# Properties

 Label 925.a.925.1 Conductor 925 Discriminant 925 Mordell-Weil group $$\Z/{8}\Z$$ 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, 0, -1, -1, 2, -1], R![1, 1]);

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

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

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, 2, -3, -4, 8, -4]))

 $y^2 + (x + 1)y = -x^5 + 2x^4 - x^3 - x^2$ (homogenize, simplify) $y^2 + (xz^2 + z^3)y = -x^5z + 2x^4z^2 - x^3z^3 - x^2z^4$ (dehomogenize, simplify) $y^2 = -4x^5 + 8x^4 - 4x^3 - 3x^2 + 2x + 1$ (minimize, homogenize)

## Invariants

 $$N$$ = $$925$$ = $$5^{2} \cdot 37$$ magma: Conductor(LSeries(C)); Factorization($1); $$\Delta$$ = $$925$$ = $$5^{2} \cdot 37$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$160$$ = $$2^{5} \cdot 5$$ $$I_4$$ = $$-15104$$ = $$- 2^{8} \cdot 59$$ $$I_6$$ = $$-903488$$ = $$- 2^{6} \cdot 19 \cdot 743$$ $$I_{10}$$ = $$3788800$$ = $$2^{12} \cdot 5^{2} \cdot 37$$ $$J_2$$ = $$20$$ = $$2^{2} \cdot 5$$ $$J_4$$ = $$174$$ = $$2 \cdot 3 \cdot 29$$ $$J_6$$ = $$713$$ = $$23 \cdot 31$$ $$J_8$$ = $$-4004$$ = $$- 2^{2} \cdot 7 \cdot 11 \cdot 13$$ $$J_{10}$$ = $$925$$ = $$5^{2} \cdot 37$$ $$g_1$$ = $$128000/37$$ $$g_2$$ = $$55680/37$$ $$g_3$$ = $$11408/37$$

## 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,-1,1],C![1,0,0]];

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

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

Number of rational Weierstrass points: $$2$$

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: $$\Z/{8}\Z$$

Generator Height Order
$$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0$$ $$8$$

## BSD invariants

 Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$1$$ Regulator: $$1$$ Real period: $$20.87893$$ Tamagawa product: $$1$$ Torsion order: $$8$$ Leading coefficient: $$0.326233$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$5$$ $$2$$ $$2$$ $$1$$ $$( 1 - T )( 1 + T )$$
$$37$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 - 6 T + 37 T^{2} )$$

## 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 $$\Q$$.