# Properties

 Label 1311.a.814131.1 Conductor 1311 Discriminant -814131 Mordell-Weil group $$\Z/{2}\Z \times \Z/{8}\Z$$ Sato-Tate group $\mathrm{USp}(4)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type no

# Related objects

Show commands for: Magma / SageMath

## Simplified equation

 $y^2 + xy = x^5 + 5x^4 + 5x^3 + 4x^2 + x$ (homogenize, simplify) $y^2 + xz^2y = x^5z + 5x^4z^2 + 5x^3z^3 + 4x^2z^4 + xz^5$ (dehomogenize, simplify) $y^2 = 4x^5 + 20x^4 + 20x^3 + 17x^2 + 4x$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, 4, 5, 5, 1], R![0, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, 4, 5, 5, 1]), R([0, 1]));

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([0, 4, 17, 20, 20, 4]))

## Invariants

 Conductor: $$N$$ = $$1311$$ = $$3 \cdot 19 \cdot 23$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$-814131$$ = $$- 3^{4} \cdot 19 \cdot 23^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$-2400$$ = $$- 2^{5} \cdot 3 \cdot 5^{2}$$ $$I_4$$ = $$32640$$ = $$2^{7} \cdot 3 \cdot 5 \cdot 17$$ $$I_6$$ = $$-55062336$$ = $$- 2^{6} \cdot 3 \cdot 7 \cdot 53 \cdot 773$$ $$I_{10}$$ = $$-3334680576$$ = $$- 2^{12} \cdot 3^{4} \cdot 19 \cdot 23^{2}$$ $$J_2$$ = $$-300$$ = $$- 2^{2} \cdot 3 \cdot 5^{2}$$ $$J_4$$ = $$3410$$ = $$2 \cdot 5 \cdot 11 \cdot 31$$ $$J_6$$ = $$4761$$ = $$3^{2} \cdot 23^{2}$$ $$J_8$$ = $$-3264100$$ = $$- 2^{2} \cdot 5^{2} \cdot 7 \cdot 4663$$ $$J_{10}$$ = $$-814131$$ = $$- 3^{4} \cdot 19 \cdot 23^{2}$$ $$g_1$$ = $$30000000000/10051$$ $$g_2$$ = $$3410000000/30153$$ $$g_3$$ = $$-10000/19$$

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

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

## 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: $$(1 : 0 : 0),\, (0 : 0 : 1),\, (-1 : -1 : 1),\, (-1 : 2 : 1),\, (-4 : 2 : 1)$$

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

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: $$\Z/{2}\Z \times \Z/{8}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

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

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$2$$ Regulator: $$1$$ Real period: $$13.75349$$ Tamagawa product: $$8$$ Torsion order: $$16$$ Leading coefficient: $$0.429796$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$3$$ $$4$$ $$1$$ $$4$$ $$( 1 - T )( 1 + 3 T^{2} )$$
$$19$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 - 4 T + 19 T^{2} )$$
$$23$$ $$2$$ $$1$$ $$2$$ $$( 1 + T )( 1 - 8 T + 23 T^{2} )$$

## Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $\mathrm{USp}(4)$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{USp}(4)$$

## Decomposition of the Jacobian

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

## Endomorphisms of the Jacobian

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

Endomorphism ring over $$\Q$$:

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

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