# Properties

 Label 100076.a.200152.1 Conductor 100076 Discriminant 200152 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: SageMath / Magma

## Simplified equation

 $y^2 + (x^3 + x + 1)y = -6x^4 + 10x^3 - 2x^2$ (homogenize, simplify) $y^2 + (x^3 + xz^2 + z^3)y = -6x^4z^2 + 10x^3z^3 - 2x^2z^4$ (dehomogenize, simplify) $y^2 = x^6 - 22x^4 + 42x^3 - 7x^2 + 2x + 1$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, -2, 10, -6]), R([1, 1, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, -2, 10, -6], R![1, 1, 0, 1]);

sage: X = HyperellipticCurve(R([1, 2, -7, 42, -22, 0, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$100076$$ $$=$$ $$2^{2} \cdot 127 \cdot 197$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$200152$$ $$=$$ $$2^{3} \cdot 127 \cdot 197$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$7880$$ $$=$$ $$2^{3} \cdot 5 \cdot 197$$ $$I_4$$ $$=$$ $$-345596$$ $$=$$ $$- 2^{2} \cdot 86399$$ $$I_6$$ $$=$$ $$-1468066040$$ $$=$$ $$- 2^{3} \cdot 5 \cdot 7 \cdot 5243093$$ $$I_{10}$$ $$=$$ $$819822592$$ $$=$$ $$2^{15} \cdot 127 \cdot 197$$ $$J_2$$ $$=$$ $$985$$ $$=$$ $$5 \cdot 197$$ $$J_4$$ $$=$$ $$44026$$ $$=$$ $$2 \cdot 22013$$ $$J_6$$ $$=$$ $$3775940$$ $$=$$ $$2^{2} \cdot 5 \cdot 7^{2} \cdot 3853$$ $$J_8$$ $$=$$ $$445253056$$ $$=$$ $$2^{6} \cdot 67 \cdot 103837$$ $$J_{10}$$ $$=$$ $$200152$$ $$=$$ $$2^{3} \cdot 127 \cdot 197$$ $$g_1$$ $$=$$ $$4706682753125/1016$$ $$g_2$$ $$=$$ $$106787814625/508$$ $$g_3$$ $$=$$ $$4649126125/254$$

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

Known points
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(0 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(2 : -3 : 1)$$ $$(1 : -1 : 5)$$
$$(2 : -8 : 1)$$ $$(3 : -13 : 1)$$ $$(3 : -18 : 1)$$ $$(1 : -150 : 5)$$

magma: [C![0,-1,1],C![0,0,1],C![1,-150,5],C![1,-1,0],C![1,-1,5],C![1,0,0],C![2,-8,1],C![2,-3,1],C![3,-18,1],C![3,-13,1]];

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
$$(0 : -1 : 1) - (1 : 0 : 0)$$ $$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3 - z^3$$ $$0.479087$$ $$\infty$$
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2 - 3xz + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-4xz^2$$ $$0.077183$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$2$$ Mordell-Weil rank: $$2$$ 2-Selmer rank: $$2$$ Regulator: $$0.035573$$ Real period: $$13.92968$$ Tamagawa product: $$3$$ Torsion order: $$1$$ Leading coefficient: $$1.486586$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$2$$ $$2$$ $$3$$ $$3$$ $$1 + T + T^{2}$$
$$127$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 - 16 T + 127 T^{2} )$$
$$197$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 - 24 T + 197 T^{2} )$$

## 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 $$\Q$$.