# Properties

 Label 4428.a.239112.1 Conductor 4428 Discriminant 239112 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: SageMath / Magma

## Simplified equation

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

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

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

sage: X = HyperellipticCurve(R([1, -2, 1, 2, -2, -8, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$4428$$ $$=$$ $$2^{2} \cdot 3^{3} \cdot 41$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$239112$$ $$=$$ $$2^{3} \cdot 3^{6} \cdot 41$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$456$$ $$=$$ $$2^{3} \cdot 3 \cdot 19$$ $$I_4$$ $$=$$ $$-11196$$ $$=$$ $$- 2^{2} \cdot 3^{2} \cdot 311$$ $$I_6$$ $$=$$ $$-3802680$$ $$=$$ $$- 2^{3} \cdot 3^{3} \cdot 5 \cdot 7 \cdot 503$$ $$I_{10}$$ $$=$$ $$979402752$$ $$=$$ $$2^{15} \cdot 3^{6} \cdot 41$$ $$J_2$$ $$=$$ $$57$$ $$=$$ $$3 \cdot 19$$ $$J_4$$ $$=$$ $$252$$ $$=$$ $$2^{2} \cdot 3^{2} \cdot 7$$ $$J_6$$ $$=$$ $$5184$$ $$=$$ $$2^{6} \cdot 3^{4}$$ $$J_8$$ $$=$$ $$57996$$ $$=$$ $$2^{2} \cdot 3^{4} \cdot 179$$ $$J_{10}$$ $$=$$ $$239112$$ $$=$$ $$2^{3} \cdot 3^{6} \cdot 41$$ $$g_1$$ $$=$$ $$2476099/984$$ $$g_2$$ $$=$$ $$48013/246$$ $$g_3$$ $$=$$ $$2888/41$$

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
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(0 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(-1 : -1 : 1)$$ $$(-1 : 2 : 1)$$
$$(1 : -5 : 2)$$ $$(1 : -8 : 2)$$ $$(-5 : 1462 : 17)$$ $$(-5 : -4805 : 17)$$

magma: [C![-5,-4805,17],C![-5,1462,17],C![-1,-1,1],C![-1,2,1],C![0,-1,1],C![0,0,1],C![1,-8,2],C![1,-5,2],C![1,-1,0],C![1,0,0]];

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
$$(1 : -5 : 2) - (1 : 0 : 0)$$ $$z (2x - z)$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$-2x^3 - z^3$$ $$0.003691$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$1$$ Regulator: $$0.003691$$ Real period: $$11.60915$$ Tamagawa product: $$12$$ Torsion order: $$1$$ Leading coefficient: $$0.514252$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$2$$ $$3$$ $$3$$ $$1 + T + T^{2}$$
$$3$$ $$3$$ $$6$$ $$4$$ $$1 + T$$
$$41$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 3 T + 41 T^{2} )$$

## 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 $$\Q$$.