# Properties

 Label 64829.b.64829.1 Conductor 64829 Discriminant 64829 Mordell-Weil group trivial 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

## Simplified equation

 $y^2 + (x^2 + 1)y = x^5 - 2x^4 + 4x^3 + x^2 - 2x - 1$ (homogenize, simplify) $y^2 + (x^2z + z^3)y = x^5z - 2x^4z^2 + 4x^3z^3 + x^2z^4 - 2xz^5 - z^6$ (dehomogenize, simplify) $y^2 = 4x^5 - 7x^4 + 16x^3 + 6x^2 - 8x - 3$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, -2, 1, 4, -2, 1], R![1, 0, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, -2, 1, 4, -2, 1]), R([1, 0, 1]));

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([-3, -8, 6, 16, -7, 4]))

## Invariants

 Conductor: $$N$$ = $$64829$$ = $$241 \cdot 269$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$64829$$ = $$241 \cdot 269$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$928$$ = $$2^{5} \cdot 29$$ $$I_4$$ = $$-595712$$ = $$- 2^{8} \cdot 13 \cdot 179$$ $$I_6$$ = $$-157539136$$ = $$- 2^{6} \cdot 17 \cdot 29 \cdot 4993$$ $$I_{10}$$ = $$265539584$$ = $$2^{12} \cdot 241 \cdot 269$$ $$J_2$$ = $$116$$ = $$2^{2} \cdot 29$$ $$J_4$$ = $$6766$$ = $$2 \cdot 17 \cdot 199$$ $$J_6$$ = $$77169$$ = $$3 \cdot 29 \cdot 887$$ $$J_8$$ = $$-9206788$$ = $$- 2^{2} \cdot 113 \cdot 20369$$ $$J_{10}$$ = $$64829$$ = $$241 \cdot 269$$ $$g_1$$ = $$21003416576/64829$$ $$g_2$$ = $$10561022336/64829$$ $$g_3$$ = $$1038386064/64829$$

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

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

## 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: $$(1 : 0 : 0)$$

magma: [C![1,0,0]];

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: trivial

magma: MordellWeilGroupGenus2(Jacobian(C));

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$0$$ Regulator: $$1$$ Real period: $$3.186146$$ Tamagawa product: $$1$$ Torsion order: $$1$$ Leading coefficient: $$3.186146$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$241$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 13 T + 241 T^{2} )$$
$$269$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 8 T + 269 T^{2} )$$

## Sato-Tate group

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

## Decomposition of the Jacobian

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

## 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 $$\Q$$.