# Properties

 Label 126859.b.126859.1 Conductor $126859$ Discriminant $-126859$ Mordell-Weil group $$\Z$$ 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: SageMath / Magma

## Simplified equation

 $y^2 + y = x^5 - 8x^4 - 38x^3 - 29x^2 - 8x - 1$ (homogenize, simplify) $y^2 + z^3y = x^5z - 8x^4z^2 - 38x^3z^3 - 29x^2z^4 - 8xz^5 - z^6$ (dehomogenize, simplify) $y^2 = 4x^5 - 32x^4 - 152x^3 - 116x^2 - 32x - 3$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, -8, -29, -38, -8, 1]), R([1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, -8, -29, -38, -8, 1], R![1]);

sage: X = HyperellipticCurve(R([-3, -32, -116, -152, -32, 4]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$126859$$ $$=$$ $$126859$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-126859$$ $$=$$ $$-126859$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$18528$$ $$=$$ $$2^{5} \cdot 3 \cdot 193$$ $$I_4$$ $$=$$ $$46416$$ $$=$$ $$2^{4} \cdot 3 \cdot 967$$ $$I_6$$ $$=$$ $$283353360$$ $$=$$ $$2^{4} \cdot 3 \cdot 5 \cdot 367 \cdot 3217$$ $$I_{10}$$ $$=$$ $$-507436$$ $$=$$ $$- 2^{2} \cdot 126859$$ $$J_2$$ $$=$$ $$9264$$ $$=$$ $$2^{4} \cdot 3 \cdot 193$$ $$J_4$$ $$=$$ $$3568168$$ $$=$$ $$2^{3} \cdot 577 \cdot 773$$ $$J_6$$ $$=$$ $$1828822192$$ $$=$$ $$2^{4} \cdot 17 \cdot 79 \cdot 85109$$ $$J_8$$ $$=$$ $$1052596477616$$ $$=$$ $$2^{4} \cdot 137 \cdot 197 \cdot 2437559$$ $$J_{10}$$ $$=$$ $$-126859$$ $$=$$ $$-126859$$ $$g_1$$ $$=$$ $$-68232727503987277824/126859$$ $$g_2$$ $$=$$ $$-2836879788910804992/126859$$ $$g_3$$ $$=$$ $$-156952622199877632/126859$$

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: $$(1 : 0 : 0)$$
All points: $$(1 : 0 : 0)$$
All points: $$(1 : 0 : 0)$$

magma: [C![1,0,0]]; // minimal model

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

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: $$\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$x^2 + 3xz + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$xz^2$$ $$1.302295$$ $$\infty$$
Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$x^2 + 3xz + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$xz^2$$ $$1.302295$$ $$\infty$$
Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$x^2 + 3xz + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$2xz^2 + z^3$$ $$1.302295$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$1$$ Regulator: $$1.302295$$ Real period: $$3.803713$$ Tamagawa product: $$1$$ Torsion order: $$1$$ Leading coefficient: $$4.953558$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$126859$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 - 551 T + 126859 T^{2} )$$

## Galois representations

For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not the maximal image $\GSp(4,\F_\ell)$

Prime $$\ell$$ mod-$$\ell$$ image
$$2$$ 2.6.1

## 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 $$\Q$$.