# Properties

 Label 6982.a.13964.1 Conductor 6982 Discriminant -13964 Mordell-Weil group $$\Z \times \Z$$ 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

 $y^2 + (x^3 + 1)y = x^5 - 3x^3 + 2x$ (homogenize, simplify) $y^2 + (x^3 + z^3)y = x^5z - 3x^3z^3 + 2xz^5$ (dehomogenize, simplify) $y^2 = x^6 + 4x^5 - 10x^3 + 8x + 1$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 2, 0, -3, 0, 1], R![1, 0, 0, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 2, 0, -3, 0, 1]), R([1, 0, 0, 1]));

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([1, 8, 0, -10, 0, 4, 1]))

## Invariants

 Conductor: $$N$$ = $$6982$$ = $$2 \cdot 3491$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$-13964$$ = $$- 2^{2} \cdot 3491$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$1640$$ = $$2^{3} \cdot 5 \cdot 41$$ $$I_4$$ = $$50020$$ = $$2^{2} \cdot 5 \cdot 41 \cdot 61$$ $$I_6$$ = $$23554280$$ = $$2^{3} \cdot 5 \cdot 263 \cdot 2239$$ $$I_{10}$$ = $$-57196544$$ = $$- 2^{14} \cdot 3491$$ $$J_2$$ = $$205$$ = $$5 \cdot 41$$ $$J_4$$ = $$1230$$ = $$2 \cdot 3 \cdot 5 \cdot 41$$ $$J_6$$ = $$8720$$ = $$2^{4} \cdot 5 \cdot 109$$ $$J_8$$ = $$68675$$ = $$5^{2} \cdot 41 \cdot 67$$ $$J_{10}$$ = $$-13964$$ = $$- 2^{2} \cdot 3491$$ $$g_1$$ = $$-362050628125/13964$$ $$g_2$$ = $$-5298301875/6982$$ $$g_3$$ = $$-91614500/3491$$

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

Known points
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(0 : 0 : 1)$$ $$(-1 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(1 : 0 : 1)$$
$$(1 : -2 : 1)$$ $$(-2 : 3 : 1)$$ $$(-2 : 4 : 1)$$ $$(-3 : 12 : 1)$$ $$(-3 : 14 : 1)$$ $$(13 : 408 : 5)$$
$$(13 : -2730 : 5)$$

magma: [C![-3,12,1],C![-3,14,1],C![-2,3,1],C![-2,4,1],C![-1,0,1],C![0,-1,1],C![0,0,1],C![1,-2,1],C![1,-1,0],C![1,0,0],C![1,0,1],C![13,-2730,5],C![13,408,5]];

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: $$\Z \times \Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(1 : -2 : 1) - (1 : 0 : 0)$$ $$z (x - z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3 - z^3$$ $$0.474383$$ $$\infty$$
$$(-1 : 0 : 1) - (1 : 0 : 0)$$ $$z (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3 - z^3$$ $$0.015759$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$2$$ Mordell-Weil rank: $$2$$ 2-Selmer rank: $$2$$ Regulator: $$0.007467$$ Real period: $$23.52533$$ Tamagawa product: $$2$$ Torsion order: $$1$$ Leading coefficient: $$0.351346$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$2$$ $$2$$ $$1$$ $$2$$ $$( 1 + T )( 1 + T + 2 T^{2} )$$
$$3491$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 92 T + 3491 T^{2} )$$

## 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 $$\Q$$.