# Properties

 Label 10020.a.160320.1 Conductor $10020$ Discriminant $-160320$ Mordell-Weil group $$\Z \times \Z/{3}\Z$$ 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

 $y^2 + (x^3 + x + 1)y = -x^4 - 4x^3 + 7x + 4$ (homogenize, simplify) $y^2 + (x^3 + xz^2 + z^3)y = -x^4z^2 - 4x^3z^3 + 7xz^5 + 4z^6$ (dehomogenize, simplify) $y^2 = x^6 - 2x^4 - 14x^3 + x^2 + 30x + 17$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([4, 7, 0, -4, -1]), R([1, 1, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![4, 7, 0, -4, -1], R![1, 1, 0, 1]);

sage: X = HyperellipticCurve(R([17, 30, 1, -14, -2, 0, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$10020$$ $$=$$ $$2^{2} \cdot 3 \cdot 5 \cdot 167$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-160320$$ $$=$$ $$- 2^{6} \cdot 3 \cdot 5 \cdot 167$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$1436$$ $$=$$ $$2^{2} \cdot 359$$ $$I_4$$ $$=$$ $$278545$$ $$=$$ $$5 \cdot 17 \cdot 29 \cdot 113$$ $$I_6$$ $$=$$ $$179457607$$ $$=$$ $$7 \cdot 4933 \cdot 5197$$ $$I_{10}$$ $$=$$ $$20520960$$ $$=$$ $$2^{13} \cdot 3 \cdot 5 \cdot 167$$ $$J_2$$ $$=$$ $$359$$ $$=$$ $$359$$ $$J_4$$ $$=$$ $$-6236$$ $$=$$ $$- 2^{2} \cdot 1559$$ $$J_6$$ $$=$$ $$-1227984$$ $$=$$ $$- 2^{4} \cdot 3 \cdot 25583$$ $$J_8$$ $$=$$ $$-119933488$$ $$=$$ $$- 2^{4} \cdot 53 \cdot 233 \cdot 607$$ $$J_{10}$$ $$=$$ $$160320$$ $$=$$ $$2^{6} \cdot 3 \cdot 5 \cdot 167$$ $$g_1$$ $$=$$ $$5963102065799/160320$$ $$g_2$$ $$=$$ $$-72132246961/40080$$ $$g_3$$ $$=$$ $$-3297162623/3340$$

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

magma: [C![-1,0,1],C![-1,1,1],C![1,-1,0],C![1,0,0],C![2,-6,1],C![2,-5,1]]; // minimal model

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

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: $$\Z \times \Z/{3}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : -1 : 0)$$ $$z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0.113711$$ $$\infty$$
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2 - xz - z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2$$ $$0$$ $$3$$
Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : -1 : 0)$$ $$z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0.113711$$ $$\infty$$
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2 - xz - z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2$$ $$0$$ $$3$$
Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : -1 : 0)$$ $$z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3 + xz^2 + z^3$$ $$0.113711$$ $$\infty$$
$$D_0 - (1 : -1 : 0) - (1 : 1 : 0)$$ $$x^2 - xz - z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3 - xz^2 + z^3$$ $$0$$ $$3$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$1$$ Regulator: $$0.113711$$ Real period: $$19.20449$$ Tamagawa product: $$3$$ Torsion order: $$3$$ Leading coefficient: $$0.727926$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$2$$ $$6$$ $$3$$ $$1 + T + T^{2}$$
$$3$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 2 T + 3 T^{2} )$$
$$5$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 5 T^{2} )$$
$$167$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 - 18 T + 167 T^{2} )$$

## 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 $$\Q$$.