LMFDB - Genus 2 curve 11664.a.11664.1

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 $	$=$	$-32000/81$

(Igusa-Clebsch invariants, (Igusa invariants, Igusa invariants) G2 invariants)

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

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

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	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($$\Q$$)$
$\End (J_{F}) \otimes \R$	$\simeq$	$\mathrm{M}_2 (\R)$