# Properties

 Label 100017.a.300051.1 Conductor 100017 Discriminant 300051 Mordell-Weil group $$\Z/{3}\Z$$ 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: Magma / SageMath

## Simplified equation

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

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

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

magma: X,pi:= SimplifiedModel(C);

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

## Invariants

 Conductor: $$N$$ = $$100017$$ = $$3^{2} \cdot 11113$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$300051$$ = $$3^{3} \cdot 11113$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$904$$ = $$2^{3} \cdot 113$$ $$I_4$$ = $$-5084$$ = $$- 2^{2} \cdot 31 \cdot 41$$ $$I_6$$ = $$7140520$$ = $$2^{3} \cdot 5 \cdot 178513$$ $$I_{10}$$ = $$1229008896$$ = $$2^{12} \cdot 3^{3} \cdot 11113$$ $$J_2$$ = $$113$$ = $$113$$ $$J_4$$ = $$585$$ = $$3^{2} \cdot 5 \cdot 13$$ $$J_6$$ = $$-10719$$ = $$- 3^{3} \cdot 397$$ $$J_8$$ = $$-388368$$ = $$- 2^{4} \cdot 3^{3} \cdot 29 \cdot 31$$ $$J_{10}$$ = $$300051$$ = $$3^{3} \cdot 11113$$ $$g_1$$ = $$18424351793/300051$$ $$g_2$$ = $$93788305/33339$$ $$g_3$$ = $$-5069293/11113$$

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

magma: [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: $$\Z/{3}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : -1 : 0)$$ $$z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$3$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$0$$ Regulator: $$1$$ Real period: $$8.897841$$ Tamagawa product: $$3$$ Torsion order: $$3$$ Leading coefficient: $$2.965947$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$3$$ $$3$$ $$2$$ $$3$$ $$( 1 - T )( 1 + T )$$
$$11113$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 145 T + 11113 T^{2} )$$

## 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 $$\Q$$.