# Properties

 Label 59107.a.59107.1 Conductor 59107 Discriminant 59107 Mordell-Weil group $$\Z \times \Z \times \Z$$ 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

 $y^2 + (x^3 + x^2 + 1)y = 2x^4 - 3x^2 - x$ (homogenize, simplify) $y^2 + (x^3 + x^2z + z^3)y = 2x^4z^2 - 3x^2z^4 - xz^5$ (dehomogenize, simplify) $y^2 = x^6 + 2x^5 + 9x^4 + 2x^3 - 10x^2 - 4x + 1$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, -3, 0, 2], R![1, 0, 1, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, -3, 0, 2]), R([1, 0, 1, 1]));

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([1, -4, -10, 2, 9, 2, 1]))

## Invariants

 Conductor: $$N$$ = $$59107$$ = $$59107$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$59107$$ = $$59107$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$904$$ = $$2^{3} \cdot 113$$ $$I_4$$ = $$81892$$ = $$2^{2} \cdot 59 \cdot 347$$ $$I_6$$ = $$10385896$$ = $$2^{3} \cdot 149 \cdot 8713$$ $$I_{10}$$ = $$242102272$$ = $$2^{12} \cdot 59107$$ $$J_2$$ = $$113$$ = $$113$$ $$J_4$$ = $$-321$$ = $$- 3 \cdot 107$$ $$J_6$$ = $$12085$$ = $$5 \cdot 2417$$ $$J_8$$ = $$315641$$ = $$439 \cdot 719$$ $$J_{10}$$ = $$59107$$ = $$59107$$ $$g_1$$ = $$18424351793/59107$$ $$g_2$$ = $$-463169937/59107$$ $$g_3$$ = $$154313365/59107$$

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

Known points
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(0 : 0 : 1)$$ $$(-1 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(-1 : -1 : 1)$$
$$(1 : -1 : 1)$$ $$(-1 : -1 : 2)$$ $$(1 : -2 : 1)$$ $$(-3 : -6 : 1)$$ $$(-1 : -8 : 2)$$ $$(-2 : -9 : 3)$$
$$(-2 : -22 : 3)$$ $$(-3 : 23 : 1)$$ $$(-4 : 36 : 3)$$ $$(-4 : -47 : 3)$$ $$(11 : -616 : 10)$$ $$(11 : -2925 : 10)$$

magma: [C![-4,-47,3],C![-4,36,3],C![-3,-6,1],C![-3,23,1],C![-2,-22,3],C![-2,-9,3],C![-1,-8,2],C![-1,-1,1],C![-1,-1,2],C![-1,0,1],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![11,-2925,10],C![11,-616,10]];

Number of rational Weierstrass points: $$0$$

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: $$\Z \times \Z \times \Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(0 : 0 : 1) - (1 : -1 : 0)$$ $$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0.514009$$ $$\infty$$
$$(-1 : 0 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2 - z^3$$ $$0.232060$$ $$\infty$$
$$(-1 : -1 : 1) - (1 : -1 : 0)$$ $$z (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0.269174$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$3$$   (upper bound) Mordell-Weil rank: $$3$$ 2-Selmer rank: $$3$$ Regulator: $$0.027565$$ Real period: $$20.50443$$ Tamagawa product: $$1$$ Torsion order: $$1$$ Leading coefficient: $$0.565206$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$59107$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 - 200 T + 59107 T^{2} )$$

## 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 $$\Q$$.