def native_two_isogeny_descent_work(E, two_tor_rk):
"""
Prepares the output from two-descent by two-isogeny.
INPUT:
- ``E`` - an elliptic curve
- ``two_tor_rk`` - its two-torsion rank
OUTPUT:
- a lower bound on the rank
- an upper bound on the rank
- a lower bound on the rank of Sha[2]
- an upper bound on the rank of Sha[2]
- a list of the generators found (currently None, since we don't store them)
EXAMPLES::
sage: from sage.schemes.elliptic_curves.BSD import native_two_isogeny_descent_work
sage: E = EllipticCurve('14a')
sage: native_two_isogeny_descent_work(E, E.two_torsion_rank())
(0, 0, 0, 0, None)
sage: E = EllipticCurve('65a')
sage: native_two_isogeny_descent_work(E, E.two_torsion_rank())
(1, 1, 0, 0, None)
"""
from sage.schemes.elliptic_curves.descent_two_isogeny import two_descent_by_two_isogeny
n1, n2, n1p, n2p = two_descent_by_two_isogeny(E)
# bring n1 and n1p up to the nearest power of two
two = ZZ(2) # otherwise "log" is symbolic >.<
e1 = ceil(ZZ(n1).log(two))
e1p = ceil(ZZ(n1p).log(two))
e2 = ZZ(n2).log(two)
e2p = ZZ(n2p).log(two)
rank_lower_bd = e1 + e1p - 2
rank_upper_bd = e2 + e2p - 2
sha_upper_bd = e2 + e2p - e1 - e1p
gens = None # right now, we are not keeping track of them
return rank_lower_bd, rank_upper_bd, 0, sha_upper_bd, gens
def native_two_isogeny_descent_work(E, two_tor_rk):
"""
Prepares the output from two-descent by two-isogeny.
INPUT:
- ``E`` - an elliptic curve
- ``two_tor_rk`` - its two-torsion rank
OUTPUT:
- a lower bound on the rank
- an upper bound on the rank
- a lower bound on the rank of Sha[2]
- an upper bound on the rank of Sha[2]
- a list of the generators found (currently None, since we don't store them)
EXAMPLES::
sage: from sage.schemes.elliptic_curves.BSD import native_two_isogeny_descent_work
sage: E = EllipticCurve('14a')
sage: native_two_isogeny_descent_work(E, E.two_torsion_rank())
(0, 0, 0, 0, None)
sage: E = EllipticCurve('65a')
sage: native_two_isogeny_descent_work(E, E.two_torsion_rank())
(1, 1, 0, 0, None)
"""
from sage.schemes.elliptic_curves.descent_two_isogeny import two_descent_by_two_isogeny
n1, n2, n1p, n2p = two_descent_by_two_isogeny(E)
# bring n1 and n1p up to the nearest power of two
two = ZZ(2) # otherwise "log" is symbolic >.<
e1 = ceil(ZZ(n1).log(two))
e1p = ceil(ZZ(n1p).log(two))
e2 = ZZ(n2).log(two)
e2p = ZZ(n2p).log(two)
rank_lower_bd = e1 + e1p - 2
rank_upper_bd = e2 + e2p - 2
sha_upper_bd = e2 + e2p - e1 - e1p
gens = None # right now, we are not keeping track of them
return rank_lower_bd, rank_upper_bd, 0, sha_upper_bd, gens
