# Properties

 Label 932.a.3728.1 Conductor $932$ Discriminant $-3728$ Mordell-Weil group $$\Z$$ 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

 $y^2 + y = x^6 - 2x^5 + x^4 + x^2 - x$ (homogenize, simplify) $y^2 + z^3y = x^6 - 2x^5z + x^4z^2 + x^2z^4 - xz^5$ (dehomogenize, simplify) $y^2 = 4x^6 - 8x^5 + 4x^4 + 4x^2 - 4x + 1$ (minimize, homogenize)

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

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

sage: X = HyperellipticCurve(R([1, -4, 4, 0, 4, -8, 4]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$932$$ $$=$$ $$2^{2} \cdot 233$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-3728$$ $$=$$ $$- 2^{4} \cdot 233$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$8$$ $$=$$ $$2^{3}$$ $$I_4$$ $$=$$ $$229$$ $$=$$ $$229$$ $$I_6$$ $$=$$ $$527$$ $$=$$ $$17 \cdot 31$$ $$I_{10}$$ $$=$$ $$-466$$ $$=$$ $$- 2 \cdot 233$$ $$J_2$$ $$=$$ $$8$$ $$=$$ $$2^{3}$$ $$J_4$$ $$=$$ $$-150$$ $$=$$ $$- 2 \cdot 3 \cdot 5^{2}$$ $$J_6$$ $$=$$ $$-128$$ $$=$$ $$- 2^{7}$$ $$J_8$$ $$=$$ $$-5881$$ $$=$$ $$-5881$$ $$J_{10}$$ $$=$$ $$-3728$$ $$=$$ $$- 2^{4} \cdot 233$$ $$g_1$$ $$=$$ $$-2048/233$$ $$g_2$$ $$=$$ $$4800/233$$ $$g_3$$ $$=$$ $$512/233$$

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
$$(1 : -1 : 0)$$ $$(1 : 1 : 0)$$ $$(0 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(1 : 0 : 1)$$ $$(1 : -1 : 1)$$
$$(-1 : 2 : 1)$$ $$(-1 : -3 : 1)$$ $$(1 : -3 : 2)$$ $$(1 : -5 : 2)$$
All points
$$(1 : -1 : 0)$$ $$(1 : 1 : 0)$$ $$(0 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(1 : 0 : 1)$$ $$(1 : -1 : 1)$$
$$(-1 : 2 : 1)$$ $$(-1 : -3 : 1)$$ $$(1 : -3 : 2)$$ $$(1 : -5 : 2)$$
All points
$$(1 : -2 : 0)$$ $$(1 : 2 : 0)$$ $$(0 : -1 : 1)$$ $$(0 : 1 : 1)$$ $$(1 : -1 : 1)$$ $$(1 : 1 : 1)$$
$$(1 : -2 : 2)$$ $$(1 : 2 : 2)$$ $$(-1 : -5 : 1)$$ $$(-1 : 5 : 1)$$

magma: [C![-1,-3,1],C![-1,2,1],C![0,-1,1],C![0,0,1],C![1,-5,2],C![1,-3,2],C![1,-1,0],C![1,-1,1],C![1,0,1],C![1,1,0]]; // minimal model

magma: [C![-1,-5,1],C![-1,5,1],C![0,-1,1],C![0,1,1],C![1,-2,2],C![1,2,2],C![1,-2,0],C![1,-1,1],C![1,1,1],C![1,2,0]]; // simplified model

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
$$(1 : -1 : 1) - (1 : 1 : 0)$$ $$z (x - z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3$$ $$0.002249$$ $$\infty$$
Generator $D_0$ Height Order
$$(1 : -1 : 1) - (1 : 1 : 0)$$ $$z (x - z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3$$ $$0.002249$$ $$\infty$$
Generator $D_0$ Height Order
$$(1 : -1 : 1) - (1 : 2 : 0)$$ $$z (x - z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-2x^3 + z^3$$ $$0.002249$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$1$$ Regulator: $$0.002249$$ Real period: $$25.16836$$ Tamagawa product: $$3$$ Torsion order: $$1$$ Leading coefficient: $$0.169871$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$2$$ $$4$$ $$3$$ $$1 + 2 T + 2 T^{2}$$
$$233$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 19 T + 233 T^{2} )$$

## 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 $$\Q$$.