# Properties

 Label 1070.a.2140.1 Conductor $1070$ Discriminant $-2140$ Mordell-Weil group $$\Z \times \Z/{4}\Z$$ 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

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

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

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

sage: X = HyperellipticCurve(R([1, -4, 0, 6, 0, 0, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$1070$$ $$=$$ $$2 \cdot 5 \cdot 107$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-2140$$ $$=$$ $$- 2^{2} \cdot 5 \cdot 107$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$12$$ $$=$$ $$2^{2} \cdot 3$$ $$I_4$$ $$=$$ $$3321$$ $$=$$ $$3^{4} \cdot 41$$ $$I_6$$ $$=$$ $$141939$$ $$=$$ $$3^{3} \cdot 7 \cdot 751$$ $$I_{10}$$ $$=$$ $$273920$$ $$=$$ $$2^{9} \cdot 5 \cdot 107$$ $$J_2$$ $$=$$ $$3$$ $$=$$ $$3$$ $$J_4$$ $$=$$ $$-138$$ $$=$$ $$- 2 \cdot 3 \cdot 23$$ $$J_6$$ $$=$$ $$-1856$$ $$=$$ $$- 2^{6} \cdot 29$$ $$J_8$$ $$=$$ $$-6153$$ $$=$$ $$- 3 \cdot 7 \cdot 293$$ $$J_{10}$$ $$=$$ $$2140$$ $$=$$ $$2^{2} \cdot 5 \cdot 107$$ $$g_1$$ $$=$$ $$243/2140$$ $$g_2$$ $$=$$ $$-1863/1070$$ $$g_3$$ $$=$$ $$-4176/535$$

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
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(0 : 0 : 1)$$ $$(-1 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(1 : 0 : 1)$$
$$(-2 : 1 : 1)$$ $$(1 : -2 : 1)$$ $$(-2 : 6 : 1)$$
All points
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(0 : 0 : 1)$$ $$(-1 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(1 : 0 : 1)$$
$$(-2 : 1 : 1)$$ $$(1 : -2 : 1)$$ $$(-2 : 6 : 1)$$
All points
$$(1 : -1 : 0)$$ $$(1 : 1 : 0)$$ $$(-1 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(0 : 1 : 1)$$ $$(1 : -2 : 1)$$
$$(1 : 2 : 1)$$ $$(-2 : -5 : 1)$$ $$(-2 : 5 : 1)$$

magma: [C![-2,1,1],C![-2,6,1],C![-1,0,1],C![0,-1,1],C![0,0,1],C![1,-2,1],C![1,-1,0],C![1,0,0],C![1,0,1]]; // minimal model

magma: [C![-2,-5,1],C![-2,5,1],C![-1,0,1],C![0,-1,1],C![0,1,1],C![1,-2,1],C![1,-1,0],C![1,1,0],C![1,2,1]]; // 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: $$\Z \times \Z/{4}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(-1 : 0 : 1) - (1 : 0 : 0)$$ $$z (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3 - z^3$$ $$0.054348$$ $$\infty$$
$$(0 : 0 : 1) - (1 : -1 : 0)$$ $$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$4$$
Generator $D_0$ Height Order
$$(-1 : 0 : 1) - (1 : 0 : 0)$$ $$z (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3 - z^3$$ $$0.054348$$ $$\infty$$
$$(0 : 0 : 1) - (1 : -1 : 0)$$ $$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$4$$
Generator $D_0$ Height Order
$$(-1 : 0 : 1) - (1 : 1 : 0)$$ $$z (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3 - z^3$$ $$0.054348$$ $$\infty$$
$$(0 : 1 : 1) - (1 : -1 : 0)$$ $$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3 + z^3$$ $$0$$ $$4$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$2$$ Regulator: $$0.054348$$ Real period: $$27.74393$$ Tamagawa product: $$2$$ Torsion order: $$4$$ Leading coefficient: $$0.188481$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$1$$ $$2$$ $$2$$ $$( 1 + T )( 1 + T + 2 T^{2} )$$
$$5$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 4 T + 5 T^{2} )$$
$$107$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 - 16 T + 107 T^{2} )$$

## 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 $$\Q$$.