# Properties

 Label 1104.b.141312.1 Conductor $1104$ Discriminant $-141312$ Mordell-Weil group $$\Z/{2}\Z$$ 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

 $y^2 + (x^3 + x)y = -x^6 - 3x^4 + 29x^2 - 46$ (homogenize, simplify) $y^2 + (x^3 + xz^2)y = -x^6 - 3x^4z^2 + 29x^2z^4 - 46z^6$ (dehomogenize, simplify) $y^2 = -3x^6 - 10x^4 + 117x^2 - 184$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-46, 0, 29, 0, -3, 0, -1]), R([0, 1, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-46, 0, 29, 0, -3, 0, -1], R![0, 1, 0, 1]);

sage: X = HyperellipticCurve(R([-184, 0, 117, 0, -10, 0, -3]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$1104$$ $$=$$ $$2^{4} \cdot 3 \cdot 23$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-141312$$ $$=$$ $$- 2^{11} \cdot 3 \cdot 23$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$14220$$ $$=$$ $$2^{2} \cdot 3^{2} \cdot 5 \cdot 79$$ $$I_4$$ $$=$$ $$9418737$$ $$=$$ $$3 \cdot 19 \cdot 149 \cdot 1109$$ $$I_6$$ $$=$$ $$54280328031$$ $$=$$ $$3^{2} \cdot 503 \cdot 11990353$$ $$I_{10}$$ $$=$$ $$17664$$ $$=$$ $$2^{8} \cdot 3 \cdot 23$$ $$J_2$$ $$=$$ $$14220$$ $$=$$ $$2^{2} \cdot 3^{2} \cdot 5 \cdot 79$$ $$J_4$$ $$=$$ $$2146192$$ $$=$$ $$2^{4} \cdot 31 \cdot 4327$$ $$J_6$$ $$=$$ $$-16790479872$$ $$=$$ $$- 2^{10} \cdot 3 \cdot 23 \cdot 71 \cdot 3347$$ $$J_8$$ $$=$$ $$-60841690970176$$ $$=$$ $$- 2^{6} \cdot 950651421409$$ $$J_{10}$$ $$=$$ $$141312$$ $$=$$ $$2^{11} \cdot 3 \cdot 23$$ $$g_1$$ $$=$$ $$189267815942240625/46$$ $$g_2$$ $$=$$ $$2008843709918625/46$$ $$g_3$$ $$=$$ $$-24026098775400$$

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

 This curve has no rational points. This curve has no rational points. This curve has no rational points.

magma: []; // minimal model

magma: []; // simplified model

Number of rational Weierstrass points: $$0$$

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));

This curve is locally solvable except over $\Q_{2}$.

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: $$\Z/{2}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - D_\infty$$ $$3x^2 - 8z^2$$ $$=$$ $$0,$$ $$6y$$ $$=$$ $$-11xz^2$$ $$0$$ $$2$$
Generator $D_0$ Height Order
$$D_0 - D_\infty$$ $$3x^2 - 8z^2$$ $$=$$ $$0,$$ $$6y$$ $$=$$ $$-11xz^2$$ $$0$$ $$2$$
Generator $D_0$ Height Order
$$D_0 - D_\infty$$ $$3x^2 - 8z^2$$ $$=$$ $$0,$$ $$6y$$ $$=$$ $$x^3 - 21xz^2$$ $$0$$ $$2$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$2$$ Regulator: $$1$$ Real period: $$0.712625$$ Tamagawa product: $$1$$ Torsion order: $$2$$ Leading coefficient: $$0.356312$$ Analytic order of Ш: $$2$$   (rounded) Order of Ш: twice a square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$4$$ $$11$$ $$1$$ $$1 + T$$
$$3$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 3 T^{2} )$$
$$23$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 8 T + 23 T^{2} )$$

## 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 46.a
Elliptic curve isogeny class 24.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 $$\Q$$.