# Properties

 Label 363.a.43923.1 Conductor $363$ Discriminant $-43923$ Mordell-Weil group $$\Z/{10}\Z$$ 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

# Related objects

Show commands: Magma / SageMath

## Simplified equation

 $y^2 + x^2y = 11x^5 - 13x^4 - 7x^3 + 10x^2 + x - 2$ (homogenize, simplify) $y^2 + x^2zy = 11x^5z - 13x^4z^2 - 7x^3z^3 + 10x^2z^4 + xz^5 - 2z^6$ (dehomogenize, simplify) $y^2 = 44x^5 - 51x^4 - 28x^3 + 40x^2 + 4x - 8$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-2, 1, 10, -7, -13, 11]), R([0, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-2, 1, 10, -7, -13, 11], R![0, 0, 1]);

sage: X = HyperellipticCurve(R([-8, 4, 40, -28, -51, 44]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$363$$ $$=$$ $$3 \cdot 11^{2}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-43923$$ $$=$$ $$- 3 \cdot 11^{4}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$11096$$ $$=$$ $$2^{3} \cdot 19 \cdot 73$$ $$I_4$$ $$=$$ $$25612$$ $$=$$ $$2^{2} \cdot 19 \cdot 337$$ $$I_6$$ $$=$$ $$88274095$$ $$=$$ $$5 \cdot 7 \cdot 13 \cdot 19 \cdot 10211$$ $$I_{10}$$ $$=$$ $$-175692$$ $$=$$ $$- 2^{2} \cdot 3 \cdot 11^{4}$$ $$J_2$$ $$=$$ $$5548$$ $$=$$ $$2^{2} \cdot 19 \cdot 73$$ $$J_4$$ $$=$$ $$1278244$$ $$=$$ $$2^{2} \cdot 11^{2} \cdot 19 \cdot 139$$ $$J_6$$ $$=$$ $$392069161$$ $$=$$ $$11^{2} \cdot 19 \cdot 170539$$ $$J_8$$ $$=$$ $$135322995423$$ $$=$$ $$3 \cdot 7 \cdot 11^{2} \cdot 19^{2} \cdot 29 \cdot 5087$$ $$J_{10}$$ $$=$$ $$-43923$$ $$=$$ $$- 3 \cdot 11^{4}$$ $$g_1$$ $$=$$ $$-5256325630316243968/43923$$ $$g_2$$ $$=$$ $$-1804005053317888/363$$ $$g_3$$ $$=$$ $$-99735603013264/363$$

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

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

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

Number of rational Weierstrass points: $$1$$

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: $$\Z/{10}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$11x^2 - 3xz - 6z^2$$ $$=$$ $$0,$$ $$11y$$ $$=$$ $$-2xz^2 - 4z^3$$ $$0$$ $$10$$
Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$11x^2 - 3xz - 6z^2$$ $$=$$ $$0,$$ $$11y$$ $$=$$ $$-2xz^2 - 4z^3$$ $$0$$ $$10$$
Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$11x^2 - 3xz - 6z^2$$ $$=$$ $$0,$$ $$11y$$ $$=$$ $$x^2z - 4xz^2 - 8z^3$$ $$0$$ $$10$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$1$$ Regulator: $$1$$ Real period: $$3.794119$$ Tamagawa product: $$5$$ Torsion order: $$10$$ Leading coefficient: $$0.189705$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$3$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + T + 3 T^{2} )$$
$$11$$ $$2$$ $$4$$ $$5$$ $$( 1 - T )^{2}$$

## 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?
$$2$$ 2.60.1 yes
$$3$$ 3.80.4 no

## 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 33.a
Elliptic curve isogeny class 11.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 $$3$$ 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 $$\Q$$.

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);