# Properties

 Label 100035.b.300105.1 Conductor 100035 Discriminant 300105 Mordell-Weil group $$\Z \times \Z/{2}\Z$$ 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

 $y^2 + (x^3 + 1)y = x^5 - 3x^4 - 5x^3 + 11x^2 - 5$ (homogenize, simplify) $y^2 + (x^3 + z^3)y = x^5z - 3x^4z^2 - 5x^3z^3 + 11x^2z^4 - 5z^6$ (dehomogenize, simplify) $y^2 = x^6 + 4x^5 - 12x^4 - 18x^3 + 44x^2 - 19$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-5, 0, 11, -5, -3, 1], R![1, 0, 0, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-5, 0, 11, -5, -3, 1]), R([1, 0, 0, 1]));

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([-19, 0, 44, -18, -12, 4, 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

 $$I_2$$ = $$14952$$ = $$2^{3} \cdot 3 \cdot 7 \cdot 89$$ $$I_4$$ = $$928836$$ = $$2^{2} \cdot 3^{2} \cdot 25801$$ $$I_6$$ = $$4462396488$$ = $$2^{3} \cdot 3^{3} \cdot 11 \cdot 241 \cdot 7793$$ $$I_{10}$$ = $$1229230080$$ = $$2^{12} \cdot 3^{5} \cdot 5 \cdot 13 \cdot 19$$ $$J_2$$ = $$1869$$ = $$3 \cdot 7 \cdot 89$$ $$J_4$$ = $$135873$$ = $$3^{2} \cdot 31 \cdot 487$$ $$J_6$$ = $$12388689$$ = $$3^{2} \cdot 911 \cdot 1511$$ $$J_8$$ = $$1173246903$$ = $$3^{3} \cdot 19 \cdot 2287031$$ $$J_{10}$$ = $$300105$$ = $$3^{5} \cdot 5 \cdot 13 \cdot 19$$ $$g_1$$ = $$93851287159343/1235$$ $$g_2$$ = $$3650520528599/1235$$ $$g_3$$ = $$534267719209/3705$$

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

magma: [C![1,-1,0],C![1,-1,1],C![1,0,0]];

Number of rational Weierstrass points: $$1$$

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: $$\Z \times \Z/{2}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(1 : -1 : 1) - (1 : 0 : 0)$$ $$z (x - z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3$$ $$0.246223$$ $$\infty$$
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2 - xz - z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2 - z^3$$ $$0$$ $$2$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$2$$ Regulator: $$0.246223$$ Real period: $$15.99319$$ Tamagawa product: $$2$$ Torsion order: $$2$$ Leading coefficient: $$1.968946$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

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

## 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 $$\Q$$.