# Properties

 Label 574.a.293888.1 Conductor 574 Discriminant -293888 Mordell-Weil group $$\Z/{2}\Z \times \Z/{10}\Z$$ 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![1, 1, -3, 0, -1, 1], R![0, 1, 1]);

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

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

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([4, 4, -11, 2, -3, 4]))

 $y^2 + (x^2 + x)y = x^5 - x^4 - 3x^2 + x + 1$ (homogenize, simplify) $y^2 + (x^2z + xz^2)y = x^5z - x^4z^2 - 3x^2z^4 + xz^5 + z^6$ (dehomogenize, simplify) $y^2 = 4x^5 - 3x^4 + 2x^3 - 11x^2 + 4x + 4$ (minimize, homogenize)

## Invariants

 $$N$$ = $$574$$ = $$2 \cdot 7 \cdot 41$$ magma: Conductor(LSeries(C)); Factorization($1); $$\Delta$$ = $$-293888$$ = $$- 2^{10} \cdot 7 \cdot 41$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$136$$ = $$2^{3} \cdot 17$$ $$I_4$$ = $$-223292$$ = $$- 2^{2} \cdot 55823$$ $$I_6$$ = $$-7647160$$ = $$- 2^{3} \cdot 5 \cdot 37 \cdot 5167$$ $$I_{10}$$ = $$-1203765248$$ = $$- 2^{22} \cdot 7 \cdot 41$$ $$J_2$$ = $$17$$ = $$17$$ $$J_4$$ = $$2338$$ = $$2 \cdot 7 \cdot 167$$ $$J_6$$ = $$2304$$ = $$2^{8} \cdot 3^{2}$$ $$J_8$$ = $$-1356769$$ = $$- 331 \cdot 4099$$ $$J_{10}$$ = $$-293888$$ = $$- 2^{10} \cdot 7 \cdot 41$$ $$g_1$$ = $$-1419857/293888$$ $$g_2$$ = $$-820471/20992$$ $$g_3$$ = $$-2601/1148$$

## 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![0,-1,1],C![0,1,1],C![1,-1,1],C![1,0,0],C![2,-7,1],C![2,1,1]];

Points: $$(1 : 0 : 0),\, (0 : -1 : 1),\, (0 : 1 : 1),\, (1 : -1 : 1),\, (2 : 1 : 1),\, (2 : -7 : 1)$$

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

Number of rational Weierstrass points: $$2$$

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: $$\Z/{2}\Z \times \Z/{10}\Z$$

Generator Height Order
$$4x^2 - 3xz - 2z^2$$ $$=$$ $$0,$$ $$8y$$ $$=$$ $$-7xz^2 - 2z^3$$ $$0$$ $$2$$
$$(-x + z) x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-2xz^2 + z^3$$ $$0$$ $$10$$

## BSD invariants

 Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$2$$ Regulator: $$1$$ Real period: $$11.54635$$ Tamagawa product: $$10$$ Torsion order: $$20$$ Leading coefficient: $$0.288658$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$2$$ $$10$$ $$1$$ $$10$$ $$( 1 - T )( 1 + T + 2 T^{2} )$$
$$7$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 2 T + 7 T^{2} )$$
$$41$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 - 2 T + 41 T^{2} )$$

## 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 $$\Q$$.