# Properties

 Label 936.a.1872.1 Conductor $936$ Discriminant $-1872$ Mordell-Weil group $$\Z/{2}\Z \times \Z/{4}\Z$$ Sato-Tate group $\mathrm{SU}(2)\times\mathrm{SU}(2)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q \times \Q$$ $$\End(J) \otimes \Q$$ $$\Q \times \Q$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Learn more

Show commands: SageMath / Magma

## Simplified equation

 $y^2 + (x^3 + x)y = -x^6 - 9x^4 - 32x^2 - 39$ (homogenize, simplify) $y^2 + (x^3 + xz^2)y = -x^6 - 9x^4z^2 - 32x^2z^4 - 39z^6$ (dehomogenize, simplify) $y^2 = -3x^6 - 34x^4 - 127x^2 - 156$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-39, 0, -32, 0, -9, 0, -1]), R([0, 1, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-39, 0, -32, 0, -9, 0, -1], R![0, 1, 0, 1]);

sage: X = HyperellipticCurve(R([-156, 0, -127, 0, -34, 0, -3]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$936$$ $$=$$ $$2^{3} \cdot 3^{2} \cdot 13$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-1872$$ $$=$$ $$- 2^{4} \cdot 3^{2} \cdot 13$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$45352$$ $$=$$ $$2^{3} \cdot 5669$$ $$I_4$$ $$=$$ $$11224$$ $$=$$ $$2^{3} \cdot 23 \cdot 61$$ $$I_6$$ $$=$$ $$169415364$$ $$=$$ $$2^{2} \cdot 3 \cdot 14117947$$ $$I_{10}$$ $$=$$ $$7488$$ $$=$$ $$2^{6} \cdot 3^{2} \cdot 13$$ $$J_2$$ $$=$$ $$22676$$ $$=$$ $$2^{2} \cdot 5669$$ $$J_4$$ $$=$$ $$21423170$$ $$=$$ $$2 \cdot 5 \cdot 29 \cdot 31 \cdot 2383$$ $$J_6$$ $$=$$ $$26983749312$$ $$=$$ $$2^{6} \cdot 3^{2} \cdot 13 \cdot 1069 \cdot 3371$$ $$J_8$$ $$=$$ $$38232821637503$$ $$=$$ $$38232821637503$$ $$J_{10}$$ $$=$$ $$1872$$ $$=$$ $$2^{4} \cdot 3^{2} \cdot 13$$ $$g_1$$ $$=$$ $$374724646811252438336/117$$ $$g_2$$ $$=$$ $$15612163699641478120/117$$ $$g_3$$ $$=$$ $$7411896491650496$$

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_2^2$magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);

## Rational points

 This curve has no rational points. This curve has no rational points. This curve has no rational points.

magma: []; // minimal model

magma: []; // simplified model

Number of rational Weierstrass points: $$0$$

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));

This curve is locally solvable except over $\R$.

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: $$\Z/{2}\Z \times \Z/{4}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - D_\infty$$ $$x^2 + 3z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$xz^2$$ $$0$$ $$2$$
$$D_0 - D_\infty$$ $$x^2 + 5z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$2xz^2 + z^3$$ $$0$$ $$4$$
Generator $D_0$ Height Order
$$D_0 - D_\infty$$ $$x^2 + 3z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$xz^2$$ $$0$$ $$2$$
$$D_0 - D_\infty$$ $$x^2 + 5z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$2xz^2 + z^3$$ $$0$$ $$4$$
Generator $D_0$ Height Order
$$D_0 - D_\infty$$ $$x^2 + 3z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3 + 3xz^2$$ $$0$$ $$2$$
$$D_0 - D_\infty$$ $$x^2 + 5z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3 + 5xz^2 + 2z^3$$ $$0$$ $$4$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$3$$ Regulator: $$1$$ Real period: $$7.131061$$ Tamagawa product: $$2$$ Torsion order: $$8$$ Leading coefficient: $$0.445691$$ Analytic order of Ш: $$2$$   (rounded) Order of Ш: twice a square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$3$$ $$4$$ $$2$$ $$1 - T + 2 T^{2}$$
$$3$$ $$2$$ $$2$$ $$1$$ $$( 1 + T )^{2}$$
$$13$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 2 T + 13 T^{2} )$$

## Sato-Tate group

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

## 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 24.a
Elliptic curve isogeny class 39.a

## Endomorphisms of the Jacobian

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

Endomorphism ring over $$\Q$$:

 $$\End (J_{})$$ $$\simeq$$ an order of index $$2$$ in $$\Z \times \Z$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\times$$ $$\Q$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R \times \R$$

All $$\overline{\Q}$$-endomorphisms of the Jacobian are defined over $$\Q$$.