# Properties

 Label 3391.a.3391.1 Conductor 3391 Discriminant -3391 Mordell-Weil group $$\Z$$ 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

 $y^2 + (x^3 + x + 1)y = -2x^4 + x^3 + 6x^2 + 2x$ (homogenize, simplify) $y^2 + (x^3 + xz^2 + z^3)y = -2x^4z^2 + x^3z^3 + 6x^2z^4 + 2xz^5$ (dehomogenize, simplify) $y^2 = x^6 - 6x^4 + 6x^3 + 25x^2 + 10x + 1$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 2, 6, 1, -2], R![1, 1, 0, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 2, 6, 1, -2]), R([1, 1, 0, 1]));

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([1, 10, 25, 6, -6, 0, 1]))

## Invariants

 Conductor: $$N$$ = $$3391$$ = $$3391$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$-3391$$ = $$- 3391$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$2376$$ = $$2^{3} \cdot 3^{3} \cdot 11$$ $$I_4$$ = $$442596$$ = $$2^{2} \cdot 3 \cdot 7 \cdot 11 \cdot 479$$ $$I_6$$ = $$251628264$$ = $$2^{3} \cdot 3^{2} \cdot 3494837$$ $$I_{10}$$ = $$-13889536$$ = $$- 2^{12} \cdot 3391$$ $$J_2$$ = $$297$$ = $$3^{3} \cdot 11$$ $$J_4$$ = $$-935$$ = $$- 5 \cdot 11 \cdot 17$$ $$J_6$$ = $$4145$$ = $$5 \cdot 829$$ $$J_8$$ = $$89210$$ = $$2 \cdot 5 \cdot 11 \cdot 811$$ $$J_{10}$$ = $$-3391$$ = $$- 3391$$ $$g_1$$ = $$-2310905821257/3391$$ $$g_2$$ = $$24495198255/3391$$ $$g_3$$ = $$-365626305/3391$$

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
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(0 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(-2 : 4 : 1)$$ $$(-2 : 5 : 1)$$
$$(1 : 32 : 4)$$ $$(1 : -113 : 4)$$

magma: [C![-2,4,1],C![-2,5,1],C![0,-1,1],C![0,0,1],C![1,-113,4],C![1,-1,0],C![1,0,0],C![1,32,4]];

Number of rational Weierstrass points: $$0$$

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: $$\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(0 : -1 : 1) - (1 : -1 : 0)$$ $$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0.013972$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$1$$ Regulator: $$0.013972$$ Real period: $$30.84522$$ Tamagawa product: $$1$$ Torsion order: $$1$$ Leading coefficient: $$0.430985$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$3391$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 - 14 T + 3391 T^{2} )$$

## 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 $$\Q$$.