# Properties

 Label 997.b.997.1 Conductor 997 Discriminant 997 Sato-Tate group $\mathrm{USp}(4)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type no

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Show commands for: Magma / SageMath

The Jacobian of this curve is an apparently paramodular abelian surface appearing as entry "997b" in the table of Brumer and Kramer [MR:3165645].

## Minimal equation

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

$y^2 + y = x^5 - 2x^4 + 2x^3 - x^2$

## Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$997$$ = $$997$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$997$$ = $$997$$

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$-128$$ = $$-1 \cdot 2^{7}$$ $$I_4$$ = $$256$$ = $$2^{8}$$ $$I_6$$ = $$107520$$ = $$2^{10} \cdot 3 \cdot 5 \cdot 7$$ $$I_{10}$$ = $$4083712$$ = $$2^{12} \cdot 997$$ $$J_2$$ = $$-16$$ = $$-1 \cdot 2^{4}$$ $$J_4$$ = $$8$$ = $$2^{3}$$ $$J_6$$ = $$-208$$ = $$-1 \cdot 2^{4} \cdot 13$$ $$J_8$$ = $$816$$ = $$2^{4} \cdot 3 \cdot 17$$ $$J_{10}$$ = $$997$$ = $$997$$ $$g_1$$ = $$-1048576/997$$ $$g_2$$ = $$-32768/997$$ $$g_3$$ = $$-53248/997$$
Alternative geometric invariants: G2

## Automorphism group

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

## Rational points

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);

This curve is locally solvable everywhere.

magma: [C![0,-1,1],C![0,0,1],C![1,-1,1],C![1,0,0],C![1,0,1],C![2,-4,1],C![2,3,1]];

All rational points: (0 : -1 : 1), (0 : 0 : 1), (1 : -1 : 1), (1 : 0 : 0), (1 : 0 : 1), (2 : -4 : 1), (2 : 3 : 1)

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));

Number of rational Weierstrass points: $$1$$

## Invariants of the Jacobian:

Analytic rank: $$1$$

magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1); 2-Selmer rank: $$1$$ magma: HasSquareSha(Jacobian(C)); Order of Ш*: square Tamagawa numbers: 1 (p = 997) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

Torsion: $$\Z/{3}\Z$$

### Sato-Tate group

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

### Decomposition

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

### Endomorphisms

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 $$\Q$$.