# Properties

 Label 100035.a.300105.1 Conductor 100035 Discriminant 300105 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

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

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

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

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([9, -6, -3, 6, -2, 0, 1]))

## Invariants

 Conductor: $$N$$ = $$100035$$ = $$3^{4} \cdot 5 \cdot 13 \cdot 19$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$300105$$ = $$3^{5} \cdot 5 \cdot 13 \cdot 19$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$-2040$$ = $$- 2^{3} \cdot 3 \cdot 5 \cdot 17$$ $$I_4$$ = $$165060$$ = $$2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 131$$ $$I_6$$ = $$-93586104$$ = $$- 2^{3} \cdot 3^{8} \cdot 1783$$ $$I_{10}$$ = $$1229230080$$ = $$2^{12} \cdot 3^{5} \cdot 5 \cdot 13 \cdot 19$$ $$J_2$$ = $$-255$$ = $$- 3 \cdot 5 \cdot 17$$ $$J_4$$ = $$990$$ = $$2 \cdot 3^{2} \cdot 5 \cdot 11$$ $$J_6$$ = $$2304$$ = $$2^{8} \cdot 3^{2}$$ $$J_8$$ = $$-391905$$ = $$- 3^{3} \cdot 5 \cdot 2903$$ $$J_{10}$$ = $$300105$$ = $$3^{5} \cdot 5 \cdot 13 \cdot 19$$ $$g_1$$ = $$-887410625/247$$ $$g_2$$ = $$-13510750/247$$ $$g_3$$ = $$369920/741$$

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: $$(1 : 0 : 0),\, (1 : -1 : 0),\, (0 : 1 : 1),\, (0 : -2 : 1)$$

magma: [C![0,-2,1],C![0,1,1],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
$$(0 : 1 : 1) - (1 : -1 : 0)$$ $$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$z^3$$ $$0.057802$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$1$$ Regulator: $$0.057802$$ Real period: $$13.60235$$ Tamagawa product: $$2$$ Torsion order: $$1$$ Leading coefficient: $$1.572500$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$3$$ $$5$$ $$4$$ $$2$$ $$1 - T$$
$$5$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 5 T^{2} )$$
$$13$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 13 T^{2} )$$
$$19$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + T + 19 T^{2} )$$

## 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 $$\Q$$.