# Properties

 Label 64237.a.64237.1 Conductor 64237 Discriminant 64237 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

# Related objects

Show commands for: Magma / SageMath

## Minimal equation

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

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

$y^2 + (x^3 + x^2 + 1)y = x^3 - 3x$

## Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$64237$$ = $$64237$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$64237$$ = $$64237$$

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$-1016$$ = $$-1 \cdot 2^{3} \cdot 127$$ $$I_4$$ = $$17956$$ = $$2^{2} \cdot 67^{2}$$ $$I_6$$ = $$-34763352$$ = $$-1 \cdot 2^{3} \cdot 3 \cdot 13 \cdot 67 \cdot 1663$$ $$I_{10}$$ = $$263114752$$ = $$2^{12} \cdot 64237$$ $$J_2$$ = $$-127$$ = $$-1 \cdot 127$$ $$J_4$$ = $$485$$ = $$5 \cdot 97$$ $$J_6$$ = $$49013$$ = $$23 \cdot 2131$$ $$J_8$$ = $$-1614969$$ = $$-1 \cdot 3^{2} \cdot 179441$$ $$J_{10}$$ = $$64237$$ = $$64237$$ $$g_1$$ = $$-33038369407/64237$$ $$g_2$$ = $$-993465755/64237$$ $$g_3$$ = $$790530677/64237$$
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![-23,1472,12],C![-23,2619,12],C![-3,-8,2],C![-3,9,2],C![-2,-54,3],C![-2,1,1],C![-2,2,1],C![-2,23,3],C![-1,-2,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-2,1],C![1,-1,0],C![1,-1,1],C![1,0,0],C![25,-17550,3],C![25,23,3]];

Known rational points: (-23 : 1472 : 12), (-23 : 2619 : 12), (-3 : -8 : 2), (-3 : 9 : 2), (-2 : -54 : 3), (-2 : 1 : 1), (-2 : 2 : 1), (-2 : 23 : 3), (-1 : -2 : 1), (-1 : 1 : 1), (0 : -1 : 1), (0 : 0 : 1), (1 : -2 : 1), (1 : -1 : 0), (1 : -1 : 1), (1 : 0 : 0), (25 : -17550 : 3), (25 : 23 : 3)

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

Number of rational Weierstrass points: $$0$$

## Invariants of the Jacobian:

Analytic rank*: $$3$$

magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1); 2-Selmer rank: $$3$$ magma: HasSquareSha(Jacobian(C)); Order of Ш*: square Regulator: 0.0349941715097 Real period: 18.153996138825115199360644140 Tamagawa numbers: 1 (p = 64237) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

Torsion: $$\mathrm{trivial}$$

### 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 $$\Q$$.