# Properties

 Label 19881.b.536787.1 Conductor 19881 Discriminant -536787 Mordell-Weil group $$\Z \times \Z$$ 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

 $y^2 + y = x^6 - 3x^5 + 2x^4 + x^3 - 2x^2 + x$ (homogenize, simplify) $y^2 + z^3y = x^6 - 3x^5z + 2x^4z^2 + x^3z^3 - 2x^2z^4 + xz^5$ (dehomogenize, simplify) $y^2 = 4x^6 - 12x^5 + 8x^4 + 4x^3 - 8x^2 + 4x + 1$ (minimize, homogenize)

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

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

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([1, 4, -8, 4, 8, -12, 4]))

## Invariants

 Conductor: $$N$$ = $$19881$$ = $$3^{2} \cdot 47^{2}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$-536787$$ = $$- 3^{5} \cdot 47^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$-1760$$ = $$- 2^{5} \cdot 5 \cdot 11$$ $$I_4$$ = $$16192$$ = $$2^{6} \cdot 11 \cdot 23$$ $$I_6$$ = $$-27226112$$ = $$- 2^{12} \cdot 17^{2} \cdot 23$$ $$I_{10}$$ = $$-2198679552$$ = $$- 2^{12} \cdot 3^{5} \cdot 47^{2}$$ $$J_2$$ = $$-220$$ = $$- 2^{2} \cdot 5 \cdot 11$$ $$J_4$$ = $$1848$$ = $$2^{3} \cdot 3 \cdot 7 \cdot 11$$ $$J_6$$ = $$12312$$ = $$2^{3} \cdot 3^{4} \cdot 19$$ $$J_8$$ = $$-1530936$$ = $$- 2^{3} \cdot 3^{2} \cdot 11 \cdot 1933$$ $$J_{10}$$ = $$-536787$$ = $$- 3^{5} \cdot 47^{2}$$ $$g_1$$ = $$515363200000/536787$$ $$g_2$$ = $$6559168000/178929$$ $$g_3$$ = $$-7356800/6627$$

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
$$(1 : -1 : 0)$$ $$(1 : 1 : 0)$$ $$(0 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(1 : 0 : 1)$$ $$(-1 : 1 : 1)$$
$$(1 : -1 : 1)$$ $$(-1 : -2 : 1)$$ $$(2 : 1 : 1)$$ $$(2 : -2 : 1)$$ $$(-1 : -164 : 11)$$ $$(12 : -164 : 11)$$
$$(-1 : -1167 : 11)$$ $$(12 : -1167 : 11)$$

magma: [C![-1,-1167,11],C![-1,-164,11],C![-1,-2,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-1,0],C![1,-1,1],C![1,0,1],C![1,1,0],C![2,-2,1],C![2,1,1],C![12,-1167,11],C![12,-164,11]];

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: $$\Z \times \Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(0 : 0 : 1) + (2 : -2 : 1) - (1 : -1 : 0) - (1 : 1 : 0)$$ $$(-x + 2z) x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2$$ $$0.116491$$ $$\infty$$
$$D_0 - (1 : -1 : 0) - (1 : 1 : 0)$$ $$x^2 - xz + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$xz^2 - z^3$$ $$0.068973$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$2$$ Mordell-Weil rank: $$2$$ 2-Selmer rank: $$2$$ Regulator: $$0.006845$$ Real period: $$14.07745$$ Tamagawa product: $$7$$ Torsion order: $$1$$ Leading coefficient: $$0.674566$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$3$$ $$5$$ $$2$$ $$7$$ $$( 1 - T )( 1 + T )$$
$$47$$ $$2$$ $$2$$ $$1$$ $$( 1 - T )( 1 + T )$$

## 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 141.a1
Elliptic curve 141.d1

## 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 $$\Q$$.