# Properties

 Label 100036.a.200072.1 Conductor 100036 Discriminant 200072 Mordell-Weil group $$\Z \times \Z$$ 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

 $y^2 + (x^3 + x^2 + 1)y = x^5 - 8x^3 - x^2 + 12x - 6$ (homogenize, simplify) $y^2 + (x^3 + x^2z + z^3)y = x^5z - 8x^3z^3 - x^2z^4 + 12xz^5 - 6z^6$ (dehomogenize, simplify) $y^2 = x^6 + 6x^5 + x^4 - 30x^3 - 2x^2 + 48x - 23$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-6, 12, -1, -8, 0, 1], R![1, 0, 1, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-6, 12, -1, -8, 0, 1]), R([1, 0, 1, 1]));

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([-23, 48, -2, -30, 1, 6, 1]))

## Invariants

 Conductor: $$N$$ = $$100036$$ = $$2^{2} \cdot 89 \cdot 281$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$200072$$ = $$2^{3} \cdot 89 \cdot 281$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$22472$$ = $$2^{3} \cdot 53^{2}$$ $$I_4$$ = $$346180$$ = $$2^{2} \cdot 5 \cdot 19 \cdot 911$$ $$I_6$$ = $$2597903816$$ = $$2^{3} \cdot 1019 \cdot 318683$$ $$I_{10}$$ = $$819494912$$ = $$2^{15} \cdot 89 \cdot 281$$ $$J_2$$ = $$2809$$ = $$53^{2}$$ $$J_4$$ = $$325164$$ = $$2^{2} \cdot 3 \cdot 7^{3} \cdot 79$$ $$J_6$$ = $$49609856$$ = $$2^{7} \cdot 387577$$ $$J_8$$ = $$8405614652$$ = $$2^{2} \cdot 2101403663$$ $$J_{10}$$ = $$200072$$ = $$2^{3} \cdot 89 \cdot 281$$ $$g_1$$ = $$174887470365513049/200072$$ $$g_2$$ = $$1801763080537539/50018$$ $$g_3$$ = $$48930703272592/25009$$

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

Known points
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(1 : -1 : 1)$$ $$(-2 : 1 : 1)$$ $$(1 : -2 : 1)$$ $$(-2 : 2 : 1)$$
$$(4 : -61 : 3)$$ $$(4 : -78 : 3)$$

magma: [C![-2,1,1],C![-2,2,1],C![1,-2,1],C![1,-1,0],C![1,-1,1],C![1,0,0],C![4,-78,3],C![4,-61,3]];

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: $$\Z \times \Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(-2 : 2 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)$$ $$(x - z) (x + 2z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2$$ $$0.301384$$ $$\infty$$
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2 + xz - 3z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2 - z^3$$ $$0.100952$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$2$$ Mordell-Weil rank: $$2$$ 2-Selmer rank: $$2$$ Regulator: $$0.028791$$ Real period: $$14.12244$$ Tamagawa product: $$3$$ Torsion order: $$1$$ Leading coefficient: $$1.219825$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$2$$ $$3$$ $$2$$ $$3$$ $$1 + T + T^{2}$$
$$89$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 - 14 T + 89 T^{2} )$$
$$281$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 - 20 T + 281 T^{2} )$$

## 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 $$\Q$$.