# Properties

 Label 676.a.562432.1 Conductor $676$ Discriminant $562432$ 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

This is a model for the modular curve $X_0(26)$.

## Simplified equation

 $y^2 + (x^3 + 1)y = 2x^5 + 2x^4 + 4x^3 + 2x^2 + 2x$ (homogenize, simplify) $y^2 + (x^3 + z^3)y = 2x^5z + 2x^4z^2 + 4x^3z^3 + 2x^2z^4 + 2xz^5$ (dehomogenize, simplify) $y^2 = x^6 + 8x^5 + 8x^4 + 18x^3 + 8x^2 + 8x + 1$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 2, 2, 4, 2, 2]), R([1, 0, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 2, 2, 4, 2, 2], R![1, 0, 0, 1]);

sage: X = HyperellipticCurve(R([1, 8, 8, 18, 8, 8, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$676$$ $$=$$ $$2^{2} \cdot 13^{2}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$562432$$ $$=$$ $$2^{8} \cdot 13^{3}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$1620$$ $$=$$ $$2^{2} \cdot 3^{4} \cdot 5$$ $$I_4$$ $$=$$ $$52953$$ $$=$$ $$3 \cdot 19 \cdot 929$$ $$I_6$$ $$=$$ $$29527389$$ $$=$$ $$3^{3} \cdot 251 \cdot 4357$$ $$I_{10}$$ $$=$$ $$71991296$$ $$=$$ $$2^{15} \cdot 13^{3}$$ $$J_2$$ $$=$$ $$405$$ $$=$$ $$3^{4} \cdot 5$$ $$J_4$$ $$=$$ $$4628$$ $$=$$ $$2^{2} \cdot 13 \cdot 89$$ $$J_6$$ $$=$$ $$-8112$$ $$=$$ $$- 2^{4} \cdot 3 \cdot 13^{2}$$ $$J_8$$ $$=$$ $$-6175936$$ $$=$$ $$- 2^{6} \cdot 13^{2} \cdot 571$$ $$J_{10}$$ $$=$$ $$562432$$ $$=$$ $$2^{8} \cdot 13^{3}$$ $$g_1$$ $$=$$ $$10896201253125/562432$$ $$g_2$$ $$=$$ $$5912281125/10816$$ $$g_3$$ $$=$$ $$-492075/208$$

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

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

magma: [C![0,-1,1],C![0,1,1],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
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$2x^2 + xz + 3z^2$$ $$=$$ $$0,$$ $$4y$$ $$=$$ $$xz^2 - 3z^3$$ $$0$$ $$21$$
Generator $D_0$ Height Order
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$2x^2 + xz + 3z^2$$ $$=$$ $$0,$$ $$4y$$ $$=$$ $$xz^2 - 3z^3$$ $$0$$ $$21$$
Generator $D_0$ Height Order
$$D_0 - (1 : -1 : 0) - (1 : 1 : 0)$$ $$2x^2 + xz + 3z^2$$ $$=$$ $$0,$$ $$4y$$ $$=$$ $$x^3 + 2xz^2 - 5z^3$$ $$0$$ $$21$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$0$$ Regulator: $$1$$ Real period: $$6.723259$$ Tamagawa product: $$21$$ 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
$$2$$ $$2$$ $$8$$ $$7$$ $$( 1 - T )( 1 + T )$$
$$13$$ $$2$$ $$3$$ $$3$$ $$( 1 - T )( 1 + T )$$

## 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
$$3$$ 3.2160.20 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);