# Properties

 Label 22112.b.353792.1 Conductor 22112 Discriminant 353792 Mordell-Weil group $$\Z \times \Z$$ 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: Magma / SageMath

## Simplified equation

 $y^2 + y = x^6 - 6x^4 - 9x^3 - 4x^2$ (homogenize, simplify) $y^2 + z^3y = x^6 - 6x^4z^2 - 9x^3z^3 - 4x^2z^4$ (dehomogenize, simplify) $y^2 = 4x^6 - 24x^4 - 36x^3 - 16x^2 + 1$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, -4, -9, -6, 0, 1], R![1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, -4, -9, -6, 0, 1]), R([1]));

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([1, 0, -16, -36, -24, 0, 4]))

## Invariants

 Conductor: $$N$$ = $$22112$$ = $$2^{5} \cdot 691$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$353792$$ = $$2^{9} \cdot 691$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$672$$ = $$2^{5} \cdot 3 \cdot 7$$ $$I_4$$ = $$71232$$ = $$2^{6} \cdot 3 \cdot 7 \cdot 53$$ $$I_6$$ = $$6339072$$ = $$2^{9} \cdot 3 \cdot 4127$$ $$I_{10}$$ = $$1449132032$$ = $$2^{21} \cdot 691$$ $$J_2$$ = $$84$$ = $$2^{2} \cdot 3 \cdot 7$$ $$J_4$$ = $$-448$$ = $$- 2^{6} \cdot 7$$ $$J_6$$ = $$7680$$ = $$2^{9} \cdot 3 \cdot 5$$ $$J_8$$ = $$111104$$ = $$2^{9} \cdot 7 \cdot 31$$ $$J_{10}$$ = $$353792$$ = $$2^{9} \cdot 691$$ $$g_1$$ = $$8168202/691$$ $$g_2$$ = $$-518616/691$$ $$g_3$$ = $$105840/691$$

magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);

sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]

## 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

Known points
$$(1 : -1 : 0)$$ $$(1 : 1 : 0)$$ $$(0 : 0 : 1)$$ $$(-1 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(-1 : -1 : 1)$$
$$(-1 : -3 : 2)$$ $$(-1 : -5 : 2)$$ $$(-2 : -8 : 3)$$ $$(-3 : 9 : 2)$$ $$(-3 : -17 : 2)$$ $$(-2 : -19 : 3)$$
$$(1 : -51 : 5)$$ $$(1 : -74 : 5)$$

magma: [C![-3,-17,2],C![-3,9,2],C![-2,-19,3],C![-2,-8,3],C![-1,-5,2],C![-1,-3,2],C![-1,-1,1],C![-1,0,1],C![0,-1,1],C![0,0,1],C![1,-74,5],C![1,-51,5],C![1,-1,0],C![1,1,0]];

Number of rational Weierstrass points: $$0$$

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: $$\Z \times \Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(-1 : -3 : 2) + (-1 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)$$ $$(x + z) (2x + z)$$ $$=$$ $$0,$$ $$4y$$ $$=$$ $$-3xz^2 - 3z^3$$ $$0.220586$$ $$\infty$$
$$(-1 : -1 : 1) + (0 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)$$ $$x (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$xz^2$$ $$0.031601$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$2$$ Mordell-Weil rank: $$2$$ 2-Selmer rank: $$2$$ Regulator: $$0.006891$$ Real period: $$15.84805$$ Tamagawa product: $$8$$ Torsion order: $$1$$ Leading coefficient: $$0.873731$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$2$$ $$9$$ $$5$$ $$8$$ $$1$$
$$691$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 38 T + 691 T^{2} )$$

## 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 $$\Q$$.