# Properties

 Label 604.a.9664.2 Conductor 604 Discriminant 9664 Mordell-Weil group $$\Z/{27}\Z$$ 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

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

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

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

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, -4, 4, 6, -4, 0, 1]))

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

## Invariants

 $$N$$ = $$604$$ = $$2^{2} \cdot 151$$ magma: Conductor(LSeries(C)); Factorization($1); $$\Delta$$ = $$9664$$ = $$2^{6} \cdot 151$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$232$$ = $$2^{3} \cdot 29$$ $$I_4$$ = $$25060$$ = $$2^{2} \cdot 5 \cdot 7 \cdot 179$$ $$I_6$$ = $$762216$$ = $$2^{3} \cdot 3 \cdot 7 \cdot 13 \cdot 349$$ $$I_{10}$$ = $$39583744$$ = $$2^{18} \cdot 151$$ $$J_2$$ = $$29$$ = $$29$$ $$J_4$$ = $$-226$$ = $$- 2 \cdot 113$$ $$J_6$$ = $$836$$ = $$2^{2} \cdot 11 \cdot 19$$ $$J_8$$ = $$-6708$$ = $$- 2^{2} \cdot 3 \cdot 13 \cdot 43$$ $$J_{10}$$ = $$9664$$ = $$2^{6} \cdot 151$$ $$g_1$$ = $$20511149/9664$$ $$g_2$$ = $$-2755957/4832$$ $$g_3$$ = $$175769/2416$$

## Automorphism group

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

## Rational points

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

Points: $$(0 : 0 : 1),\, (1 : 0 : 0),\, (-1 : 0 : 1),\, (0 : -1 : 1),\, (1 : -1 : 0),\, (1 : 0 : 1),\, (1 : -2 : 1)$$

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

Number of rational Weierstrass points: $$1$$

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.

## Mordell-Weil group of the Jacobian:

magma: MordellWeilGroupGenus2(Jacobian(C));

Group structure: $$\Z/{27}\Z$$

Generator Height Order
$$x (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$27$$

## BSD invariants

 Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$0$$ Regulator: $$1$$ Real period: $$23.63483$$ Tamagawa product: $$9$$ Torsion order: $$27$$ Leading coefficient: $$0.291788$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$2$$ $$6$$ $$2$$ $$9$$ $$( 1 - T )( 1 + T )$$
$$151$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 10 T + 151 T^{2} )$$

## 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 $$\Q$$.