# Properties

 Label 3462.a.560844.1 Conductor 3462 Discriminant -560844 Mordell-Weil group $$\Z \times \Z/{2}\Z$$ 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

 $y^2 + (x + 1)y = x^5 - 3x^3 + 4x^2 - 3x$ (homogenize, simplify) $y^2 + (xz^2 + z^3)y = x^5z - 3x^3z^3 + 4x^2z^4 - 3xz^5$ (dehomogenize, simplify) $y^2 = 4x^5 - 12x^3 + 17x^2 - 10x + 1$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -3, 4, -3, 0, 1]), R([1, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -3, 4, -3, 0, 1], R![1, 1]);

sage: X = HyperellipticCurve(R([1, -10, 17, -12, 0, 4]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$3462$$ $$=$$ $$2 \cdot 3 \cdot 577$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-560844$$ $$=$$ $$- 2^{2} \cdot 3^{5} \cdot 577$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$-736$$ $$=$$ $$- 2^{5} \cdot 23$$ $$I_4$$ $$=$$ $$-87296$$ $$=$$ $$- 2^{8} \cdot 11 \cdot 31$$ $$I_6$$ $$=$$ $$9360576$$ $$=$$ $$2^{6} \cdot 3^{3} \cdot 5417$$ $$I_{10}$$ $$=$$ $$-2297217024$$ $$=$$ $$- 2^{14} \cdot 3^{5} \cdot 577$$ $$J_2$$ $$=$$ $$-92$$ $$=$$ $$- 2^{2} \cdot 23$$ $$J_4$$ $$=$$ $$1262$$ $$=$$ $$2 \cdot 631$$ $$J_6$$ $$=$$ $$5185$$ $$=$$ $$5 \cdot 17 \cdot 61$$ $$J_8$$ $$=$$ $$-517416$$ $$=$$ $$- 2^{3} \cdot 3 \cdot 21559$$ $$J_{10}$$ $$=$$ $$-560844$$ $$=$$ $$- 2^{2} \cdot 3^{5} \cdot 577$$ $$g_1$$ $$=$$ $$1647703808/140211$$ $$g_2$$ $$=$$ $$245676064/140211$$ $$g_3$$ $$=$$ $$-10971460/140211$$

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
$$(1 : 0 : 0)$$ $$(0 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(1 : -1 : 1)$$ $$(-1 : -3 : 1)$$ $$(-1 : 3 : 1)$$
$$(2 : 3 : 1)$$ $$(2 : -6 : 1)$$ $$(17 : -2652 : 16)$$ $$(17 : -5796 : 16)$$

magma: [C![-1,-3,1],C![-1,3,1],C![0,-1,1],C![0,0,1],C![1,-1,1],C![1,0,0],C![2,-6,1],C![2,3,1],C![17,-5796,16],C![17,-2652,16]];

Number of rational Weierstrass points: $$2$$

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

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(0 : 0 : 1) + (2 : -6 : 1) - 2 \cdot(1 : 0 : 0)$$ $$(-x + 2z) x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-3xz^2$$ $$0.015775$$ $$\infty$$
$$(1 : -1 : 1) - (1 : 0 : 0)$$ $$x - z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0$$ $$2$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$2$$ Regulator: $$0.015775$$ Real period: $$11.41134$$ Tamagawa product: $$10$$ Torsion order: $$2$$ Leading coefficient: $$0.450053$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$1$$ $$2$$ $$2$$ $$( 1 + T )( 1 + 2 T^{2} )$$
$$3$$ $$1$$ $$5$$ $$5$$ $$( 1 - T )( 1 + 2 T + 3 T^{2} )$$
$$577$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 - 2 T + 577 T^{2} )$$

## 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 $$\Q$$.