# Properties

 Label 5968.a.381952.1 Conductor 5968 Discriminant 381952 Mordell-Weil group $$\Z/{13}\Z$$ 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

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

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

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

sage: X = HyperellipticCurve(R([1, 2, 5, 0, 0, 8, 4]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$5968$$ $$=$$ $$2^{4} \cdot 373$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$381952$$ $$=$$ $$2^{10} \cdot 373$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$-320$$ $$=$$ $$- 2^{6} \cdot 5$$ $$I_4$$ $$=$$ $$90880$$ $$=$$ $$2^{8} \cdot 5 \cdot 71$$ $$I_6$$ $$=$$ $$-9692160$$ $$=$$ $$- 2^{10} \cdot 3 \cdot 5 \cdot 631$$ $$I_{10}$$ $$=$$ $$1564475392$$ $$=$$ $$2^{22} \cdot 373$$ $$J_2$$ $$=$$ $$-40$$ $$=$$ $$- 2^{3} \cdot 5$$ $$J_4$$ $$=$$ $$-880$$ $$=$$ $$- 2^{4} \cdot 5 \cdot 11$$ $$J_6$$ $$=$$ $$6160$$ $$=$$ $$2^{4} \cdot 5 \cdot 7 \cdot 11$$ $$J_8$$ $$=$$ $$-255200$$ $$=$$ $$- 2^{5} \cdot 5^{2} \cdot 11 \cdot 29$$ $$J_{10}$$ $$=$$ $$381952$$ $$=$$ $$2^{10} \cdot 373$$ $$g_1$$ $$=$$ $$-100000/373$$ $$g_2$$ $$=$$ $$55000/373$$ $$g_3$$ $$=$$ $$9625/373$$

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

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

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: $$\Z/{13}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(0 : 0 : 1) - (1 : -1 : 0)$$ $$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3$$ $$0$$ $$13$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$0$$ Regulator: $$1$$ Real period: $$14.24857$$ Tamagawa product: $$13$$ Torsion order: $$13$$ Leading coefficient: $$1.096044$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$4$$ $$10$$ $$13$$ $$1 - T$$
$$373$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 - 10 T + 373 T^{2} )$$

## 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 $$\Q$$.