# Properties

 Label 676.a.5408.1 Conductor $676$ Discriminant $-5408$ Mordell-Weil group $$\Z/{21}\Z$$ Sato-Tate group $\mathrm{SU}(2)\times\mathrm{SU}(2)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q \times \Q$$ $$\End(J) \otimes \Q$$ $$\Q \times \Q$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Related objects

Show commands: Magma / SageMath

## Simplified equation

 $y^2 + (x^3 + x^2 + x)y = x^3 + 3x^2 + 3x + 1$ (homogenize, simplify) $y^2 + (x^3 + x^2z + xz^2)y = x^3z^3 + 3x^2z^4 + 3xz^5 + z^6$ (dehomogenize, simplify) $y^2 = x^6 + 2x^5 + 3x^4 + 6x^3 + 13x^2 + 12x + 4$ (homogenize, minimize)

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

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

sage: X = HyperellipticCurve(R([4, 12, 13, 6, 3, 2, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$676$$ $$=$$ $$2^{2} \cdot 13^{2}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-5408$$ $$=$$ $$- 2^{5} \cdot 13^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$204$$ $$=$$ $$2^{2} \cdot 3 \cdot 17$$ $$I_4$$ $$=$$ $$3273$$ $$=$$ $$3 \cdot 1091$$ $$I_6$$ $$=$$ $$161211$$ $$=$$ $$3 \cdot 17 \cdot 29 \cdot 109$$ $$I_{10}$$ $$=$$ $$692224$$ $$=$$ $$2^{12} \cdot 13^{2}$$ $$J_2$$ $$=$$ $$51$$ $$=$$ $$3 \cdot 17$$ $$J_4$$ $$=$$ $$-28$$ $$=$$ $$- 2^{2} \cdot 7$$ $$J_6$$ $$=$$ $$0$$ $$=$$ $$0$$ $$J_8$$ $$=$$ $$-196$$ $$=$$ $$- 2^{2} \cdot 7^{2}$$ $$J_{10}$$ $$=$$ $$5408$$ $$=$$ $$2^{5} \cdot 13^{2}$$ $$g_1$$ $$=$$ $$345025251/5408$$ $$g_2$$ $$=$$ $$-928557/1352$$ $$g_3$$ $$=$$ $$0$$

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

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

## Automorphism group

 $$\mathrm{Aut}(X)$$ $$\simeq$$ $C_2^2$ magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$$C_2^2$magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);

## Rational points

All points: $$(1 : 0 : 0),\, (1 : -1 : 0),\, (-1 : 0 : 1),\, (0 : -1 : 1),\, (0 : 1 : 1),\, (-1 : 1 : 1)$$
All points: $$(1 : 0 : 0),\, (1 : -1 : 0),\, (-1 : 0 : 1),\, (0 : -1 : 1),\, (0 : 1 : 1),\, (-1 : 1 : 1)$$
All points: $$(1 : -1 : 0),\, (1 : 1 : 0),\, (-1 : -1 : 1),\, (-1 : 1 : 1),\, (0 : -2 : 1),\, (0 : 2 : 1)$$

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

magma: [C![-1,-1,1],C![-1,1,1],C![0,-2,1],C![0,2,1],C![1,-1,0],C![1,1,0]]; // 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: $$\Z/{21}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(-1 : 1 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-2xz^2 - z^3$$ $$0$$ $$21$$
Generator $D_0$ Height Order
$$(-1 : 1 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-2xz^2 - z^3$$ $$0$$ $$21$$
Generator $D_0$ Height Order
$$(-1 : 1 : 1) + (0 : -2 : 1) - (1 : -1 : 0) - (1 : 1 : 0)$$ $$x (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3 + x^2z - 3xz^2 - 2z^3$$ $$0$$ $$21$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$0$$ Regulator: $$1$$ Real period: $$20.16977$$ Tamagawa product: $$7$$ Torsion order: $$21$$ Leading coefficient: $$0.320155$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$2$$ $$5$$ $$7$$ $$( 1 - T )( 1 + T )$$
$$13$$ $$2$$ $$2$$ $$1$$ $$( 1 - T )( 1 + T )$$

## Galois representations

For primes $\ell \ge 5$ the Galois representation data has not been computed for this curve since it is not generic.

For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not all of $\GSp(4,\F_\ell)$.

Prime $$\ell$$ mod-$$\ell$$ image Is torsion prime?
$$2$$ 2.60.2 no
$$3$$ 3.2160.21 yes

## Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $\mathrm{SU}(2)\times\mathrm{SU}(2)$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{SU}(2)\times\mathrm{SU}(2)$$

## Decomposition of the Jacobian

Splits over $$\Q$$

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
Elliptic curve isogeny class 26.a
Elliptic curve isogeny class 26.b

magma: HeuristicDecompositionFactors(C);

## 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 $$\Q$$.

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);