# Properties

 Label 3092.a.98944.1 Conductor 3092 Discriminant -98944 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: SageMath / Magma

## Simplified equation

 $y^2 + (x^3 + 1)y = -x^4 + 3x^2 + x$ (homogenize, simplify) $y^2 + (x^3 + z^3)y = -x^4z^2 + 3x^2z^4 + xz^5$ (dehomogenize, simplify) $y^2 = x^6 - 4x^4 + 2x^3 + 12x^2 + 4x + 1$ (minimize, homogenize)

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

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

sage: X = HyperellipticCurve(R([1, 4, 12, 2, -4, 0, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$3092$$ $$=$$ $$2^{2} \cdot 773$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-98944$$ $$=$$ $$- 2^{7} \cdot 773$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$552$$ $$=$$ $$2^{3} \cdot 3 \cdot 23$$ $$I_4$$ $$=$$ $$78180$$ $$=$$ $$2^{2} \cdot 3 \cdot 5 \cdot 1303$$ $$I_6$$ $$=$$ $$10230504$$ $$=$$ $$2^{3} \cdot 3 \cdot 29 \cdot 14699$$ $$I_{10}$$ $$=$$ $$-405274624$$ $$=$$ $$- 2^{19} \cdot 773$$ $$J_2$$ $$=$$ $$69$$ $$=$$ $$3 \cdot 23$$ $$J_4$$ $$=$$ $$-616$$ $$=$$ $$- 2^{3} \cdot 7 \cdot 11$$ $$J_6$$ $$=$$ $$-1392$$ $$=$$ $$- 2^{4} \cdot 3 \cdot 29$$ $$J_8$$ $$=$$ $$-118876$$ $$=$$ $$- 2^{2} \cdot 113 \cdot 263$$ $$J_{10}$$ $$=$$ $$-98944$$ $$=$$ $$- 2^{7} \cdot 773$$ $$g_1$$ $$=$$ $$-1564031349/98944$$ $$g_2$$ $$=$$ $$25295193/12368$$ $$g_3$$ $$=$$ $$414207/6184$$

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
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(0 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(-1 : -1 : 1)$$ $$(-1 : 1 : 1)$$
$$(1 : 1 : 1)$$ $$(-2 : 1 : 1)$$ $$(1 : -3 : 1)$$ $$(-2 : 6 : 1)$$

magma: [C![-2,1,1],C![-2,6,1],C![-1,-1,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-3,1],C![1,-1,0],C![1,0,0],C![1,1,1]];

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
$$(-1 : -1 : 1) - (1 : 0 : 0)$$ $$z (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3 - 2z^3$$ $$0.002123$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$1$$ Regulator: $$0.002123$$ Real period: $$19.59106$$ Tamagawa product: $$11$$ Torsion order: $$1$$ Leading coefficient: $$0.457654$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$2$$ $$7$$ $$11$$ $$( 1 - T )( 1 + T )$$
$$773$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 - 46 T + 773 T^{2} )$$

## 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 $$\Q$$.