# Properties

 Label 5501.a.5501.1 Conductor 5501 Discriminant 5501 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

 $y^2 + (x^3 + x^2 + 1)y = -2x^3 - 3x^2 + x + 2$ (homogenize, simplify) $y^2 + (x^3 + x^2z + z^3)y = -2x^3z^3 - 3x^2z^4 + xz^5 + 2z^6$ (dehomogenize, simplify) $y^2 = x^6 + 2x^5 + x^4 - 6x^3 - 10x^2 + 4x + 9$ (minimize, homogenize)

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

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

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([9, 4, -10, -6, 1, 2, 1]))

## Invariants

 Conductor: $$N$$ = $$5501$$ = $$5501$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$5501$$ = $$5501$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$-1464$$ = $$- 2^{3} \cdot 3 \cdot 61$$ $$I_4$$ = $$88164$$ = $$2^{2} \cdot 3^{2} \cdot 31 \cdot 79$$ $$I_6$$ = $$-50339736$$ = $$- 2^{3} \cdot 3^{2} \cdot 743 \cdot 941$$ $$I_{10}$$ = $$22532096$$ = $$2^{12} \cdot 5501$$ $$J_2$$ = $$-183$$ = $$- 3 \cdot 61$$ $$J_4$$ = $$477$$ = $$3^{2} \cdot 53$$ $$J_6$$ = $$26525$$ = $$5^{2} \cdot 1061$$ $$J_8$$ = $$-1270401$$ = $$- 3 \cdot 11 \cdot 137 \cdot 281$$ $$J_{10}$$ = $$5501$$ = $$5501$$ $$g_1$$ = $$-205236901143/5501$$ $$g_2$$ = $$-2923288299/5501$$ $$g_3$$ = $$888295725/5501$$

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
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(-1 : 0 : 1)$$ $$(0 : 1 : 1)$$ $$(-1 : -1 : 1)$$ $$(1 : -1 : 1)$$
$$(0 : -2 : 1)$$ $$(-2 : -1 : 1)$$ $$(1 : -2 : 1)$$ $$(-2 : 4 : 1)$$ $$(-13 : -128 : 12)$$ $$(-13 : -1431 : 12)$$

magma: [C![-13,-1431,12],C![-13,-128,12],C![-2,-1,1],C![-2,4,1],C![-1,-1,1],C![-1,0,1],C![0,-2,1],C![0,1,1],C![1,-2,1],C![1,-1,0],C![1,-1,1],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: $$\Z \times \Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(-1 : -1 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)$$ $$(x - z) (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0.176934$$ $$\infty$$
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2 - 2z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2 - z^3$$ $$0.057232$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$2$$ Mordell-Weil rank: $$2$$ 2-Selmer rank: $$2$$ Regulator: $$0.010011$$ Real period: $$26.16477$$ Tamagawa product: $$1$$ Torsion order: $$1$$ Leading coefficient: $$0.261938$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$5501$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 9 T + 5501 T^{2} )$$

## 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 $$\Q$$.