# Properties

 Label 3319.a.3319.1 Conductor $3319$ Discriminant $3319$ Mordell-Weil group $$\Z \oplus \Z$$ Sato-Tate group $\mathrm{USp}(4)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q$$ $$\End(J) \otimes \Q$$ $$\Q$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type no

# Related objects

Show commands: Magma / SageMath

## Simplified equation

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

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

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

sage: X = HyperellipticCurve(R([1, 2, 1, 6, 10, 4, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$3319$$ $$=$$ $$3319$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$3319$$ $$=$$ $$3319$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$68$$ $$=$$ $$2^{2} \cdot 17$$ $$I_4$$ $$=$$ $$3673$$ $$=$$ $$3673$$ $$I_6$$ $$=$$ $$38093$$ $$=$$ $$11 \cdot 3463$$ $$I_{10}$$ $$=$$ $$424832$$ $$=$$ $$2^{7} \cdot 3319$$ $$J_2$$ $$=$$ $$17$$ $$=$$ $$17$$ $$J_4$$ $$=$$ $$-141$$ $$=$$ $$- 3 \cdot 47$$ $$J_6$$ $$=$$ $$205$$ $$=$$ $$5 \cdot 41$$ $$J_8$$ $$=$$ $$-4099$$ $$=$$ $$-4099$$ $$J_{10}$$ $$=$$ $$3319$$ $$=$$ $$3319$$ $$g_1$$ $$=$$ $$1419857/3319$$ $$g_2$$ $$=$$ $$-692733/3319$$ $$g_3$$ $$=$$ $$59245/3319$$

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

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

## 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
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(0 : 0 : 1)$$ $$(-1 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(-1 : 1 : 1)$$
$$(1 : 1 : 1)$$ $$(-2 : 1 : 1)$$ $$(-1 : -1 : 2)$$ $$(-1 : -2 : 2)$$ $$(1 : -4 : 1)$$ $$(-2 : 8 : 1)$$
Known points
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(0 : 0 : 1)$$ $$(-1 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(-1 : 1 : 1)$$
$$(1 : 1 : 1)$$ $$(-2 : 1 : 1)$$ $$(-1 : -1 : 2)$$ $$(-1 : -2 : 2)$$ $$(1 : -4 : 1)$$ $$(-2 : 8 : 1)$$
Known points
$$(1 : -1 : 0)$$ $$(1 : 1 : 0)$$ $$(0 : -1 : 1)$$ $$(0 : 1 : 1)$$ $$(-1 : -1 : 1)$$ $$(-1 : 1 : 1)$$
$$(-1 : -1 : 2)$$ $$(-1 : 1 : 2)$$ $$(1 : -5 : 1)$$ $$(1 : 5 : 1)$$ $$(-2 : -7 : 1)$$ $$(-2 : 7 : 1)$$

magma: [C![-2,1,1],C![-2,8,1],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,-4,1],C![1,-1,0],C![1,0,0],C![1,1,1]]; // minimal model

magma: [C![-2,-7,1],C![-2,7,1],C![-1,-1,2],C![-1,1,2],C![-1,-1,1],C![-1,1,1],C![0,-1,1],C![0,1,1],C![1,-5,1],C![1,-1,0],C![1,1,0],C![1,5,1]]; // simplified model

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: $$\Z \oplus \Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(-1 : 0 : 1) - (1 : -1 : 0)$$ $$z (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0.205416$$ $$\infty$$
$$(-1 : 0 : 1) + (0 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0.036927$$ $$\infty$$
Generator $D_0$ Height Order
$$(-1 : 0 : 1) - (1 : -1 : 0)$$ $$z (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0.205416$$ $$\infty$$
$$(-1 : 0 : 1) + (0 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0.036927$$ $$\infty$$
Generator $D_0$ Height Order
$$(-1 : -1 : 1) - (1 : -1 : 0)$$ $$z (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3 + xz^2 + z^3$$ $$0.205416$$ $$\infty$$
$$(-1 : -1 : 1) + (0 : 1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)$$ $$x (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3 + xz^2 + z^3$$ $$0.036927$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$2$$ Mordell-Weil rank: $$2$$ 2-Selmer rank: $$2$$ Regulator: $$0.007585$$ Real period: $$25.35935$$ Tamagawa product: $$1$$ Torsion order: $$1$$ Leading coefficient: $$0.192363$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$3319$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 - 74 T + 3319 T^{2} )$$

## Galois representations

The mod-$\ell$ Galois representation has maximal image $$\GSp(4,\F_\ell)$$ for all primes $$\ell$$ .

## Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $\mathrm{USp}(4)$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{USp}(4)$$

## Decomposition of the Jacobian

Simple over $$\overline{\Q}$$

magma: HeuristicDecompositionFactors(C);

## 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 $$\Q$$.

magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

magma: HeuristicIsGL2(C); HeuristicEndomorphismDescription(C); HeuristicEndomorphismFieldOfDefinition(C);

magma: HeuristicIsGL2(C : Geometric := true); HeuristicEndomorphismDescription(C : Geometric := true); HeuristicEndomorphismLatticeDescription(C);

The conductor $3319$ of the Jacobian of this curve is the smallest known for a genus $2$ curve with analytic rank $2$.