# Properties

 Label 100096.b.100096.1 Conductor 100096 Discriminant -100096 Mordell-Weil group $$\Z/{2}\Z$$ 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

 $y^2 = x^5 - 9x^3 + 5x^2 + 21x - 20$ (homogenize, simplify) $y^2 = x^5z - 9x^3z^3 + 5x^2z^4 + 21xz^5 - 20z^6$ (dehomogenize, simplify) $y^2 = x^5 - 9x^3 + 5x^2 + 21x - 20$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-20, 21, 5, -9, 0, 1], R![]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-20, 21, 5, -9, 0, 1]), R([]));

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([-20, 21, 5, -9, 0, 1]))

## Invariants

 Conductor: $$N$$ = $$100096$$ = $$2^{8} \cdot 17 \cdot 23$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$-100096$$ = $$- 2^{8} \cdot 17 \cdot 23$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$21216$$ = $$2^{5} \cdot 3 \cdot 13 \cdot 17$$ $$I_4$$ = $$-344064$$ = $$- 2^{14} \cdot 3 \cdot 7$$ $$I_6$$ = $$-2446712832$$ = $$- 2^{13} \cdot 3 \cdot 29 \cdot 3433$$ $$I_{10}$$ = $$-409993216$$ = $$- 2^{20} \cdot 17 \cdot 23$$ $$J_2$$ = $$2652$$ = $$2^{2} \cdot 3 \cdot 13 \cdot 17$$ $$J_4$$ = $$296630$$ = $$2 \cdot 5 \cdot 29663$$ $$J_6$$ = $$44782996$$ = $$2^{2} \cdot 101 \cdot 110849$$ $$J_8$$ = $$7693787123$$ = $$11 \cdot 461 \cdot 1517213$$ $$J_{10}$$ = $$-100096$$ = $$- 2^{8} \cdot 17 \cdot 23$$ $$g_1$$ = $$-30142461298716/23$$ $$g_2$$ = $$-2542592373165/46$$ $$g_3$$ = $$-289488481893/92$$

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: $$(1 : 0 : 0)$$

magma: [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: $$\Z/{2}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$x^2 + xz - 4z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$2$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$1$$ Regulator: $$1$$ Real period: $$6.972917$$ Tamagawa product: $$1$$ Torsion order: $$2$$ Leading coefficient: $$1.743229$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$2$$ $$8$$ $$8$$ $$1$$ $$1$$
$$17$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 4 T + 17 T^{2} )$$
$$23$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 2 T + 23 T^{2} )$$

## 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 $$\Q$$.