# Properties

 Label 574.a.293888.1 Conductor $574$ Discriminant $-293888$ Mordell-Weil group $$\Z/{2}\Z \oplus \Z/{10}\Z$$ Sato-Tate group $\mathrm{USp}(4)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q$$ $$\End(J) \otimes \Q$$ $$\Q$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type no

# Related objects

Show commands: Magma / SageMath

## Simplified equation

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

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: X = HyperellipticCurve(R([4, 4, -11, 2, -3, 4]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

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

### G2 invariants

 $$I_2$$ $$=$$ $$68$$ $$=$$ $$2^{2} \cdot 17$$ $$I_4$$ $$=$$ $$-55823$$ $$=$$ $$-55823$$ $$I_6$$ $$=$$ $$-955895$$ $$=$$ $$- 5 \cdot 37 \cdot 5167$$ $$I_{10}$$ $$=$$ $$-37617664$$ $$=$$ $$- 2^{17} \cdot 7 \cdot 41$$ $$J_2$$ $$=$$ $$17$$ $$=$$ $$17$$ $$J_4$$ $$=$$ $$2338$$ $$=$$ $$2 \cdot 7 \cdot 167$$ $$J_6$$ $$=$$ $$2304$$ $$=$$ $$2^{8} \cdot 3^{2}$$ $$J_8$$ $$=$$ $$-1356769$$ $$=$$ $$- 331 \cdot 4099$$ $$J_{10}$$ $$=$$ $$-293888$$ $$=$$ $$- 2^{10} \cdot 7 \cdot 41$$ $$g_1$$ $$=$$ $$-1419857/293888$$ $$g_2$$ $$=$$ $$-820471/20992$$ $$g_3$$ $$=$$ $$-2601/1148$$

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

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

## 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: $$(1 : 0 : 0),\, (0 : -1 : 1),\, (0 : 1 : 1),\, (1 : -1 : 1),\, (2 : 1 : 1),\, (2 : -7 : 1)$$
All points: $$(1 : 0 : 0),\, (0 : -1 : 1),\, (0 : 1 : 1),\, (1 : -1 : 1),\, (2 : 1 : 1),\, (2 : -7 : 1)$$
All points: $$(1 : 0 : 0),\, (1 : 0 : 1),\, (0 : -2 : 1),\, (0 : 2 : 1),\, (2 : -8 : 1),\, (2 : 8 : 1)$$

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]]; // minimal model

magma: [C![0,-2,1],C![0,2,1],C![1,0,1],C![1,0,0],C![2,-8,1],C![2,8,1]]; // simplified model

Number of rational Weierstrass points: $$2$$

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

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$4x^2 - 3xz - 2z^2$$ $$=$$ $$0,$$ $$8y$$ $$=$$ $$-7xz^2 - 2z^3$$ $$0$$ $$2$$
$$(0 : 1 : 1) + (1 : -1 : 1) - 2 \cdot(1 : 0 : 0)$$ $$x (x - z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-2xz^2 + z^3$$ $$0$$ $$10$$
Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$4x^2 - 3xz - 2z^2$$ $$=$$ $$0,$$ $$8y$$ $$=$$ $$-7xz^2 - 2z^3$$ $$0$$ $$2$$
$$(0 : 1 : 1) + (1 : -1 : 1) - 2 \cdot(1 : 0 : 0)$$ $$x (x - z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-2xz^2 + z^3$$ $$0$$ $$10$$
Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$4x^2 - 3xz - 2z^2$$ $$=$$ $$0,$$ $$8y$$ $$=$$ $$x^2z - 13xz^2 - 4z^3$$ $$0$$ $$2$$
$$(0 : 2 : 1) + (1 : 0 : 1) - 2 \cdot(1 : 0 : 0)$$ $$x (x - z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^2z - 3xz^2 + 2z^3$$ $$0$$ $$10$$

## BSD invariants

 Hasse-Weil conjecture: unverified 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 Cluster picture
$$2$$ $$1$$ $$10$$ $$10$$ $$( 1 - T )( 1 + T + 2 T^{2} )$$
$$7$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 2 T + 7 T^{2} )$$
$$41$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 - 2 T + 41 T^{2} )$$

## Galois representations

The mod-$\ell$ Galois representation has maximal image $$\GSp(4,\F_\ell)$$ for all primes $$\ell$$ except those listed.

Prime $$\ell$$ mod-$$\ell$$ image Is torsion prime?
$$2$$ 2.180.3 yes
$$5$$ not computed yes

## Sato-Tate group

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

## Decomposition of the Jacobian

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

magma: HeuristicDecompositionFactors(C);

## 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 $$\Q$$.

magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

magma: HeuristicIsGL2(C); HeuristicEndomorphismDescription(C); HeuristicEndomorphismFieldOfDefinition(C);

magma: HeuristicIsGL2(C : Geometric := true); HeuristicEndomorphismDescription(C : Geometric := true); HeuristicEndomorphismLatticeDescription(C);