# Properties

 Label 1122.b.2244.1 Conductor $1122$ Discriminant $2244$ Mordell-Weil group $$\Z/{2}\Z \times \Z/{6}\Z$$ 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

 $y^2 + (x^2 + x)y = x^5 + 7x^4 + 5x^3 - x^2 - x$ (homogenize, simplify) $y^2 + (x^2z + xz^2)y = x^5z + 7x^4z^2 + 5x^3z^3 - x^2z^4 - xz^5$ (dehomogenize, simplify) $y^2 = 4x^5 + 29x^4 + 22x^3 - 3x^2 - 4x$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, -1, 5, 7, 1]), R([0, 1, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, -1, 5, 7, 1], R![0, 1, 1]);

sage: X = HyperellipticCurve(R([0, -4, -3, 22, 29, 4]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$1122$$ $$=$$ $$2 \cdot 3 \cdot 11 \cdot 17$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$2244$$ $$=$$ $$2^{2} \cdot 3 \cdot 11 \cdot 17$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$1828$$ $$=$$ $$2^{2} \cdot 457$$ $$I_4$$ $$=$$ $$153793$$ $$=$$ $$113 \cdot 1361$$ $$I_6$$ $$=$$ $$73850145$$ $$=$$ $$3 \cdot 5 \cdot 349 \cdot 14107$$ $$I_{10}$$ $$=$$ $$287232$$ $$=$$ $$2^{9} \cdot 3 \cdot 11 \cdot 17$$ $$J_2$$ $$=$$ $$457$$ $$=$$ $$457$$ $$J_4$$ $$=$$ $$2294$$ $$=$$ $$2 \cdot 31 \cdot 37$$ $$J_6$$ $$=$$ $$8704$$ $$=$$ $$2^{9} \cdot 17$$ $$J_8$$ $$=$$ $$-321177$$ $$=$$ $$- 3 \cdot 151 \cdot 709$$ $$J_{10}$$ $$=$$ $$2244$$ $$=$$ $$2^{2} \cdot 3 \cdot 11 \cdot 17$$ $$g_1$$ $$=$$ $$19933382494057/2244$$ $$g_2$$ $$=$$ $$109474259971/1122$$ $$g_3$$ $$=$$ $$26732672/33$$

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: $$(1 : 0 : 0),\, (0 : 0 : 1),\, (-1 : -1 : 1),\, (-1 : 1 : 1)$$ All points: $$(1 : 0 : 0),\, (0 : 0 : 1),\, (-1 : -1 : 1),\, (-1 : 1 : 1)$$ All points: $$(1 : 0 : 0),\, (0 : 0 : 1),\, (-1 : -2 : 1),\, (-1 : 2 : 1)$$

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

magma: [C![-1,-2,1],C![-1,2,1],C![0,0,1],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: $$\Z/{2}\Z \times \Z/{6}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$4x^2 + xz - z^2$$ $$=$$ $$0,$$ $$8y$$ $$=$$ $$-3xz^2 - z^3$$ $$0$$ $$2$$
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$x^2 - xz - z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$3xz^2 + 2z^3$$ $$0$$ $$6$$
Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$4x^2 + xz - z^2$$ $$=$$ $$0,$$ $$8y$$ $$=$$ $$-3xz^2 - z^3$$ $$0$$ $$2$$
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$x^2 - xz - z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$3xz^2 + 2z^3$$ $$0$$ $$6$$
Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$4x^2 + xz - z^2$$ $$=$$ $$0,$$ $$8y$$ $$=$$ $$x^2z - 5xz^2 - 2z^3$$ $$0$$ $$2$$
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$x^2 - xz - z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^2z + 7xz^2 + 4z^3$$ $$0$$ $$6$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$2$$ Regulator: $$1$$ Real period: $$30.72413$$ Tamagawa product: $$2$$ Torsion order: $$12$$ Leading coefficient: $$0.426724$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$1$$ $$2$$ $$2$$ $$( 1 + T )( 1 - T + 2 T^{2} )$$
$$3$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 2 T + 3 T^{2} )$$
$$11$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 6 T + 11 T^{2} )$$
$$17$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 - 6 T + 17 T^{2} )$$

## 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 $$\Q$$.