Properties

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

Related objects

Show commands: Magma / SageMath

Simplified equation

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

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

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

sage: X = HyperellipticCurve(R([1, 0, 0, 0, 0, 0, -4]))

magma: X,pi:= SimplifiedModel(C);

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);

G2 invariants

 $$I_2$$ $$=$$ $$40$$ $$=$$ $$2^{3} \cdot 5$$ $$I_4$$ $$=$$ $$45$$ $$=$$ $$3^{2} \cdot 5$$ $$I_6$$ $$=$$ $$555$$ $$=$$ $$3 \cdot 5 \cdot 37$$ $$I_{10}$$ $$=$$ $$6$$ $$=$$ $$2 \cdot 3$$ $$J_2$$ $$=$$ $$120$$ $$=$$ $$2^{3} \cdot 3 \cdot 5$$ $$J_4$$ $$=$$ $$330$$ $$=$$ $$2 \cdot 3 \cdot 5 \cdot 11$$ $$J_6$$ $$=$$ $$-320$$ $$=$$ $$- 2^{6} \cdot 5$$ $$J_8$$ $$=$$ $$-36825$$ $$=$$ $$- 3 \cdot 5^{2} \cdot 491$$ $$J_{10}$$ $$=$$ $$11664$$ $$=$$ $$2^{4} \cdot 3^{6}$$ $$g_1$$ $$=$$ $$6400000/3$$ $$g_2$$ $$=$$ $$440000/9$$ $$g_3$$ $$=$$ $$-32000/81$$

sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]

magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);

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

magma: [C![0,-1,1],C![0,0,1]]; // minimal model

magma: [C![0,-1,1],C![0,1,1]]; // simplified model

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: $$\Z \oplus \Z/{3}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

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

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 Cluster picture
$$2$$ $$4$$ $$4$$ $$1$$ $$1 + 2 T^{2}$$
$$3$$ $$6$$ $$6$$ $$1$$ $$1$$

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?
$$2$$ 2.120.2 no
$$3$$ 3.8640.7 yes

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 curve isogeny classes:
Elliptic curve isogeny class 432.d
Elliptic curve isogeny class 27.a

magma: HeuristicDecompositionFactors(C);

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

magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

magma: HeuristicIsGL2(C); HeuristicEndomorphismDescription(C); HeuristicEndomorphismFieldOfDefinition(C);

magma: HeuristicIsGL2(C : Geometric := true); HeuristicEndomorphismDescription(C : Geometric := true); HeuristicEndomorphismLatticeDescription(C);