# Properties

 Label 68209.a.68209.1 Conductor 68209 Discriminant 68209 Mordell-Weil group $$\Z \times \Z \times \Z$$ Sato-Tate group $\mathrm{USp}(4)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type no

# Related objects

Show commands for: Magma / SageMath

## Simplified equation

 $y^2 + (x^3 + 1)y = x^5 - 3x^4 + 3x^3 - x$ (homogenize, simplify) $y^2 + (x^3 + z^3)y = x^5z - 3x^4z^2 + 3x^3z^3 - xz^5$ (dehomogenize, simplify) $y^2 = x^6 + 4x^5 - 12x^4 + 14x^3 - 4x + 1$ (minimize, homogenize)

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

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

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([1, -4, 0, 14, -12, 4, 1]))

## Invariants

 Conductor: $$N$$ = $$68209$$ = $$68209$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$68209$$ = $$68209$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$296$$ = $$2^{3} \cdot 37$$ $$I_4$$ = $$17572$$ = $$2^{2} \cdot 23 \cdot 191$$ $$I_6$$ = $$-5910616$$ = $$- 2^{3} \cdot 738827$$ $$I_{10}$$ = $$279384064$$ = $$2^{12} \cdot 68209$$ $$J_2$$ = $$37$$ = $$37$$ $$J_4$$ = $$-126$$ = $$- 2 \cdot 3^{2} \cdot 7$$ $$J_6$$ = $$12260$$ = $$2^{2} \cdot 5 \cdot 613$$ $$J_8$$ = $$109436$$ = $$2^{2} \cdot 109 \cdot 251$$ $$J_{10}$$ = $$68209$$ = $$68209$$ $$g_1$$ = $$69343957/68209$$ $$g_2$$ = $$-6382278/68209$$ $$g_3$$ = $$16783940/68209$$

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

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

## Automorphism group

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

## Rational points

Known points
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(0 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(1 : 0 : 1)$$ $$(-1 : -1 : 2)$$
$$(1 : -2 : 1)$$ $$(1 : -3 : 2)$$ $$(-1 : -6 : 2)$$ $$(1 : -6 : 2)$$ $$(2 : -6 : 3)$$ $$(2 : -29 : 3)$$
$$(2 : -55 : 5)$$ $$(2 : -78 : 5)$$

magma: [C![-1,-6,2],C![-1,-1,2],C![0,-1,1],C![0,0,1],C![1,-6,2],C![1,-3,2],C![1,-2,1],C![1,-1,0],C![1,0,0],C![1,0,1],C![2,-78,5],C![2,-55,5],C![2,-29,3],C![2,-6,3]];

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: $$\Z \times \Z \times \Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(0 : 0 : 1) + (1 : -2 : 1) - (1 : -1 : 0) - (1 : 0 : 0)$$ $$(-x + z) x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-2xz^2$$ $$0.404526$$ $$\infty$$
$$2 \cdot(0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$xz^2 - z^3$$ $$0.326742$$ $$\infty$$
$$(0 : -1 : 1) - (1 : 0 : 0)$$ $$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3 - z^3$$ $$0.375390$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$3$$   (upper bound) Mordell-Weil rank: $$3$$ 2-Selmer rank: $$3$$ Regulator: $$0.044349$$ Real period: $$18.00406$$ Tamagawa product: $$1$$ Torsion order: $$1$$ Leading coefficient: $$0.798465$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$68209$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 - 164 T + 68209 T^{2} )$$

## 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 $$\Q$$.