# Properties

 Label 1148.a.47068.1 Conductor $1148$ Discriminant $-47068$ Mordell-Weil group $$\Z/{10}\Z$$ Sato-Tate group $\mathrm{USp}(4)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q$$ $$\End(J) \otimes \Q$$ $$\Q$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type no

# Related objects

Show commands: SageMath / Magma

## Simplified equation

 $y^2 + (x^2 + x + 1)y = x^5 + 2x^4 - 5x^3 + x$ (homogenize, simplify) $y^2 + (x^2z + xz^2 + z^3)y = x^5z + 2x^4z^2 - 5x^3z^3 + xz^5$ (dehomogenize, simplify) $y^2 = 4x^5 + 9x^4 - 18x^3 + 3x^2 + 6x + 1$ (minimize, homogenize)

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

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

sage: X = HyperellipticCurve(R([1, 6, 3, -18, 9, 4]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$1148$$ $$=$$ $$2^{2} \cdot 7 \cdot 41$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-47068$$ $$=$$ $$- 2^{2} \cdot 7 \cdot 41^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$1236$$ $$=$$ $$2^{2} \cdot 3 \cdot 103$$ $$I_4$$ $$=$$ $$129537$$ $$=$$ $$3^{2} \cdot 37 \cdot 389$$ $$I_6$$ $$=$$ $$36025137$$ $$=$$ $$3^{2} \cdot 1907 \cdot 2099$$ $$I_{10}$$ $$=$$ $$-6024704$$ $$=$$ $$- 2^{9} \cdot 7 \cdot 41^{2}$$ $$J_2$$ $$=$$ $$309$$ $$=$$ $$3 \cdot 103$$ $$J_4$$ $$=$$ $$-1419$$ $$=$$ $$- 3 \cdot 11 \cdot 43$$ $$J_6$$ $$=$$ $$31221$$ $$=$$ $$3^{2} \cdot 3469$$ $$J_8$$ $$=$$ $$1908432$$ $$=$$ $$2^{4} \cdot 3^{2} \cdot 29 \cdot 457$$ $$J_{10}$$ $$=$$ $$-47068$$ $$=$$ $$- 2^{2} \cdot 7 \cdot 41^{2}$$ $$g_1$$ $$=$$ $$-2817036000549/47068$$ $$g_2$$ $$=$$ $$41865649551/47068$$ $$g_3$$ $$=$$ $$-2981012301/47068$$

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: $$(1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (-1 : -26 : 4)$$ All points: $$(1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (-1 : -26 : 4)$$ All points: $$(1 : 0 : 0),\, (0 : -1 : 1),\, (0 : 1 : 1),\, (-1 : 0 : 4)$$

magma: [C![-1,-26,4],C![0,-1,1],C![0,0,1],C![1,0,0]]; // minimal model

magma: [C![-1,0,4],C![0,-1,1],C![0,1,1],C![1,0,0]]; // simplified model

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: $$\Z/{10}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(-1 : -26 : 4) + (0 : 0 : 1) - 2 \cdot(1 : 0 : 0)$$ $$x (4x + z)$$ $$=$$ $$0,$$ $$8y$$ $$=$$ $$13xz^2$$ $$0$$ $$10$$
Generator $D_0$ Height Order
$$(-1 : -26 : 4) + (0 : 0 : 1) - 2 \cdot(1 : 0 : 0)$$ $$x (4x + z)$$ $$=$$ $$0,$$ $$8y$$ $$=$$ $$13xz^2$$ $$0$$ $$10$$
Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$x (4x + z)$$ $$=$$ $$0,$$ $$8y$$ $$=$$ $$x^2z + 27xz^2 + z^3$$ $$0$$ $$10$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$1$$ Regulator: $$1$$ Real period: $$22.53131$$ Tamagawa product: $$2$$ Torsion order: $$10$$ Leading coefficient: $$0.450626$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$2$$ $$2$$ $$1$$ $$1 + T^{2}$$
$$7$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 2 T + 7 T^{2} )$$
$$41$$ $$1$$ $$2$$ $$2$$ $$( 1 - T )( 1 + 8 T + 41 T^{2} )$$

## 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 $$\Q$$.