Properties

 Label 20736.l.373248.1 Conductor $20736$ Discriminant $373248$ Mordell-Weil group $$\Z/{2}\Z \oplus \Z/{6}\Z$$ Sato-Tate group $J(E_4)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\mathrm{M}_2(\R)$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathsf{QM}$$ $$\End(J) \otimes \Q$$ $$\Q$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type no

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Show commands: Magma / SageMath

Simplified equation

 $y^2 + y = 6x^5 + 9x^4 - x^3 - 3x^2$ (homogenize, simplify) $y^2 + z^3y = 6x^5z + 9x^4z^2 - x^3z^3 - 3x^2z^4$ (dehomogenize, simplify) $y^2 = 24x^5 + 36x^4 - 4x^3 - 12x^2 + 1$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, -3, -1, 9, 6]), R([1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, -3, -1, 9, 6], R![1]);

sage: X = HyperellipticCurve(R([1, 0, -12, -4, 36, 24]))

magma: X,pi:= SimplifiedModel(C);

Invariants

 Conductor: $$N$$ $$=$$ $$20736$$ $$=$$ $$2^{8} \cdot 3^{4}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$373248$$ $$=$$ $$2^{9} \cdot 3^{6}$$ magma: Discriminant(C); Factorization(Integers()!$1);

G2 invariants

 $$I_2$$ $$=$$ $$146$$ $$=$$ $$2 \cdot 73$$ $$I_4$$ $$=$$ $$738$$ $$=$$ $$2 \cdot 3^{2} \cdot 41$$ $$I_6$$ $$=$$ $$29472$$ $$=$$ $$2^{5} \cdot 3 \cdot 307$$ $$I_{10}$$ $$=$$ $$6$$ $$=$$ $$2 \cdot 3$$ $$J_2$$ $$=$$ $$876$$ $$=$$ $$2^{2} \cdot 3 \cdot 73$$ $$J_4$$ $$=$$ $$14262$$ $$=$$ $$2 \cdot 3 \cdot 2377$$ $$J_6$$ $$=$$ $$207364$$ $$=$$ $$2^{2} \cdot 47 \cdot 1103$$ $$J_8$$ $$=$$ $$-5438445$$ $$=$$ $$- 3 \cdot 5 \cdot 37 \cdot 41 \cdot 239$$ $$J_{10}$$ $$=$$ $$373248$$ $$=$$ $$2^{9} \cdot 3^{6}$$ $$g_1$$ $$=$$ $$4146143186/3$$ $$g_2$$ $$=$$ $$924693409/36$$ $$g_3$$ $$=$$ $$276260689/648$$

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

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

Automorphism group

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

Rational points

All points: $$(1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (-1 : -4 : 2)$$
All points: $$(1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (-1 : -4 : 2)$$
All points: $$(1 : 0 : 0),\, (0 : -1 : 1),\, (0 : 1 : 1),\, (-1 : 0 : 2)$$

magma: [C![-1,-4,2],C![0,-1,1],C![0,0,1],C![1,0,0]]; // minimal model

magma: [C![-1,0,2],C![0,-1,1],C![0,1,1],C![1,0,0]]; // simplified model

Number of rational Weierstrass points: $$2$$

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: $$\Z/{2}\Z \oplus \Z/{6}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(-1 : -4 : 2) - (1 : 0 : 0)$$ $$2x + z$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$-z^3$$ $$0$$ $$2$$
$$(0 : -1 : 1) - (1 : 0 : 0)$$ $$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0$$ $$6$$
Generator $D_0$ Height Order
$$(-1 : -4 : 2) - (1 : 0 : 0)$$ $$2x + z$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$-z^3$$ $$0$$ $$2$$
$$(0 : -1 : 1) - (1 : 0 : 0)$$ $$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0$$ $$6$$
Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$2x + z$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$-z^3$$ $$0$$ $$2$$
$$(0 : -1 : 1) - (1 : 0 : 0)$$ $$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0$$ $$6$$

BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$2$$ Regulator: $$1$$ Real period: $$18.60015$$ Tamagawa product: $$8$$ Torsion order: $$12$$ Leading coefficient: $$1.033342$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$8$$ $$9$$ $$2$$ $$1 + 2 T^{2}$$
$$3$$ $$4$$ $$6$$ $$4$$ $$1$$

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?
$$2$$ 2.180.3 yes
$$3$$ 3.480.3 yes

Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $J(E_4)$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{SU}(2)$$

Decomposition of the Jacobian

Simple over $$\overline{\Q}$$

magma: HeuristicDecompositionFactors(C);

Endomorphisms of the Jacobian

Not of $$\GL_2$$-type over $$\Q$$

Endomorphism ring over $$\Q$$:

 $$\End (J_{})$$ $$\simeq$$ $$\Z$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R$$

Smallest field over which all endomorphisms are defined:
Galois number field $$K = \Q (a) \simeq$$ 8.0.339738624.10 with defining polynomial $$x^{8} + 4 x^{6} + 10 x^{4} + 24 x^{2} + 36$$

