# Properties

 Label 1038.a.1038.2 Conductor 1038 Discriminant -1038 Mordell-Weil group $$\Z/{6}\Z$$ 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![1, 1, 2, 0, 1], R![1, 0, 0, 1]);

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

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

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([5, 4, 8, 2, 4, 0, 1]))

 $y^2 + (x^3 + 1)y = x^4 + 2x^2 + x + 1$ (homogenize, simplify) $y^2 + (x^3 + z^3)y = x^4z^2 + 2x^2z^4 + xz^5 + z^6$ (dehomogenize, simplify) $y^2 = x^6 + 4x^4 + 2x^3 + 8x^2 + 4x + 5$ (minimize, homogenize)

## Invariants

 $$N$$ = $$1038$$ = $$2 \cdot 3 \cdot 173$$ magma: Conductor(LSeries(C)); Factorization($1); $$\Delta$$ = $$-1038$$ = $$- 2 \cdot 3 \cdot 173$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$-1688$$ = $$- 2^{3} \cdot 211$$ $$I_4$$ = $$16516$$ = $$2^{2} \cdot 4129$$ $$I_6$$ = $$-9071864$$ = $$- 2^{3} \cdot 127 \cdot 8929$$ $$I_{10}$$ = $$-4251648$$ = $$- 2^{13} \cdot 3 \cdot 173$$ $$J_2$$ = $$-211$$ = $$- 211$$ $$J_4$$ = $$1683$$ = $$3^{2} \cdot 11 \cdot 17$$ $$J_6$$ = $$-16079$$ = $$- 7 \cdot 2297$$ $$J_8$$ = $$140045$$ = $$5 \cdot 37 \cdot 757$$ $$J_{10}$$ = $$-1038$$ = $$- 2 \cdot 3 \cdot 173$$ $$g_1$$ = $$418227202051/1038$$ $$g_2$$ = $$5269995291/346$$ $$g_3$$ = $$715853159/1038$$

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

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

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: $$\Z/{6}\Z$$

Generator Height Order
$$z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$6$$

## BSD invariants

 Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$1$$ Regulator: $$1$$ Real period: $$15.39734$$ Tamagawa product: $$1$$ Torsion order: $$6$$ Leading coefficient: $$0.427704$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$2$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 - T + 2 T^{2} )$$
$$3$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 - T + 3 T^{2} )$$
$$173$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 - 6 T + 173 T^{2} )$$

## 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 $$\Q$$.