# Properties

 Label 5280.a.633600.1 Conductor 5280 Discriminant -633600 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

 $y^2 + (x^2 + 1)y = x^5 + 12x^4 + 5x^3 + 4x^2 + 2x$ (homogenize, simplify) $y^2 + (x^2z + z^3)y = x^5z + 12x^4z^2 + 5x^3z^3 + 4x^2z^4 + 2xz^5$ (dehomogenize, simplify) $y^2 = 4x^5 + 49x^4 + 20x^3 + 18x^2 + 8x + 1$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 2, 4, 5, 12, 1]), R([1, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 2, 4, 5, 12, 1], R![1, 0, 1]);

sage: X = HyperellipticCurve(R([1, 8, 18, 20, 49, 4]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$5280$$ $$=$$ $$2^{5} \cdot 3 \cdot 5 \cdot 11$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-633600$$ $$=$$ $$- 2^{8} \cdot 3^{2} \cdot 5^{2} \cdot 11$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$-10432$$ $$=$$ $$- 2^{6} \cdot 163$$ $$I_4$$ $$=$$ $$3710464$$ $$=$$ $$2^{9} \cdot 7247$$ $$I_6$$ $$=$$ $$-15373442048$$ $$=$$ $$- 2^{10} \cdot 15013127$$ $$I_{10}$$ $$=$$ $$-2595225600$$ $$=$$ $$- 2^{20} \cdot 3^{2} \cdot 5^{2} \cdot 11$$ $$J_2$$ $$=$$ $$-1304$$ $$=$$ $$- 2^{3} \cdot 163$$ $$J_4$$ $$=$$ $$32200$$ $$=$$ $$2^{3} \cdot 5^{2} \cdot 7 \cdot 23$$ $$J_6$$ $$=$$ $$7557136$$ $$=$$ $$2^{4} \cdot 19 \cdot 24859$$ $$J_8$$ $$=$$ $$-2722836336$$ $$=$$ $$- 2^{4} \cdot 3 \cdot 11 \cdot 47 \cdot 109721$$ $$J_{10}$$ $$=$$ $$-633600$$ $$=$$ $$- 2^{8} \cdot 3^{2} \cdot 5^{2} \cdot 11$$ $$g_1$$ $$=$$ $$14728142981504/2475$$ $$g_2$$ $$=$$ $$11156004272/99$$ $$g_3$$ $$=$$ $$-50196386596/2475$$

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
$$(1 : 0 : 0)$$ $$(0 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(-1 : 2 : 1)$$ $$(-1 : -4 : 1)$$ $$(1 : 4 : 1)$$
$$(1 : -6 : 1)$$ $$(-1 : -34 : 4)$$ $$(-2 : -348 : 9)$$ $$(-2 : -417 : 9)$$

magma: [C![-2,-417,9],C![-2,-348,9],C![-1,-34,4],C![-1,-4,1],C![-1,2,1],C![0,-1,1],C![0,0,1],C![1,-6,1],C![1,0,0],C![1,4,1]];

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
$$(-1 : 2 : 1) + (0 : -1 : 1) - 2 \cdot(1 : 0 : 0)$$ $$x (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-3xz^2 - z^3$$ $$0.008231$$ $$\infty$$
$$(-1 : -34 : 4) - (1 : 0 : 0)$$ $$4x + z$$ $$=$$ $$0,$$ $$32y$$ $$=$$ $$-17z^3$$ $$0$$ $$2$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$2$$ Regulator: $$0.008231$$ Real period: $$16.18182$$ Tamagawa product: $$16$$ Torsion order: $$2$$ Leading coefficient: $$0.532825$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$5$$ $$8$$ $$4$$ $$1 + T$$
$$3$$ $$1$$ $$2$$ $$2$$ $$( 1 + T )( 1 + T + 3 T^{2} )$$
$$5$$ $$1$$ $$2$$ $$2$$ $$( 1 + T )( 1 + T + 5 T^{2} )$$
$$11$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 6 T + 11 T^{2} )$$

## 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 $$\Q$$.