# Properties

 Label 1083.a.20577.1 Conductor $1083$ Discriminant $20577$ Mordell-Weil group $$\Z \oplus \Z/{3}\Z$$ Sato-Tate group $\mathrm{SU}(2)\times\mathrm{SU}(2)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q \times \Q$$ $$\End(J) \otimes \Q$$ $$\Q \times \Q$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Related objects

Show commands: Magma / SageMath

## Simplified equation

 $y^2 + x^3y = x^5 - 5x^4 + 11x^3 - 13x^2 + 9x - 3$ (homogenize, simplify) $y^2 + x^3y = x^5z - 5x^4z^2 + 11x^3z^3 - 13x^2z^4 + 9xz^5 - 3z^6$ (dehomogenize, simplify) $y^2 = x^6 + 4x^5 - 20x^4 + 44x^3 - 52x^2 + 36x - 12$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-3, 9, -13, 11, -5, 1]), R([0, 0, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-3, 9, -13, 11, -5, 1], R![0, 0, 0, 1]);

sage: X = HyperellipticCurve(R([-12, 36, -52, 44, -20, 4, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$1083$$ $$=$$ $$3 \cdot 19^{2}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$20577$$ $$=$$ $$3 \cdot 19^{3}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$904$$ $$=$$ $$2^{3} \cdot 113$$ $$I_4$$ $$=$$ $$13684$$ $$=$$ $$2^{2} \cdot 11 \cdot 311$$ $$I_6$$ $$=$$ $$4578992$$ $$=$$ $$2^{4} \cdot 11 \cdot 26017$$ $$I_{10}$$ $$=$$ $$82308$$ $$=$$ $$2^{2} \cdot 3 \cdot 19^{3}$$ $$J_2$$ $$=$$ $$452$$ $$=$$ $$2^{2} \cdot 113$$ $$J_4$$ $$=$$ $$6232$$ $$=$$ $$2^{3} \cdot 19 \cdot 41$$ $$J_6$$ $$=$$ $$-8664$$ $$=$$ $$- 2^{3} \cdot 3 \cdot 19^{2}$$ $$J_8$$ $$=$$ $$-10688488$$ $$=$$ $$- 2^{3} \cdot 19^{2} \cdot 3701$$ $$J_{10}$$ $$=$$ $$20577$$ $$=$$ $$3 \cdot 19^{3}$$ $$g_1$$ $$=$$ $$18866536236032/20577$$ $$g_2$$ $$=$$ $$30289293824/1083$$ $$g_3$$ $$=$$ $$-1634432/19$$

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

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

## Automorphism group

 $$\mathrm{Aut}(X)$$ $$\simeq$$ $C_2^2$ magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$$C_2^2$magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);

## Rational points

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

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

magma: [C![1,-1,0],C![1,-1,1],C![1,1,0],C![1,1,1]]; // 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: $$\Z \oplus \Z/{3}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2 - xz + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$z^3$$ $$0.075149$$ $$\infty$$
$$(1 : -1 : 1) - (1 : -1 : 0)$$ $$z (x - z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0$$ $$3$$
Generator $D_0$ Height Order
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2 - xz + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$z^3$$ $$0.075149$$ $$\infty$$
$$(1 : -1 : 1) - (1 : -1 : 0)$$ $$z (x - z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0$$ $$3$$
Generator $D_0$ Height Order
$$D_0 - (1 : -1 : 0) - (1 : 1 : 0)$$ $$x^2 - xz + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3 + 2z^3$$ $$0.075149$$ $$\infty$$
$$(1 : -1 : 1) - (1 : -1 : 0)$$ $$z (x - z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3 - 2z^3$$ $$0$$ $$3$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$1$$ Regulator: $$0.075149$$ Real period: $$7.554151$$ Tamagawa product: $$3$$ Torsion order: $$3$$ Leading coefficient: $$0.189229$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$3$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 2 T + 3 T^{2} )$$
$$19$$ $$2$$ $$3$$ $$3$$ $$( 1 - T )( 1 + T )$$

## Galois representations

For primes $\ell \ge 5$ the Galois representation data has not been computed for this curve since it is not generic.

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

Prime $$\ell$$ mod-$$\ell$$ image Is torsion prime?
$$2$$ 2.15.2 no
$$3$$ 3.2160.20 yes

## Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $\mathrm{SU}(2)\times\mathrm{SU}(2)$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{SU}(2)\times\mathrm{SU}(2)$$

## Decomposition of the Jacobian

Splits over $$\Q$$

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
Elliptic curve isogeny class 19.a
Elliptic curve isogeny class 57.a

magma: HeuristicDecompositionFactors(C);

## Endomorphisms of the Jacobian

Of $$\GL_2$$-type over $$\Q$$

Endomorphism ring over $$\Q$$:

 $$\End (J_{})$$ $$\simeq$$ an order of index $$2$$ in $$\Z \times \Z$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\times$$ $$\Q$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R \times \R$$

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);