# Properties

 Label 196.a.21952.1 Conductor $196$ Discriminant $-21952$ Mordell-Weil group $$\Z/{6}\Z \oplus \Z/{6}\Z$$ Sato-Tate group $E_1$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\mathrm{M}_2(\R)$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathrm{M}_2(\Q)$$ $$\End(J) \otimes \Q$$ $$\mathrm{M}_2(\Q)$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type no

# Related objects

Show commands: Magma / SageMath

This is a model for the modular curve $X_0(28)$.

## Simplified equation

 $y^2 + (x^2 + x)y = x^6 + 3x^5 + 6x^4 + 7x^3 + 6x^2 + 3x + 1$ (homogenize, simplify) $y^2 + (x^2z + xz^2)y = x^6 + 3x^5z + 6x^4z^2 + 7x^3z^3 + 6x^2z^4 + 3xz^5 + z^6$ (dehomogenize, simplify) $y^2 = 4x^6 + 12x^5 + 25x^4 + 30x^3 + 25x^2 + 12x + 4$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, 3, 6, 7, 6, 3, 1]), R([0, 1, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![1, 3, 6, 7, 6, 3, 1], R![0, 1, 1]);

sage: X = HyperellipticCurve(R([4, 12, 25, 30, 25, 12, 4]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$196$$ $$=$$ $$2^{2} \cdot 7^{2}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-21952$$ $$=$$ $$- 2^{6} \cdot 7^{3}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$1340$$ $$=$$ $$2^{2} \cdot 5 \cdot 67$$ $$I_4$$ $$=$$ $$1345$$ $$=$$ $$5 \cdot 269$$ $$I_6$$ $$=$$ $$149855$$ $$=$$ $$5 \cdot 17 \cdot 41 \cdot 43$$ $$I_{10}$$ $$=$$ $$2809856$$ $$=$$ $$2^{13} \cdot 7^{3}$$ $$J_2$$ $$=$$ $$335$$ $$=$$ $$5 \cdot 67$$ $$J_4$$ $$=$$ $$4620$$ $$=$$ $$2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11$$ $$J_6$$ $$=$$ $$90160$$ $$=$$ $$2^{4} \cdot 5 \cdot 7^{2} \cdot 23$$ $$J_8$$ $$=$$ $$2214800$$ $$=$$ $$2^{4} \cdot 5^{2} \cdot 7^{2} \cdot 113$$ $$J_{10}$$ $$=$$ $$21952$$ $$=$$ $$2^{6} \cdot 7^{3}$$ $$g_1$$ $$=$$ $$4219140959375/21952$$ $$g_2$$ $$=$$ $$6203236875/784$$ $$g_3$$ $$=$$ $$12905875/28$$

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

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

## Automorphism group

 $$\mathrm{Aut}(X)$$ $$\simeq$$ $D_6$ magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$$D_6$magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);

## Rational points

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

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

magma: [C![-1,-2,1],C![-1,2,1],C![0,-2,1],C![0,2,1],C![1,-2,0],C![1,2,0]]; // 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: $$\Z/{6}\Z \oplus \Z/{6}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(-1 : 1 : 1) - (1 : 1 : 0)$$ $$z (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3$$ $$0$$ $$6$$
$$D_0 - (1 : -1 : 0) - (1 : 1 : 0)$$ $$3x^2 + 5xz + 4z^2$$ $$=$$ $$0,$$ $$9y$$ $$=$$ $$8xz^2 + 13z^3$$ $$0$$ $$6$$
Generator $D_0$ Height Order
$$(-1 : 1 : 1) - (1 : 1 : 0)$$ $$z (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3$$ $$0$$ $$6$$
$$D_0 - (1 : -1 : 0) - (1 : 1 : 0)$$ $$3x^2 + 5xz + 4z^2$$ $$=$$ $$0,$$ $$9y$$ $$=$$ $$8xz^2 + 13z^3$$ $$0$$ $$6$$
Generator $D_0$ Height Order
$$(-1 : 2 : 1) - (1 : 2 : 0)$$ $$z (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-2x^3 + x^2z + xz^2$$ $$0$$ $$6$$
$$D_0 - (1 : -2 : 0) - (1 : 2 : 0)$$ $$3x^2 + 5xz + 4z^2$$ $$=$$ $$0,$$ $$9y$$ $$=$$ $$x^2z + 17xz^2 + 26z^3$$ $$0$$ $$6$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$2$$ Regulator: $$1$$ Real period: $$11.77714$$ Tamagawa product: $$12$$ Torsion order: $$36$$ Leading coefficient: $$0.109047$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$2$$ $$6$$ $$4$$ $$( 1 + T )^{2}$$
$$7$$ $$2$$ $$3$$ $$3$$ $$( 1 - T )^{2}$$

## 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?
$$2$$ 2.360.3 yes
$$3$$ 3.17280.1 yes

## Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $E_1$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{SU}(2)$$

## Decomposition of the Jacobian

Splits over $$\Q$$

Decomposes up to isogeny as the square of the elliptic curve isogeny class:
Elliptic curve isogeny class 14.a

magma: HeuristicDecompositionFactors(C);

## Endomorphisms of the Jacobian

Not of $$\GL_2$$-type over $$\Q$$

Endomorphism ring over $$\Q$$:

 $$\End (J_{})$$ $$\simeq$$ an Eichler order of index $$3$$ in a maximal order of $$\End (J_{}) \otimes \Q$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\mathrm{M}_2($$$$\Q$$$$)$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\mathrm{M}_2 (\R)$$

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);