# Properties

 Label 745.a.745.1 Conductor 745 Discriminant -745 Mordell-Weil group $$\Z/{9}\Z$$ 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

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

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

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

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

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

## Invariants

 $$N$$ = $$745$$ = $$5 \cdot 149$$ magma: Conductor(LSeries(C)); Factorization($1); $$\Delta$$ = $$-745$$ = $$- 5 \cdot 149$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$-248$$ = $$- 2^{3} \cdot 31$$ $$I_4$$ = $$5668$$ = $$2^{2} \cdot 13 \cdot 109$$ $$I_6$$ = $$-310104$$ = $$- 2^{3} \cdot 3^{2} \cdot 59 \cdot 73$$ $$I_{10}$$ = $$-3051520$$ = $$- 2^{12} \cdot 5 \cdot 149$$ $$J_2$$ = $$-31$$ = $$- 31$$ $$J_4$$ = $$-19$$ = $$- 19$$ $$J_6$$ = $$-39$$ = $$- 3 \cdot 13$$ $$J_8$$ = $$212$$ = $$2^{2} \cdot 53$$ $$J_{10}$$ = $$-745$$ = $$- 5 \cdot 149$$ $$g_1$$ = $$28629151/745$$ $$g_2$$ = $$-566029/745$$ $$g_3$$ = $$37479/745$$

## Automorphism group

 magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X)$$ $$\simeq$$$C_2$magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$ $C_2$

## Rational points

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

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

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));

Number of rational Weierstrass points: $$0$$

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);

This curve is locally solvable everywhere.

## Mordell-Weil group of the Jacobian:

magma: MordellWeilGroupGenus2(Jacobian(C));

Group structure: $$\Z/{9}\Z$$

Generator Height Order
$$z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$9$$

## BSD invariants

 Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$0$$ Regulator: $$1$$ Real period: $$24.57283$$ Tamagawa product: $$1$$ Torsion order: $$9$$ Leading coefficient: $$0.303368$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$5$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 3 T + 5 T^{2} )$$
$$149$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 3 T + 149 T^{2} )$$

## 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 $$\Q$$.