# Properties

 Label 388.a.776.1 Conductor $388$ Discriminant $776$ Mordell-Weil group $$\Z/{21}\Z$$ Sato-Tate group $\mathrm{USp}(4)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q$$ $$\End(J) \otimes \Q$$ $$\Q$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type no

# Related objects

Show commands: SageMath / Magma

## Simplified equation

 $y^2 + (x^3 + x + 1)y = -x^4 + 2x^2 + x$ (homogenize, simplify) $y^2 + (x^3 + xz^2 + z^3)y = -x^4z^2 + 2x^2z^4 + xz^5$ (dehomogenize, simplify) $y^2 = x^6 - 2x^4 + 2x^3 + 9x^2 + 6x + 1$ (homogenize, minimize)

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

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

sage: X = HyperellipticCurve(R([1, 6, 9, 2, -2, 0, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$388$$ $$=$$ $$2^{2} \cdot 97$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$776$$ $$=$$ $$2^{3} \cdot 97$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$36$$ $$=$$ $$2^{2} \cdot 3^{2}$$ $$I_4$$ $$=$$ $$1569$$ $$=$$ $$3 \cdot 523$$ $$I_6$$ $$=$$ $$-13743$$ $$=$$ $$- 3^{3} \cdot 509$$ $$I_{10}$$ $$=$$ $$99328$$ $$=$$ $$2^{10} \cdot 97$$ $$J_2$$ $$=$$ $$9$$ $$=$$ $$3^{2}$$ $$J_4$$ $$=$$ $$-62$$ $$=$$ $$- 2 \cdot 31$$ $$J_6$$ $$=$$ $$356$$ $$=$$ $$2^{2} \cdot 89$$ $$J_8$$ $$=$$ $$-160$$ $$=$$ $$- 2^{5} \cdot 5$$ $$J_{10}$$ $$=$$ $$776$$ $$=$$ $$2^{3} \cdot 97$$ $$g_1$$ $$=$$ $$59049/776$$ $$g_2$$ $$=$$ $$-22599/388$$ $$g_3$$ $$=$$ $$7209/194$$

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

magma: [C![-1,0,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-1,0],C![1,0,0]]; // minimal model

magma: [C![-1,-1,1],C![-1,1,1],C![0,-1,1],C![0,1,1],C![1,-1,0],C![1,1,0]]; // simplified model

Number of rational Weierstrass points: $$0$$

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: $$\Z/{21}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$2 \cdot(0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-2xz^2 - z^3$$ $$0$$ $$21$$
Generator $D_0$ Height Order
$$2 \cdot(0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-2xz^2 - z^3$$ $$0$$ $$21$$
Generator $D_0$ Height Order
$$2 \cdot(0 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)$$ $$x^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3 - 3xz^2 - z^3$$ $$0$$ $$21$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$0$$ Regulator: $$1$$ Real period: $$29.13550$$ Tamagawa product: $$3$$ Torsion order: $$21$$ Leading coefficient: $$0.198200$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$2$$ $$3$$ $$3$$ $$1 + T + T^{2}$$
$$97$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 - 14 T + 97 T^{2} )$$

## Galois representations

For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not the maximal image $\GSp(4,\F_\ell)$

Prime $$\ell$$ mod-$$\ell$$ image
$$2$$ 2.10.1
$$3$$ 3.80.1

## 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 $$\Q$$.