def Phi_polys(L, x, j):
r"""
Return a certain polynomial of degree `L+1` in the
indeterminate x over a finite field.
The roots of the **modular** polynomial `\Phi(L, x, j)` are the
`L`-isogenous supersingular j-invariants of j.
INPUT:
- ``L`` -- integer
- ``x`` -- indeterminate of a univariate polynomial ring defined over a
finite field with p^2 elements, where p is a prime number
- ``j`` -- supersingular j-invariant over the finite field
OUTPUT:
- polynomial -- defined over the finite field
EXAMPLES:
The following code snippet produces the modular polynomial
`\Phi_{L}(x,j_{in})`, where `j_{in}` is a supersingular j-invariant
defined over the finite field with `7^2` elements::
sage: F = GF(7^2, 'a')
sage: X = PolynomialRing(F, 'x').gen()
sage: j_in = supersingular_j(F)
sage: sage.modular.ssmod.ssmod.Phi_polys(2,X,j_in)
x^3 + 3*x^2 + 3*x + 1
sage: sage.modular.ssmod.ssmod.Phi_polys(3,X,j_in)
x^4 + 4*x^3 + 6*x^2 + 4*x + 1
sage: sage.modular.ssmod.ssmod.Phi_polys(5,X,j_in)
x^6 + 6*x^5 + x^4 + 6*x^3 + x^2 + 6*x + 1
sage: sage.modular.ssmod.ssmod.Phi_polys(7,X,j_in)
x^8 + x^7 + x + 1
sage: sage.modular.ssmod.ssmod.Phi_polys(11,X,j_in)
x^12 + 5*x^11 + 3*x^10 + 3*x^9 + 5*x^8 + x^7 + x^5 + 5*x^4 + 3*x^3 + 3*x^2 + 5*x + 1
sage: sage.modular.ssmod.ssmod.Phi_polys(13,X,j_in)
x^14 + 2*x^7 + 1
"""
r = 0
for pol in pari.polmodular(L).Vec():
r = r * x + ZZy(pol)(j)
return r
def Phi_polys(L, x, j):
r"""
This function returns a certain polynomial of degree `L+1` in the
indeterminate x over a finite field.
The roots of the **modular** polynomial `\Phi(L, x, j)` are the
`L`-isogenous supersingular j-invariants of j.
INPUT:
- ``L`` -- integer
- ``x`` -- indeterminate of a univariate polynomial ring defined over a
finite field with p^2 elements, where p is a prime number
- ``j`` -- supersingular j-invariant over the finite field
OUTPUT:
- polynomial -- defined over the finite field
EXAMPLES:
The following code snippet produces the modular polynomial
`\Phi_{L}(x,j_{in})`, where `j_{in}` is a supersingular j-invariant
defined over the finite field with `7^2` elements::
sage: F = GF(7^2, 'a')
sage: X = PolynomialRing(F, 'x').gen()
sage: j_in = supersingular_j(F)
sage: sage.modular.ssmod.ssmod.Phi_polys(2,X,j_in)
x^3 + 3*x^2 + 3*x + 1
sage: sage.modular.ssmod.ssmod.Phi_polys(3,X,j_in)
x^4 + 4*x^3 + 6*x^2 + 4*x + 1
sage: sage.modular.ssmod.ssmod.Phi_polys(5,X,j_in)
x^6 + 6*x^5 + x^4 + 6*x^3 + x^2 + 6*x + 1
sage: sage.modular.ssmod.ssmod.Phi_polys(7,X,j_in)
x^8 + x^7 + x + 1
sage: sage.modular.ssmod.ssmod.Phi_polys(11,X,j_in)
x^12 + 5*x^11 + 3*x^10 + 3*x^9 + 5*x^8 + x^7 + x^5 + 5*x^4 + 3*x^3 + 3*x^2 + 5*x + 1
sage: sage.modular.ssmod.ssmod.Phi_polys(13,X,j_in)
x^14 + 2*x^7 + 1
"""
r = 0
for pol in pari.polmodular(L).Vec():
r = r*x + ZZy(pol)(j)
return r