# Properties

 Label 1296.a.20736.1 Conductor $1296$ Discriminant $20736$ Mordell-Weil group $$\Z/{2}\Z \oplus \Z/{6}\Z$$ Sato-Tate group $E_3$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\mathrm{M}_2(\R)$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathrm{M}_2(\Q)$$ $$\End(J) \otimes \Q$$ $$\mathsf{CM}$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Related objects

Show commands: Magma / SageMath

## Simplified equation

 $y^2 = x^5 - x^4 - 3x^3 + 4x^2 - x$ (homogenize, simplify) $y^2 = x^5z - x^4z^2 - 3x^3z^3 + 4x^2z^4 - xz^5$ (dehomogenize, simplify) $y^2 = x^5 - x^4 - 3x^3 + 4x^2 - x$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, 4, -3, -1, 1]), R([]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, 4, -3, -1, 1], R![]);

sage: X = HyperellipticCurve(R([0, -1, 4, -3, -1, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$1296$$ $$=$$ $$2^{4} \cdot 3^{4}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$20736$$ $$=$$ $$2^{8} \cdot 3^{4}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$78$$ $$=$$ $$2 \cdot 3 \cdot 13$$ $$I_4$$ $$=$$ $$216$$ $$=$$ $$2^{3} \cdot 3^{3}$$ $$I_6$$ $$=$$ $$4806$$ $$=$$ $$2 \cdot 3^{3} \cdot 89$$ $$I_{10}$$ $$=$$ $$81$$ $$=$$ $$3^{4}$$ $$J_2$$ $$=$$ $$156$$ $$=$$ $$2^{2} \cdot 3 \cdot 13$$ $$J_4$$ $$=$$ $$438$$ $$=$$ $$2 \cdot 3 \cdot 73$$ $$J_6$$ $$=$$ $$-428$$ $$=$$ $$- 2^{2} \cdot 107$$ $$J_8$$ $$=$$ $$-64653$$ $$=$$ $$- 3 \cdot 23 \cdot 937$$ $$J_{10}$$ $$=$$ $$20736$$ $$=$$ $$2^{8} \cdot 3^{4}$$ $$g_1$$ $$=$$ $$4455516$$ $$g_2$$ $$=$$ $$160381/2$$ $$g_3$$ $$=$$ $$-18083/36$$

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

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

## Automorphism group

 $$\mathrm{Aut}(X)$$ $$\simeq$$ $C_6$ magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$$D_6$magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);

## Rational points

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

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

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

Number of rational Weierstrass points: $$3$$

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

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$2$$ Regulator: $$1$$ Real period: $$23.23504$$ Tamagawa product: $$3$$ Torsion order: $$12$$ Leading coefficient: $$0.484063$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

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

## 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?
$$2$$ 2.240.1 yes
$$3$$ 3.1920.3 yes

## Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $E_3$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{SU}(2)$$

## Decomposition of the Jacobian

Splits over the number field $$\Q (b) \simeq$$ $$\Q(\zeta_{9})^+$$ with defining polynomial:
$$x^{3} - 3 x - 1$$

Decomposes up to isogeny as the square of the elliptic curve isogeny class:
Elliptic curve isogeny class 3.3.81.1-64.1-a

magma: HeuristicDecompositionFactors(C);

## Endomorphisms of the Jacobian

Of $$\GL_2$$-type over $$\Q$$

Endomorphism ring over $$\Q$$:

 $$\End (J_{})$$ $$\simeq$$ $$\Z [\frac{1 + \sqrt{-3}}{2}]$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{-3})$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\C$$

Smallest field over which all endomorphisms are defined:
Galois number field $$K = \Q (a) \simeq$$ $$\Q(\zeta_{9})^+$$ with defining polynomial $$x^{3} - 3 x - 1$$

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

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

 $$\End (J_{\overline{\Q}})$$ $$\simeq$$ an Eichler order of index $$3$$ in a maximal order of $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\simeq$$ $$\mathrm{M}_2($$$$\Q$$$$)$$ $$\End (J_{\overline{\Q}}) \otimes \R$$ $$\simeq$$ $$\mathrm{M}_2 (\R)$$

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);