# Properties

 Label 1051.b.1051.2 Conductor $1051$ Discriminant $-1051$ Mordell-Weil group $$\Z/{2}\Z \times \Z/{2}\Z$$ 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

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Show commands: SageMath / Magma

## Simplified equation

 $y^2 + xy = x^5 + 8x^4 + 16x^3 + x$ (homogenize, simplify) $y^2 + xz^2y = x^5z + 8x^4z^2 + 16x^3z^3 + xz^5$ (dehomogenize, simplify) $y^2 = 4x^5 + 32x^4 + 64x^3 + x^2 + 4x$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, 0, 16, 8, 1]), R([0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, 0, 16, 8, 1], R![0, 1]);

sage: X = HyperellipticCurve(R([0, 4, 1, 64, 32, 4]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$1051$$ $$=$$ $$1051$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-1051$$ $$=$$ $$-1051$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$6176$$ $$=$$ $$2^{5} \cdot 193$$ $$I_4$$ $$=$$ $$-50240$$ $$=$$ $$- 2^{6} \cdot 5 \cdot 157$$ $$I_6$$ $$=$$ $$-103225225$$ $$=$$ $$- 5^{2} \cdot 67 \cdot 61627$$ $$I_{10}$$ $$=$$ $$-4204$$ $$=$$ $$- 2^{2} \cdot 1051$$ $$J_2$$ $$=$$ $$3088$$ $$=$$ $$2^{4} \cdot 193$$ $$J_4$$ $$=$$ $$405696$$ $$=$$ $$2^{6} \cdot 3 \cdot 2113$$ $$J_6$$ $$=$$ $$72449921$$ $$=$$ $$72449921$$ $$J_8$$ $$=$$ $$14784027908$$ $$=$$ $$2^{2} \cdot 13 \cdot 19 \cdot 313 \cdot 47807$$ $$J_{10}$$ $$=$$ $$-1051$$ $$=$$ $$-1051$$ $$g_1$$ $$=$$ $$-280793117300359168/1051$$ $$g_2$$ $$=$$ $$-11946277554880512/1051$$ $$g_3$$ $$=$$ $$-690863899476224/1051$$

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: $$(1 : 0 : 0),\, (0 : 0 : 1),\, (-4 : 2 : 1)$$ All points: $$(1 : 0 : 0),\, (0 : 0 : 1),\, (-4 : 2 : 1)$$ All points: $$(1 : 0 : 0),\, (0 : 0 : 1),\, (-4 : 0 : 1)$$

magma: [C![-4,2,1],C![0,0,1],C![1,0,0]]; // minimal model

magma: [C![-4,0,1],C![0,0,1],C![1,0,0]]; // simplified model

Number of rational Weierstrass points: $$3$$

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

magma: MordellWeilGroupGenus2(Jacobian(C));

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

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$2$$ Regulator: $$1$$ Real period: $$5.832929$$ Tamagawa product: $$1$$ Torsion order: $$4$$ Leading coefficient: $$0.364558$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$1051$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 - 28 T + 1051 T^{2} )$$

## 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 $$\Q$$.