# Properties

 Label 389.a.389.2 Conductor 389 Discriminant 389 Mordell-Weil group $$\Z/{10}\Z$$ 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

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

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

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

sage: X = HyperellipticCurve(R([1, 2, 5, 8, 8, 4]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ = $$389$$ = $$389$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$389$$ = $$389$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$64$$ = $$2^{6}$$ $$I_4$$ = $$1600$$ = $$2^{6} \cdot 5^{2}$$ $$I_6$$ = $$113600$$ = $$2^{6} \cdot 5^{2} \cdot 71$$ $$I_{10}$$ = $$1593344$$ = $$2^{12} \cdot 389$$ $$J_2$$ = $$8$$ = $$2^{3}$$ $$J_4$$ = $$-14$$ = $$- 2 \cdot 7$$ $$J_6$$ = $$-159$$ = $$- 3 \cdot 53$$ $$J_8$$ = $$-367$$ = $$- 367$$ $$J_{10}$$ = $$389$$ = $$389$$ $$g_1$$ = $$32768/389$$ $$g_2$$ = $$-7168/389$$ $$g_3$$ = $$-10176/389$$

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

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

Number of rational Weierstrass points: $$2$$

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: $$\Z/{10}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$x^2 + xz + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2 - z^3$$ $$0$$ $$10$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$1$$ Regulator: $$1$$ Real period: $$19.79862$$ Tamagawa product: $$1$$ Torsion order: $$10$$ Leading coefficient: $$0.197986$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$389$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 10 T + 389 T^{2} )$$

## 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 $$\Q$$.