# Properties

 Label 20172.b.968256.1 Conductor 20172 Discriminant -968256 Mordell-Weil group $$\Z \times \Z \times \Z/{2}\Z$$ Sato-Tate group $G_{3,3}$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q \times \Q$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Related objects

Show commands for: Magma / SageMath

## Simplified equation

 $y^2 + (x^2 + x)y = x^6 - x^5 - 2x^4 + 4x^3 - 3x + 1$ (homogenize, simplify) $y^2 + (x^2z + xz^2)y = x^6 - x^5z - 2x^4z^2 + 4x^3z^3 - 3xz^5 + z^6$ (dehomogenize, simplify) $y^2 = 4x^6 - 4x^5 - 7x^4 + 18x^3 + x^2 - 12x + 4$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![1, -3, 0, 4, -2, -1, 1], R![0, 1, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, -3, 0, 4, -2, -1, 1]), R([0, 1, 1]));

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([4, -12, 1, 18, -7, -4, 4]))

## Invariants

 Conductor: $$N$$ = $$20172$$ = $$2^{2} \cdot 3 \cdot 41^{2}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$-968256$$ = $$- 2^{6} \cdot 3^{2} \cdot 41^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$136$$ = $$2^{3} \cdot 17$$ $$I_4$$ = $$774916$$ = $$2^{2} \cdot 23 \cdot 8423$$ $$I_6$$ = $$-219321592$$ = $$- 2^{3} \cdot 7 \cdot 613 \cdot 6389$$ $$I_{10}$$ = $$-3965976576$$ = $$- 2^{18} \cdot 3^{2} \cdot 41^{2}$$ $$J_2$$ = $$17$$ = $$17$$ $$J_4$$ = $$-8060$$ = $$- 2^{2} \cdot 5 \cdot 13 \cdot 31$$ $$J_6$$ = $$418896$$ = $$2^{4} \cdot 3^{2} \cdot 2909$$ $$J_8$$ = $$-14460592$$ = $$- 2^{4} \cdot 83 \cdot 10889$$ $$J_{10}$$ = $$-968256$$ = $$- 2^{6} \cdot 3^{2} \cdot 41^{2}$$ $$g_1$$ = $$-1419857/968256$$ $$g_2$$ = $$9899695/242064$$ $$g_3$$ = $$-840701/6724$$

magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);

sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]

## Automorphism group

 $$\mathrm{Aut}(X)$$ $$\simeq$$ $C_2^2$ magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$$C_2^2$magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);

## Rational points

Known points
$$(1 : -1 : 0)$$ $$(1 : 1 : 0)$$ $$(-1 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(0 : 1 : 1)$$ $$(1 : 0 : 1)$$
$$(1 : -2 : 1)$$ $$(1 : -3 : 2)$$ $$(2 : 3 : 1)$$ $$(-3 : 5 : 2)$$ $$(2 : -9 : 1)$$ $$(-3 : -11 : 2)$$
$$(3 : -52 : 5)$$ $$(3 : -68 : 5)$$

magma: [C![-3,-11,2],C![-3,5,2],C![-1,0,1],C![0,-1,1],C![0,1,1],C![1,-3,2],C![1,-2,1],C![1,-1,0],C![1,0,1],C![1,1,0],C![2,-9,1],C![2,3,1],C![3,-68,5],C![3,-52,5]];

Number of rational Weierstrass points: $$2$$

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

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(-1 : 0 : 1) + (0 : 1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)$$ $$x (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$xz^2 + z^3$$ $$0.158713$$ $$\infty$$
$$(0 : -1 : 1) - (1 : 1 : 0)$$ $$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3 - z^3$$ $$0.112353$$ $$\infty$$
$$(-1 : 0 : 1) + (1 : -3 : 2) - (1 : -1 : 0) - (1 : 1 : 0)$$ $$(x + z) (2x - z)$$ $$=$$ $$0,$$ $$4y$$ $$=$$ $$-xz^2 - z^3$$ $$0$$ $$2$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$2$$ Mordell-Weil rank: $$2$$ 2-Selmer rank: $$3$$ Regulator: $$0.017831$$ Real period: $$16.48290$$ Tamagawa product: $$8$$ Torsion order: $$2$$ Leading coefficient: $$0.587845$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$2$$ $$6$$ $$2$$ $$4$$ $$( 1 + T )^{2}$$
$$3$$ $$2$$ $$1$$ $$2$$ $$( 1 + T )( 1 + 2 T + 3 T^{2} )$$
$$41$$ $$2$$ $$2$$ $$1$$ $$( 1 + T )^{2}$$

## Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $G_{3,3}$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{SU}(2)\times\mathrm{SU}(2)$$

## Decomposition of the Jacobian

Splits over $$\Q$$

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
Elliptic curve 246.a1
Elliptic curve 82.a2

## Endomorphisms of the Jacobian

Of $$\GL_2$$-type over $$\Q$$

Endomorphism ring over $$\Q$$:

 $$\End (J_{})$$ $$\simeq$$ an order of index $$2$$ in $$\Z \times \Z$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\times$$ $$\Q$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R \times \R$$

All $$\overline{\Q}$$-endomorphisms of the Jacobian are defined over $$\Q$$.