# Properties

 Label 100278.a.200556.1 Conductor 100278 Discriminant -200556 Mordell-Weil group $$\Z$$ 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

 $y^2 + x^2y = 2x^5 + 9x^4 + x^3 - 21x^2 + 15x - 3$ (homogenize, simplify) $y^2 + x^2zy = 2x^5z + 9x^4z^2 + x^3z^3 - 21x^2z^4 + 15xz^5 - 3z^6$ (dehomogenize, simplify) $y^2 = 8x^5 + 37x^4 + 4x^3 - 84x^2 + 60x - 12$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-3, 15, -21, 1, 9, 2], R![0, 0, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-3, 15, -21, 1, 9, 2]), R([0, 0, 1]));

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([-12, 60, -84, 4, 37, 8]))

## Invariants

 Conductor: $$N$$ = $$100278$$ = $$2 \cdot 3^{4} \cdot 619$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$-200556$$ = $$- 2^{2} \cdot 3^{4} \cdot 619$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$69024$$ = $$2^{5} \cdot 3 \cdot 719$$ $$I_4$$ = $$598464$$ = $$2^{6} \cdot 3^{2} \cdot 1039$$ $$I_6$$ = $$14277219264$$ = $$2^{6} \cdot 3^{2} \cdot 7 \cdot 11 \cdot 487 \cdot 661$$ $$I_{10}$$ = $$-821477376$$ = $$- 2^{14} \cdot 3^{4} \cdot 619$$ $$J_2$$ = $$8628$$ = $$2^{2} \cdot 3 \cdot 719$$ $$J_4$$ = $$3095532$$ = $$2^{2} \cdot 3^{2} \cdot 11 \cdot 7817$$ $$J_6$$ = $$1476933817$$ = $$379 \cdot 3896923$$ $$J_8$$ = $$790166652513$$ = $$3 \cdot 7 \cdot 37626983453$$ $$J_{10}$$ = $$-200556$$ = $$- 2^{2} \cdot 3^{4} \cdot 619$$ $$g_1$$ = $$-147572580633292032/619$$ $$g_2$$ = $$-6136499412390336/619$$ $$g_3$$ = $$-3054068731880548/5571$$

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: $$(1 : 0 : 0),\, (1 : 0 : 2),\, (1 : -2 : 2)$$

magma: [C![1,-2,2],C![1,0,0],C![1,0,2]];

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: $$\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$2x^2 + 5xz - 4z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$xz^2 - z^3$$ $$0.165292$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$1$$ Regulator: $$0.165292$$ Real period: $$7.453400$$ Tamagawa product: $$2$$ Torsion order: $$1$$ Leading coefficient: $$2.463979$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$2$$ $$2$$ $$1$$ $$2$$ $$( 1 + T )( 1 - 2 T + 2 T^{2} )$$
$$3$$ $$4$$ $$4$$ $$1$$ $$1 + 2 T + 3 T^{2}$$
$$619$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 4 T + 619 T^{2} )$$

## 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 $$\Q$$.