} + 3 x^{2} + 6 x + 1\):
\(\End (J_{F})\)\(\simeq\)a non-Eichler order of index \(12\) in a maximal order of \(\End (J_{F}) \otimes \Q\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\mathrm{M}_2(\)\(\Q\)\()\)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\mathrm{M}_2 (\R)\)
Sato Tate group: $C_{2,1}$
not of \(\GL_2\)-type, not simple

Over subfield \(F \simeq \) 6.2.559872.1 with generator \(-\frac{276}{295} a^{11} + \frac{198}{295} a^{10} + \frac{727}{295} a^{9} + \frac{1698}{295} a^{8} + \frac{266}{295} a^{7} - \frac{3844}{295} a^{6} - \frac{10358}{295} a^{5} - \frac{13497}{295} a^{4} - \frac{9638}{295} a^{3} - \frac{781}{59} a^{2} - \frac{582}{295} a - \frac{63}{59}\) with minimal polynomial \(x^{6} - 3 x^{4} - 2 x^{3} + 3 x^{2} + 6 x + 1\):
\(\End (J_{F})\)\(\simeq\)a non-Eichler order of index \(12\) in a maximal order of \(\End (J_{F}) \otimes \Q\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\mathrm{M}_2(\)\(\Q\)\()\)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\mathrm{M}_2 (\R)\)
Sato Tate group: $C_{2,1}$
not of \(\GL_2\)-type, not simple

Over subfield \(F \simeq \) 6.2.559872.1 with generator \(-a^{11} + a^{10} + 2 a^{9} + 6 a^{8} - 12 a^{6} - 35 a^{5} - 43 a^{4} - 35 a^{3} - 15 a^{2} - 5 a - 1\) with minimal polynomial \(x^{6} - 3 x^{4} - 2 x^{3} + 3 x^{2} + 6 x + 1\):
\(\End (J_{F})\)\(\simeq\)a non-Eichler order of index \(12\) in a maximal order of \(\End (J_{F}) \otimes \Q\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\mathrm{M}_2(\)\(\Q\)\()\)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\mathrm{M}_2 (\R)\)
Sato Tate group: $C_{2,1}$
not of \(\GL_2\)-type, not simple

Over subfield \(F \simeq \) 6.0.186624.1 with generator \(\frac{249}{59} a^{11} - \frac{897}{295} a^{10} - \frac{2944}{295} a^{9} - \frac{7966}{295} a^{8} - \frac{2099}{295} a^{7} + \frac{15657}{295} a^{6} + \frac{47202}{295} a^{5} + \frac{65069}{295} a^{4} + \frac{55236}{295} a^{3} + \frac{28714}{295} a^{2} + \frac{1968}{59} a + \frac{2186}{295}\) with minimal polynomial \(x^{6} - 2 x^{3} + 2\):
\(\End (J_{F})\)\(\simeq\)a non-Eichler order of index \(4\) in a maximal order of \(\End (J_{F}) \otimes \Q\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\mathrm{M}_2(\)\(\Q\)\()\)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\mathrm{M}_2 (\R)\)
Sato Tate group: $C_{2,1}$
not of \(\GL_2\)-type, not simple

Over subfield \(F \simeq \) 6.0.186624.1 with generator \(-\frac{106}{59} a^{11} + \frac{588}{295} a^{10} + \frac{1181}{295} a^{9} + \frac{2704}{295} a^{8} - \frac{399}{295} a^{7} - \frac{7273}{295} a^{6} - \frac{16913}{295} a^{5} - \frac{19196}{295} a^{4} - \frac{11739}{295} a^{3} - \frac{3081}{295} a^{2} - \frac{85}{59} a - \frac{29}{295}\) with minimal polynomial \(x^{6} - 2 x^{3} + 2\):
\(\End (J_{F})\)\(\simeq\)a non-Eichler order of index \(4\) in a maximal order of \(\End (J_{F}) \otimes \Q\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\mathrm{M}_2(\)\(\Q\)\()\)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\mathrm{M}_2 (\R)\)
Sato Tate group: $C_{2,1}$
not of \(\GL_2\)-type, not simple

Over subfield \(F \simeq \) 6.0.186624.1 with generator \(\frac{1293}{295} a^{11} - \frac{816}{295} a^{10} - \frac{709}{59} a^{9} - \frac{1583}{59} a^{8} - \frac{2312}{295} a^{7} + \frac{17839}{295} a^{6} + \frac{49401}{295} a^{5} + \frac{13414}{59} a^{4} + \frac{10559}{59} a^{3} + \frac{24904}{295} a^{2} + \frac{7671}{295} a + \frac{1856}{295}\) with minimal polynomial \(x^{6} - 2 x^{3} + 2\):
\(\End (J_{F})\)\(\simeq\)a non-Eichler order of index \(4\) in a maximal order of \(\End (J_{F}) \otimes \Q\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\mathrm{M}_2(\)\(\Q\)\()\)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\mathrm{M}_2 (\R)\)
Sato Tate group: $C_{2,1}$
not of \(\GL_2\)-type, not simple