Properties

Label 11664.a.11664.1
Conductor 11664
Discriminant 11664
Mordell-Weil group \(\Z \times \Z/{3}\Z\)
Sato-Tate group $D_{6,2}$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\mathrm{M}_2(\C)\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{M}_2(\mathrm{CM})\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

Related objects

Learn more about

Show commands for: Magma / SageMath

Minimal equation

Minimal equation

Simplified equation

$y^2 + y = -x^6$ (homogenize, simplify)
$y^2 + z^3y = -x^6$ (dehomogenize, simplify)
$y^2 = -4x^6 + 1$ (minimize, homogenize)

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]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([1, 0, 0, 0, 0, 0, -4]))
 

Invariants

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

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

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

magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
 
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 

Automorphism group

\(\mathrm{Aut}(X)\)\(\simeq\) $C_2^2$
magma: AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $C_3:D_4$
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 

Rational points

All points: \((0 : 0 : 1),\, (0 : -1 : 1)\)

magma: [C![0,-1,1],C![0,0,1]];
 

Number of rational Weierstrass points: \(0\)

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));
 

This curve is locally solvable everywhere.

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

Mordell-Weil group of the Jacobian:

Group structure: \(\Z \times \Z/{3}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(x^2 + xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(xz^2\) \(0.900641\) \(\infty\)
\(2 \cdot(0 : -1 : 1) - D_\infty\) \(x^2\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0\) \(3\)

2-torsion field: 3.1.108.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(1\)
Regulator: \( 0.900641 \)
Real period: \( 10.21623 \)
Tamagawa product: \( 1 \)
Torsion order:\( 3 \)
Leading coefficient: \( 1.022351 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(4\) \(4\) \(1\) \(1 + 2 T^{2}\)
\(3\) \(6\) \(6\) \(1\) \(1\)

Sato-Tate group

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

Decomposition of the Jacobian

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 the Jacobian

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