# Properties

 Label 277.a.277.1 Conductor $277$ Discriminant $277$ Mordell-Weil group $$\Z/{15}\Z$$ 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

This is the first proven instance of the paramodular conjecture for an abelian surface $A$ with trivial geometric endomorphism ring (meaning $\End(A_{\overline{\mathbb{Q}}})=\mathbb{Z}$); see [arXiv:1805.10873].

## Simplified equation

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

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

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

sage: X = HyperellipticCurve(R([1, -2, -1, 4, 3, 2, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$277$$ $$=$$ $$277$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$277$$ $$=$$ $$277$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$64$$ $$=$$ $$2^{6}$$ $$I_4$$ $$=$$ $$352$$ $$=$$ $$2^{5} \cdot 11$$ $$I_6$$ $$=$$ $$9552$$ $$=$$ $$2^{4} \cdot 3 \cdot 199$$ $$I_{10}$$ $$=$$ $$-1108$$ $$=$$ $$- 2^{2} \cdot 277$$ $$J_2$$ $$=$$ $$32$$ $$=$$ $$2^{5}$$ $$J_4$$ $$=$$ $$-16$$ $$=$$ $$- 2^{4}$$ $$J_6$$ $$=$$ $$-464$$ $$=$$ $$- 2^{4} \cdot 29$$ $$J_8$$ $$=$$ $$-3776$$ $$=$$ $$- 2^{6} \cdot 59$$ $$J_{10}$$ $$=$$ $$-277$$ $$=$$ $$-277$$ $$g_1$$ $$=$$ $$-33554432/277$$ $$g_2$$ $$=$$ $$524288/277$$ $$g_3$$ $$=$$ $$475136/277$$

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: $$\Z/{15}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(-1 : 0 : 1) + (0 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$15$$
Generator $D_0$ Height Order
$$(-1 : 0 : 1) + (0 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$15$$
Generator $D_0$ Height Order
$$(-1 : 0 : 1) + (0 : 1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)$$ $$x (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3 + x^2z + xz^2 + z^3$$ $$0$$ $$15$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$0$$ Regulator: $$1$$ Real period: $$32.20574$$ Tamagawa product: $$1$$ Torsion order: $$15$$ Leading coefficient: $$0.143136$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$277$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 - 8 T + 277 T^{2} )$$

## 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?
$$2$$ 2.6.1 no
$$3$$ 3.80.1 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);