# Properties

 Label 1122.a.1122.1 Conductor $1122$ Discriminant $1122$ Mordell-Weil group $$\Z/{2}\Z \times \Z/{2}\Z$$ Sato-Tate group $\mathrm{SU}(2)\times\mathrm{SU}(2)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q \times \Q$$ $$\End(J) \otimes \Q$$ $$\Q \times \Q$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Related objects

Show commands: SageMath / Magma

## Simplified equation

 $y^2 + (x^2 + x)y = x^5 + 7x^4 - 43x^2 + 51x - 17$ (homogenize, simplify) $y^2 + (x^2z + xz^2)y = x^5z + 7x^4z^2 - 43x^2z^4 + 51xz^5 - 17z^6$ (dehomogenize, simplify) $y^2 = 4x^5 + 29x^4 + 2x^3 - 171x^2 + 204x - 68$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-17, 51, -43, 0, 7, 1]), R([0, 1, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-17, 51, -43, 0, 7, 1], R![0, 1, 1]);

sage: X = HyperellipticCurve(R([-68, 204, -171, 2, 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: $$\Delta$$ $$=$$ $$1122$$ $$=$$ $$2 \cdot 3 \cdot 11 \cdot 17$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$56004$$ $$=$$ $$2^{2} \cdot 3 \cdot 13 \cdot 359$$ $$I_4$$ $$=$$ $$288321$$ $$=$$ $$3 \cdot 11 \cdot 8737$$ $$I_6$$ $$=$$ $$5331417537$$ $$=$$ $$3 \cdot 19 \cdot 93533641$$ $$I_{10}$$ $$=$$ $$143616$$ $$=$$ $$2^{8} \cdot 3 \cdot 11 \cdot 17$$ $$J_2$$ $$=$$ $$14001$$ $$=$$ $$3 \cdot 13 \cdot 359$$ $$J_4$$ $$=$$ $$8155820$$ $$=$$ $$2^{2} \cdot 5 \cdot 407791$$ $$J_6$$ $$=$$ $$6325887612$$ $$=$$ $$2^{2} \cdot 3 \cdot 11 \cdot 17 \cdot 2819023$$ $$J_8$$ $$=$$ $$5512838145803$$ $$=$$ $$198173 \cdot 27818311$$ $$J_{10}$$ $$=$$ $$1122$$ $$=$$ $$2 \cdot 3 \cdot 11 \cdot 17$$ $$g_1$$ $$=$$ $$179338702480653356667/374$$ $$g_2$$ $$=$$ $$3730727674118765970/187$$ $$g_3$$ $$=$$ $$1105214886926046$$

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

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

## Automorphism group

 $$\mathrm{Aut}(X)$$ $$\simeq$$ $C_2^2$ magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$$C_2^2$magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);

## Rational points

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

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

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

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$4x^2 + 17xz - 17z^2$$ $$=$$ $$0,$$ $$8y$$ $$=$$ $$13xz^2 - 17z^3$$ $$0$$ $$2$$
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$x^2 + 4xz - 4z^2$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$3xz^2 - 4z^3$$ $$0$$ $$2$$
Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$4x^2 + 17xz - 17z^2$$ $$=$$ $$0,$$ $$8y$$ $$=$$ $$13xz^2 - 17z^3$$ $$0$$ $$2$$
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$x^2 + 4xz - 4z^2$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$3xz^2 - 4z^3$$ $$0$$ $$2$$
Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$4x^2 + 17xz - 17z^2$$ $$=$$ $$0,$$ $$8y$$ $$=$$ $$x^2z + 27xz^2 - 34z^3$$ $$0$$ $$2$$
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$x^2 + 4xz - 4z^2$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$x^2z + 7xz^2 - 8z^3$$ $$0$$ $$2$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$2$$ Regulator: $$1$$ Real period: $$6.820718$$ Tamagawa product: $$1$$ Torsion order: $$4$$ Leading coefficient: $$0.426294$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

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

## Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $\mathrm{SU}(2)\times\mathrm{SU}(2)$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{SU}(2)\times\mathrm{SU}(2)$$

## Decomposition of the Jacobian

Splits over $$\Q$$

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
Elliptic curve isogeny class 66.b
Elliptic curve isogeny class 17.a

## Endomorphisms of the Jacobian

Of $$\GL_2$$-type over $$\Q$$

Endomorphism ring over $$\Q$$:

 $$\End (J_{})$$ $$\simeq$$ an order of index $$2$$ in $$\Z \times \Z$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\times$$ $$\Q$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R \times \R$$

All $$\overline{\Q}$$-endomorphisms of the Jacobian are defined over $$\Q$$.