# Properties

 Label 389.a.389.1 Conductor $389$ Discriminant $389$ Mordell-Weil group $$\Z/{10}\Z$$ Sato-Tate group $\mathrm{USp}(4)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q$$ $$\End(J) \otimes \Q$$ $$\Q$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type no

# Related objects

Show commands: Magma / SageMath

## Simplified equation

 $y^2 + (x^3 + x)y = x^5 - 2x^4 - 8x^3 + 16x + 7$ (homogenize, simplify) $y^2 + (x^3 + xz^2)y = x^5z - 2x^4z^2 - 8x^3z^3 + 16xz^5 + 7z^6$ (dehomogenize, simplify) $y^2 = x^6 + 4x^5 - 6x^4 - 32x^3 + x^2 + 64x + 28$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([7, 16, 0, -8, -2, 1]), R([0, 1, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![7, 16, 0, -8, -2, 1], R![0, 1, 0, 1]);

sage: X = HyperellipticCurve(R([28, 64, 1, -32, -6, 4, 1]))

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

 $$I_2$$ $$=$$ $$2440$$ $$=$$ $$2^{3} \cdot 5 \cdot 61$$ $$I_4$$ $$=$$ $$51100$$ $$=$$ $$2^{2} \cdot 5^{2} \cdot 7 \cdot 73$$ $$I_6$$ $$=$$ $$45041351$$ $$=$$ $$45041351$$ $$I_{10}$$ $$=$$ $$1556$$ $$=$$ $$2^{2} \cdot 389$$ $$J_2$$ $$=$$ $$1220$$ $$=$$ $$2^{2} \cdot 5 \cdot 61$$ $$J_4$$ $$=$$ $$53500$$ $$=$$ $$2^{2} \cdot 5^{3} \cdot 107$$ $$J_6$$ $$=$$ $$2084961$$ $$=$$ $$3 \cdot 694987$$ $$J_8$$ $$=$$ $$-79649395$$ $$=$$ $$- 5 \cdot 7 \cdot 2275697$$ $$J_{10}$$ $$=$$ $$389$$ $$=$$ $$389$$ $$g_1$$ $$=$$ $$2702708163200000/389$$ $$g_2$$ $$=$$ $$97147868000000/389$$ $$g_3$$ $$=$$ $$3103255952400/389$$

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: $$(1 : 0 : 0),\, (1 : -1 : 0),\, (-2 : 5 : 1),\, (2 : -5 : 1)$$
All points: $$(1 : 0 : 0),\, (1 : -1 : 0),\, (-2 : 5 : 1),\, (2 : -5 : 1)$$
All points: $$(1 : -1 : 0),\, (1 : 1 : 0),\, (-2 : 0 : 1),\, (2 : 0 : 1)$$

magma: [C![-2,5,1],C![1,-1,0],C![1,0,0],C![2,-5,1]]; // minimal model

magma: [C![-2,0,1],C![1,-1,0],C![1,1,0],C![2,0,1]]; // simplified model

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
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$2x^2 - 9z^2$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$-6xz^2 - z^3$$ $$0$$ $$10$$
Generator $D_0$ Height Order
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$2x^2 - 9z^2$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$-6xz^2 - z^3$$ $$0$$ $$10$$
Generator $D_0$ Height Order
$$D_0 - (1 : -1 : 0) - (1 : 1 : 0)$$ $$2x^2 - 9z^2$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$x^3 - 11xz^2 - 2z^3$$ $$0$$ $$10$$

## 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 Cluster picture
$$389$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 10 T + 389 T^{2} )$$

## Galois representations

The mod-$\ell$ Galois representation has maximal image $$\GSp(4,\F_\ell)$$ for all primes $$\ell$$ except those listed.

Prime $$\ell$$ mod-$$\ell$$ image Is torsion prime?
$$2$$ 2.30.3 yes
$$5$$ not computed yes

## Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $\mathrm{USp}(4)$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{USp}(4)$$

## Decomposition of the Jacobian

Simple over $$\overline{\Q}$$

magma: HeuristicDecompositionFactors(C);

## 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 $$\Q$$.

magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

magma: HeuristicIsGL2(C); HeuristicEndomorphismDescription(C); HeuristicEndomorphismFieldOfDefinition(C);

magma: HeuristicIsGL2(C : Geometric := true); HeuristicEndomorphismDescription(C : Geometric := true); HeuristicEndomorphismLatticeDescription(C);