# Properties

 Label 8452.a.16904.1 Conductor 8452 Discriminant 16904 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

 $y^2 + (x^3 + x + 1)y = x^5 - 2x^4 - 3x^3 + x^2 + x$ (homogenize, simplify) $y^2 + (x^3 + xz^2 + z^3)y = x^5z - 2x^4z^2 - 3x^3z^3 + x^2z^4 + xz^5$ (dehomogenize, simplify) $y^2 = x^6 + 4x^5 - 6x^4 - 10x^3 + 5x^2 + 6x + 1$ (minimize, homogenize)

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

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

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([1, 6, 5, -10, -6, 4, 1]))

## Invariants

 Conductor: $$N$$ = $$8452$$ = $$2^{2} \cdot 2113$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$16904$$ = $$2^{3} \cdot 2113$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$1800$$ = $$2^{3} \cdot 3^{2} \cdot 5^{2}$$ $$I_4$$ = $$80772$$ = $$2^{2} \cdot 3 \cdot 53 \cdot 127$$ $$I_6$$ = $$43043976$$ = $$2^{3} \cdot 3^{2} \cdot 597833$$ $$I_{10}$$ = $$69238784$$ = $$2^{15} \cdot 2113$$ $$J_2$$ = $$225$$ = $$3^{2} \cdot 5^{2}$$ $$J_4$$ = $$1268$$ = $$2^{2} \cdot 317$$ $$J_6$$ = $$4224$$ = $$2^{7} \cdot 3 \cdot 11$$ $$J_8$$ = $$-164356$$ = $$- 2^{2} \cdot 17 \cdot 2417$$ $$J_{10}$$ = $$16904$$ = $$2^{3} \cdot 2113$$ $$g_1$$ = $$576650390625/16904$$ $$g_2$$ = $$3610828125/4226$$ $$g_3$$ = $$26730000/2113$$

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
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(0 : 0 : 1)$$ $$(-1 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(-1 : 1 : 1)$$
$$(1 : -1 : 1)$$ $$(-1 : -1 : 2)$$ $$(1 : -2 : 1)$$ $$(-1 : -2 : 2)$$ $$(2 : -2 : 1)$$ $$(2 : -9 : 1)$$
$$(-4 : -2261 : 21)$$ $$(-4 : -5172 : 21)$$

magma: [C![-4,-5172,21],C![-4,-2261,21],C![-1,-2,2],C![-1,-1,2],C![-1,0,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-2,1],C![1,-1,0],C![1,-1,1],C![1,0,0],C![2,-9,1],C![2,-2,1]];

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
$$(0 : -1 : 1) - (1 : -1 : 0)$$ $$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0.105278$$ $$\infty$$
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2 - xz - z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-2xz^2 - z^3$$ $$0.060485$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$2$$ Mordell-Weil rank: $$2$$ 2-Selmer rank: $$2$$ Regulator: $$0.006277$$ Real period: $$23.65540$$ Tamagawa product: $$3$$ Torsion order: $$1$$ Leading coefficient: $$0.445471$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$2$$ $$3$$ $$2$$ $$3$$ $$1 + T + T^{2}$$
$$2113$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 34 T + 2113 T^{2} )$$

## 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 $$\Q$$.