# Properties

 Label 7927.b.7927.1 Conductor 7927 Discriminant -7927 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: SageMath / Magma

## Simplified equation

 $y^2 + x^3y = -2x^4 - x^3 + 3x^2 + x - 1$ (homogenize, simplify) $y^2 + x^3y = -2x^4z^2 - x^3z^3 + 3x^2z^4 + xz^5 - z^6$ (dehomogenize, simplify) $y^2 = x^6 - 8x^4 - 4x^3 + 12x^2 + 4x - 4$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, 1, 3, -1, -2]), R([0, 0, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, 1, 3, -1, -2], R![0, 0, 0, 1]);

sage: X = HyperellipticCurve(R([-4, 4, 12, -4, -8, 0, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$7927$$ $$=$$ $$7927$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-7927$$ $$=$$ $$-7927$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$2592$$ $$=$$ $$2^{5} \cdot 3^{4}$$ $$I_4$$ $$=$$ $$38208$$ $$=$$ $$2^{6} \cdot 3 \cdot 199$$ $$I_6$$ $$=$$ $$33408000$$ $$=$$ $$2^{10} \cdot 3^{2} \cdot 5^{3} \cdot 29$$ $$I_{10}$$ $$=$$ $$-32468992$$ $$=$$ $$- 2^{12} \cdot 7927$$ $$J_2$$ $$=$$ $$324$$ $$=$$ $$2^{2} \cdot 3^{4}$$ $$J_4$$ $$=$$ $$3976$$ $$=$$ $$2^{3} \cdot 7 \cdot 71$$ $$J_6$$ $$=$$ $$56552$$ $$=$$ $$2^{3} \cdot 7069$$ $$J_8$$ $$=$$ $$628568$$ $$=$$ $$2^{3} \cdot 78571$$ $$J_{10}$$ $$=$$ $$-7927$$ $$=$$ $$-7927$$ $$g_1$$ $$=$$ $$-3570467226624/7927$$ $$g_2$$ $$=$$ $$-135232602624/7927$$ $$g_3$$ $$=$$ $$-5936602752/7927$$

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

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

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

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
$$D_0 - 2 \cdot(1 : -1 : 0)$$ $$z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0.265853$$ $$\infty$$
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2 + xz - z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2$$ $$0.069502$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$2$$ Mordell-Weil rank: $$2$$ 2-Selmer rank: $$2$$ Regulator: $$0.018265$$ Real period: $$21.11808$$ Tamagawa product: $$1$$ Torsion order: $$1$$ Leading coefficient: $$0.385723$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$7927$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 91 T + 7927 T^{2} )$$

## 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 $$\Q$$.