Not of $$\GL_2$$-type over $$\overline{\Q}$$

Endomorphism ring over $$\overline{\Q}$$:

 $$\End (J_{\overline{\Q}})$$ $$\simeq$$ a non-Eichler order of index $$6$$ in a maximal order of $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\simeq$$ the quaternion algebra over $$\Q$$ of discriminant 6 $$\End (J_{\overline{\Q}}) \otimes \R$$ $$\simeq$$ $$\mathrm{M}_2 (\R)$$

Remainder of the endomorphism lattice by field

Over subfield $$F \simeq$$ $$\Q(\sqrt{-3})$$ with generator $$-\frac{1}{15} a^{6} - \frac{1}{6} a^{4} - \frac{2}{3} a^{2} - \frac{3}{5}$$ with minimal polynomial $$x^{2} - x + 1$$:

 $$\End (J_{F})$$ $$\simeq$$ $$\Z [3\sqrt{-1}]$$ $$\End (J_{F}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{-1})$$ $$\End (J_{F}) \otimes \R$$ $$\simeq$$ $$\C$$
Sato Tate group: E_4
Of $$\GL_2$$-type, simple

Over subfield $$F \simeq$$ $$\Q(\sqrt{-2})$$ with generator $$-\frac{1}{10} a^{6} - \frac{2}{5}$$ with minimal polynomial $$x^{2} + 2$$:

 $$\End (J_{F})$$ $$\simeq$$ $$\Z$$ $$\End (J_{F}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\End (J_{F}) \otimes \R$$ $$\simeq$$ $$\R$$
Sato Tate group: J(E_2)
Not of $$\GL_2$$-type, simple

Over subfield $$F \simeq$$ $$\Q(\sqrt{6})$$ with generator $$-\frac{1}{6} a^{6} - \frac{2}{3} a^{4} - \frac{2}{3} a^{2} - 2$$ with minimal polynomial $$x^{2} - 6$$:

 $$\End (J_{F})$$ $$\simeq$$ $$\Z$$ $$\End (J_{F}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\End (J_{F}) \otimes \R$$ $$\simeq$$ $$\R$$
Sato Tate group: J(E_2)
Not of $$\GL_2$$-type, simple

Over subfield $$F \simeq$$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ with generator $$-\frac{2}{15} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{6}{5}$$ with minimal polynomial $$x^{4} - 2 x^{2} + 4$$:

 $$\End (J_{F})$$ $$\simeq$$ $$\Z [3\sqrt{-1}]$$ $$\End (J_{F}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{-1})$$ $$\End (J_{F}) \otimes \R$$ $$\simeq$$ $$\C$$
Sato Tate group: E_2
Of $$\GL_2$$-type, simple

Over subfield $$F \simeq$$ 4.0.6144.1 with generator $$\frac{2}{45} a^{7} - \frac{1}{18} a^{5} + \frac{1}{9} a^{3} + \frac{11}{15} a$$ with minimal polynomial $$x^{4} - 4 x^{2} + 6$$:

 $$\End (J_{F})$$ $$\simeq$$ $$\Z [\sqrt{6}]$$ $$\End (J_{F}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{6})$$ $$\End (J_{F}) \otimes \R$$ $$\simeq$$ $$\R \times \R$$
Sato Tate group: J(E_1)
Of $$\GL_2$$-type, simple

Over subfield $$F \simeq$$ 4.2.18432.2 with generator $$\frac{7}{90} a^{7} + \frac{5}{18} a^{5} + \frac{4}{9} a^{3} + \frac{23}{15} a$$ with minimal polynomial $$x^{4} + 4 x^{2} - 2$$:

 $$\End (J_{F})$$ $$\simeq$$ $$\Z [\sqrt{3}]$$ $$\End (J_{F}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{3})$$ $$\End (J_{F}) \otimes \R$$ $$\simeq$$ $$\R \times \R$$
Sato Tate group: J(E_1)
Of $$\GL_2$$-type, simple

Over subfield $$F \simeq$$ 4.2.18432.2 with generator $$\frac{1}{18} a^{7} + \frac{1}{18} a^{5} - \frac{1}{9} a^{3} - \frac{1}{3} a$$ with minimal polynomial $$x^{4} + 4 x^{2} - 2$$:

 $$\End (J_{F})$$ $$\simeq$$ $$\Z [\sqrt{3}]$$ $$\End (J_{F}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{3})$$ $$\End (J_{F}) \otimes \R$$ $$\simeq$$ $$\R \times \R$$
Sato Tate group: J(E_1)
Of $$\GL_2$$-type, simple

Over subfield $$F \simeq$$ 4.0.6144.1 with generator $$-\frac{1}{15} a^{7} - \frac{1}{6} a^{5} - \frac{2}{3} a^{3} - \frac{3}{5} a$$ with minimal polynomial $$x^{4} - 4 x^{2} + 6$$:

 $$\End (J_{F})$$ $$\simeq$$ $$\Z [\sqrt{6}]$$ $$\End (J_{F}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{6})$$ $$\End (J_{F}) \otimes \R$$ $$\simeq$$ $$\R \times \R$$
Sato Tate group: J(E_1)
Of $$\GL_2$$-type, 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);

Additional information

This curve has the smallest discriminant known for a genus 2 curve with potential QM (the geometric endomorphism algebra of its Jacobian is a non-split quaternion algebra).