# Properties

 Label 249.a.249.1 Conductor $249$ Discriminant $249$ Mordell-Weil group $$\Z/{14}\Z$$ 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

The Jacobian $A$ of this curve is the first abelian surface of paramodular type (meaning $\End(A)=\Z$) that appears in the table of Brumer and Kramer [MR:3165645].

## Simplified equation

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

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

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

sage: X = HyperellipticCurve(R([1, 4, 4, 2, 0, 0, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$249$$ $$=$$ $$3 \cdot 83$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$249$$ $$=$$ $$3 \cdot 83$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$108$$ $$=$$ $$2^{2} \cdot 3^{3}$$ $$I_4$$ $$=$$ $$57$$ $$=$$ $$3 \cdot 19$$ $$I_6$$ $$=$$ $$2259$$ $$=$$ $$3^{2} \cdot 251$$ $$I_{10}$$ $$=$$ $$-31872$$ $$=$$ $$- 2^{7} \cdot 3 \cdot 83$$ $$J_2$$ $$=$$ $$27$$ $$=$$ $$3^{3}$$ $$J_4$$ $$=$$ $$28$$ $$=$$ $$2^{2} \cdot 7$$ $$J_6$$ $$=$$ $$32$$ $$=$$ $$2^{5}$$ $$J_8$$ $$=$$ $$20$$ $$=$$ $$2^{2} \cdot 5$$ $$J_{10}$$ $$=$$ $$-249$$ $$=$$ $$- 3 \cdot 83$$ $$g_1$$ $$=$$ $$-4782969/83$$ $$g_2$$ $$=$$ $$-183708/83$$ $$g_3$$ $$=$$ $$-7776/83$$

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

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

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

Number of rational Weierstrass points: $$1$$

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: $$\Z/{14}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(-1 : 0 : 1) - (1 : 0 : 0)$$ $$z (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3 - z^3$$ $$0$$ $$14$$
Generator $D_0$ Height Order
$$(-1 : 0 : 1) - (1 : 0 : 0)$$ $$z (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3 - z^3$$ $$0$$ $$14$$
Generator $D_0$ Height Order
$$(-1 : 0 : 1) - (1 : 1 : 0)$$ $$z (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3 - z^3$$ $$0$$ $$14$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$1$$ Regulator: $$1$$ Real period: $$25.78370$$ Tamagawa product: $$1$$ Torsion order: $$14$$ Leading coefficient: $$0.131549$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$3$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 3 T + 3 T^{2} )$$
$$83$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 83 T^{2} )$$

## 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?
$$2$$ 2.60.1 yes
$$7$$ 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);