# 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

# Related objects

Show commands for: Magma / SageMath

## 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); $\Delta$ = $11664$ = $2^{4} \cdot 3^{6}$

### 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: 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);

yes

## Invariants of the Jacobian:

### Analytic rank*:

$1$

### 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