def _find_alpha(self, p, k, M=None, ap=None, new_base_ring=None, ordinary=True, check=True, find_extraprec=True):
"""
Finds `alpha`, a `U_p` eigenvalue, which is found as a root of
the polynomial `x^2 - ap * x + p^(k+1)`.
INPUT:
- ``p`` -- prime
- ``k`` -- Pollack-Stevens weight
- ``M`` -- precision (default = None) of `Q_p`
- ``ap`` -- Hecke eigenvalue at p (default = None)
- ``new_base_ring`` -- field of definition of `alpha` (default = None)
- ``ordinary`` -- True if the prime is ordinary (default = True)
- ``check`` -- check to see if the prime is ordinary (default = True)
- ``find_extraprec`` -- setting this to True finds extra precision (default = True)
OUTPUT:
- ``alpha`` -- `U_p` eigenvalue
- ``new_base_ring`` -- field of definition of `alpha` with precision at least `newM`
- ``newM`` -- new precision
- ``eisenloss`` -- loss of precision
- ``q`` -- a prime not equal to p which was used to find extra precision
- ``aq`` -- the Hecke eigenvalue `aq` corresponding to `q`
EXAMPLES::
sage: from sage.modular.pollack_stevens.space import ps_modsym_from_elliptic_curve
sage: E = EllipticCurve('11a')
sage: p = 5
sage: M = 10
sage: k = 0
sage: phi = ps_modsym_from_elliptic_curve(E)
sage: phi._find_alpha(p,k,M)
(1 + 4*5 + 3*5^2 + 2*5^3 + 4*5^4 + 4*5^5 + 4*5^6 + 3*5^7 + 2*5^8 + 3*5^9 + 3*5^10 + 3*5^12 + O(5^13), 5-adic Field with capped relative precision 13, 12, 1, None, None)
"""
if ap is None:
ap = self.Tq_eigenvalue(p, check=check)
if check and ap.valuation(p) > 0:
raise ValueError("p is not ordinary")
poly = PolynomialRing(ap.parent(), 'x')([p**(k+1), -ap, 1])
if new_base_ring is None:
# These should actually be completions of disc.parent()
if p == 2:
# is this the right precision adjustment for p=2?
new_base_ring = Qp(2, M+1)
else:
new_base_ring = Qp(p, M)
set_padicbase = True
else:
set_padicbase = False
try:
verbose("finding alpha: rooting %s in %s"%(poly, new_base_ring))
(v0,e0),(v1,e1) = poly.roots(new_base_ring)
except TypeError, ValueError:
raise ValueError("new base ring must contain a root of x^2 - ap * x + p^(k+1)")
def _find_alpha(self, p, k, M=None, ap=None, new_base_ring=None, ordinary=True, check=True, find_extraprec=True):
"""
"""
if ap is None:
ap = self.Tq_eigenvalue(p, check=check)
if check and ap.valuation(p) > 0:
raise ValueError("p is not ordinary")
poly = PolynomialRing(ap.parent(), 'x')([p**(k+1), -ap, 1])
if new_base_ring is None:
# These should actually be completions of disc.parent()
if p == 2:
# is this the right precision adjustment for p=2?
new_base_ring = Qp(2, M+1)
else:
new_base_ring = Qp(p, M)
set_padicbase = True
else:
set_padicbase = False
try:
verbose("finding alpha: rooting %s in %s"%(poly, new_base_ring))
(v0,e0),(v1,e1) = poly.roots(new_base_ring)
except TypeError, ValueError:
raise ValueError("new base ring must contain a root of x^2 - ap * x + p^(k+1)")
