# Properties

 Label 295.a.295.1 Conductor 295 Discriminant -295 Mordell-Weil group $$\Z/{14}\Z$$ 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: SageMath / Magma

## Simplified equation

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

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

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

sage: X = HyperellipticCurve(R([1, 0, -4, 2, 0, 0, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$295$$ $$=$$ $$5 \cdot 59$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-295$$ $$=$$ $$- 5 \cdot 59$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$108$$ $$=$$ $$2^{2} \cdot 3^{3}$$ $$I_4$$ $$=$$ $$-39$$ $$=$$ $$- 3 \cdot 13$$ $$I_6$$ $$=$$ $$20835$$ $$=$$ $$3^{2} \cdot 5 \cdot 463$$ $$I_{10}$$ $$=$$ $$37760$$ $$=$$ $$2^{7} \cdot 5 \cdot 59$$ $$J_2$$ $$=$$ $$27$$ $$=$$ $$3^{3}$$ $$J_4$$ $$=$$ $$32$$ $$=$$ $$2^{5}$$ $$J_6$$ $$=$$ $$-256$$ $$=$$ $$- 2^{8}$$ $$J_8$$ $$=$$ $$-1984$$ $$=$$ $$- 2^{6} \cdot 31$$ $$J_{10}$$ $$=$$ $$295$$ $$=$$ $$5 \cdot 59$$ $$g_1$$ $$=$$ $$14348907/295$$ $$g_2$$ $$=$$ $$629856/295$$ $$g_3$$ $$=$$ $$-186624/295$$

sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]

magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);

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

magma: [C![0,-1,1],C![0,0,1],C![1,-1,0],C![1,-1,1],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: $$\Z/{14}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(0 : -1 : 1) - (1 : 0 : 0)$$ $$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3 - z^3$$ $$0$$ $$14$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$1$$ Regulator: $$1$$ Real period: $$29.25660$$ Tamagawa product: $$1$$ Torsion order: $$14$$ Leading coefficient: $$0.149268$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$5$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + T + 5 T^{2} )$$
$$59$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 - 4 T + 59 T^{2} )$$

## 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 $$\Q$$.