# Properties

 Label 7848.a.188352.1 Conductor 7848 Discriminant 188352 Mordell-Weil group $$\Z \times \Z/{3}\Z$$ 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

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

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

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

sage: X = HyperellipticCurve(R([4, -4, -3, 0, 2, 4, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$7848$$ $$=$$ $$2^{3} \cdot 3^{2} \cdot 109$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$188352$$ $$=$$ $$2^{6} \cdot 3^{3} \cdot 109$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$-1504$$ $$=$$ $$- 2^{5} \cdot 47$$ $$I_4$$ $$=$$ $$4864$$ $$=$$ $$2^{8} \cdot 19$$ $$I_6$$ $$=$$ $$-10466560$$ $$=$$ $$- 2^{8} \cdot 5 \cdot 13 \cdot 17 \cdot 37$$ $$I_{10}$$ $$=$$ $$771489792$$ $$=$$ $$2^{18} \cdot 3^{3} \cdot 109$$ $$J_2$$ $$=$$ $$-188$$ $$=$$ $$- 2^{2} \cdot 47$$ $$J_4$$ $$=$$ $$1422$$ $$=$$ $$2 \cdot 3^{2} \cdot 79$$ $$J_6$$ $$=$$ $$144$$ $$=$$ $$2^{4} \cdot 3^{2}$$ $$J_8$$ $$=$$ $$-512289$$ $$=$$ $$- 3^{2} \cdot 56921$$ $$J_{10}$$ $$=$$ $$188352$$ $$=$$ $$2^{6} \cdot 3^{3} \cdot 109$$ $$g_1$$ $$=$$ $$-3669520112/2943$$ $$g_2$$ $$=$$ $$-16404034/327$$ $$g_3$$ $$=$$ $$8836/327$$

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
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(-1 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(0 : 1 : 1)$$ $$(1 : 0 : 1)$$
$$(-1 : 2 : 1)$$ $$(1 : -2 : 1)$$ $$(-4 : 23 : 1)$$ $$(-4 : 45 : 1)$$

magma: [C![-4,23,1],C![-4,45,1],C![-1,0,1],C![-1,2,1],C![0,-1,1],C![0,1,1],C![1,-2,1],C![1,-1,0],C![1,0,0],C![1,0,1]];

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: $$\Z \times \Z/{3}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(0 : 1 : 1) - (1 : -1 : 0)$$ $$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$z^3$$ $$0.022214$$ $$\infty$$
$$(-1 : 2 : 1) + (1 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)$$ $$(x - z) (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2 + z^3$$ $$0$$ $$3$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$1$$ Regulator: $$0.022214$$ Real period: $$15.20359$$ Tamagawa product: $$18$$ Torsion order: $$3$$ Leading coefficient: $$0.675481$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$3$$ $$6$$ $$6$$ $$1 + T$$
$$3$$ $$2$$ $$3$$ $$3$$ $$1 + T + T^{2}$$
$$109$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 - 14 T + 109 T^{2} )$$

## 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 $$\Q$$.