# Properties

 Label 8204.a.32816.1 Conductor 8204 Discriminant 32816 Mordell-Weil group $$\Z \times \Z$$ 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

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

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

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

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([1, -4, 8, -8, 0, 0, 4]))

## Invariants

 Conductor: $$N$$ = $$8204$$ = $$2^{2} \cdot 7 \cdot 293$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$32816$$ = $$2^{4} \cdot 7 \cdot 293$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$-576$$ = $$- 2^{6} \cdot 3^{2}$$ $$I_4$$ = $$22848$$ = $$2^{6} \cdot 3 \cdot 7 \cdot 17$$ $$I_6$$ = $$-4981248$$ = $$- 2^{9} \cdot 3^{2} \cdot 23 \cdot 47$$ $$I_{10}$$ = $$134414336$$ = $$2^{16} \cdot 7 \cdot 293$$ $$J_2$$ = $$-72$$ = $$- 2^{3} \cdot 3^{2}$$ $$J_4$$ = $$-22$$ = $$- 2 \cdot 11$$ $$J_6$$ = $$3024$$ = $$2^{4} \cdot 3^{3} \cdot 7$$ $$J_8$$ = $$-54553$$ = $$- 17 \cdot 3209$$ $$J_{10}$$ = $$32816$$ = $$2^{4} \cdot 7 \cdot 293$$ $$g_1$$ = $$-120932352/2051$$ $$g_2$$ = $$513216/2051$$ $$g_3$$ = $$139968/293$$

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
$$(1 : -1 : 0)$$ $$(1 : 1 : 0)$$ $$(0 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(1 : 0 : 1)$$ $$(1 : -1 : 1)$$
$$(-1 : 2 : 1)$$ $$(-1 : -3 : 1)$$ $$(1 : -3 : 2)$$ $$(1 : -5 : 2)$$ $$(-2 : 9 : 1)$$ $$(-2 : -10 : 1)$$
$$(3 : -223 : 10)$$ $$(3 : -777 : 10)$$

magma: [C![-2,-10,1],C![-2,9,1],C![-1,-3,1],C![-1,2,1],C![0,-1,1],C![0,0,1],C![1,-5,2],C![1,-3,2],C![1,-1,0],C![1,-1,1],C![1,0,1],C![1,1,0],C![3,-777,10],C![3,-223,10]];

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
$$(0 : -1 : 1) + (1 : -5 : 2) - (1 : -1 : 0) - (1 : 1 : 0)$$ $$(-2x + z) x$$ $$=$$ $$0,$$ $$4y$$ $$=$$ $$3xz^2 - 4z^3$$ $$0.169537$$ $$\infty$$
$$(1 : -1 : 1) - (1 : 1 : 0)$$ $$z (x - z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3$$ $$0.040146$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$2$$ Mordell-Weil rank: $$2$$ 2-Selmer rank: $$2$$ Regulator: $$0.006769$$ Real period: $$18.14031$$ Tamagawa product: $$3$$ Torsion order: $$1$$ Leading coefficient: $$0.368388$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$2$$ $$4$$ $$2$$ $$3$$ $$1 + 2 T + 2 T^{2}$$
$$7$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 4 T + 7 T^{2} )$$
$$293$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 - 19 T + 293 T^{2} )$$

## 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 $$\Q$$.