# Properties

 Label 1612.a.3224.1 Conductor 1612 Discriminant 3224 Mordell-Weil group $$\Z$$ 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

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

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

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

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([1, 6, 9, 6, 2, 0, 1]))

## Invariants

 Conductor: $$N$$ = $$1612$$ = $$2^{2} \cdot 13 \cdot 31$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$3224$$ = $$2^{3} \cdot 13 \cdot 31$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$-312$$ = $$- 2^{3} \cdot 3 \cdot 13$$ $$I_4$$ = $$2436$$ = $$2^{2} \cdot 3 \cdot 7 \cdot 29$$ $$I_6$$ = $$-258168$$ = $$- 2^{3} \cdot 3 \cdot 31 \cdot 347$$ $$I_{10}$$ = $$13205504$$ = $$2^{15} \cdot 13 \cdot 31$$ $$J_2$$ = $$-39$$ = $$- 3 \cdot 13$$ $$J_4$$ = $$38$$ = $$2 \cdot 19$$ $$J_6$$ = $$36$$ = $$2^{2} \cdot 3^{2}$$ $$J_8$$ = $$-712$$ = $$- 2^{3} \cdot 89$$ $$J_{10}$$ = $$3224$$ = $$2^{3} \cdot 13 \cdot 31$$ $$g_1$$ = $$-6940323/248$$ $$g_2$$ = $$-86697/124$$ $$g_3$$ = $$1053/62$$

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

All points
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(0 : 0 : 1)$$ $$(-1 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(-1 : 1 : 1)$$
$$(1 : 1 : 1)$$ $$(1 : -4 : 1)$$

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

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: $$\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(0 : 0 : 1) - (1 : 0 : 0)$$ $$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3$$ $$0.004499$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$1$$ Regulator: $$0.004499$$ Real period: $$20.03241$$ Tamagawa product: $$3$$ Torsion order: $$1$$ Leading coefficient: $$0.270409$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$2$$ $$3$$ $$2$$ $$3$$ $$1 + T + T^{2}$$
$$13$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 3 T + 13 T^{2} )$$
$$31$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 2 T + 31 T^{2} )$$

## 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 $$\Q$$.