CoCalc -- 1430.sagews

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def ker(E, ell1, ell2):
    P, Q, R = E.gens()
    E1 = E.change_ring(GF(ell1))
    E2 = E.change_ring(GF(ell2))
    K = []
    for a, b, c in GF(3)^3:
        S = ZZ(a)*P + ZZ(b)*Q + ZZ(c)*R
        S1 = E1(S)
        if len(S1.division_points(3)) > 0:
            S2 = E2(S)
            if len(S2.division_points(3)) > 0:
                K.append(S)
    return K

E = EllipticCurve('5077a')

ker(E,3, 23, 29)

[(0 : 1 : 0), (7/9 : -44/27 : 1), (1/4 : -21/8 : 1)]

P = ker(E,3, 23, 29)[-1]; P

(1/4 : -21/8 : 1)

E.change_ring(GF(11))(P).division_points(3)

[]

def red_l(E, p, l, li):
    set = ker(E,p, li[0], li[1])
    if len(set) == p:
        P = set[-1]
        return E.change_ring(GF(l))(P).division_points(p)

red_l(E,3,11,[431,107])

[(3 : 3 : 1), (8 : 0 : 1), (10 : 7 : 1)]

K.<a> = QuadraticField(-163)
L = []
for p in prime_range(12,500):
    if len(K.factor(p))==1:
        if gcd(E.ap(p),p+1)%3 == 0:
            L.append(p)

Lpairs = []
print L
for i in range(len(L)):
    for j in range(i+1,len(L)):
        Lpairs.append([L[i],L[j]])

[23, 29, 107, 137, 149, 233, 239, 257, 293, 389, 401, 431]

Lpairs

[[23, 29], [23, 107], [23, 137], [23, 149], [23, 233], [23, 239], [23, 257], [23, 293], [23, 389], [23, 401], [23, 431], [29, 107], [29, 137], [29, 149], [29, 233], [29, 239], [29, 257], [29, 293], [29, 389], [29, 401], [29, 431], [107, 137], [107, 149], [107, 233], [107, 239], [107, 257], [107, 293], [107, 389], [107, 401], [107, 431], [137, 149], [137, 233], [137, 239], [137, 257], [137, 293], [137, 389], [137, 401], [137, 431], [149, 233], [149, 239], [149, 257], [149, 293], [149, 389], [149, 401], [149, 431], [233, 239], [233, 257], [233, 293], [233, 389], [233, 401], [233, 431], [239, 257], [239, 293], [239, 389], [239, 401], [239, 431], [257, 293], [257, 389], [257, 401], [257, 431], [293, 389], [293, 401], [293, 431], [389, 401], [389, 431], [401, 431]]

for ells in Lpairs:
    print ells, red_l(E,3,11,ells)

[23, 29] []
[23, 107] []
[23, 137] []
[23, 149] []
[23, 233] None
[23, 239] []
[23, 257] None
[23, 293] []
[23, 389] []
[23, 401] []
[23, 431] []
[29, 107] []
[29, 137] None
[29, 149] []
[29, 233] []
[29, 239] []
[29, 257] []
[29, 293] []
[29, 389] None
[29, 401] []
[29, 431] []
[107, 137] []
[107, 149] []
[107, 233] []
[107, 239] None
[107, 257] []
[107, 293] []
[107, 389] []
[107, 401] None
[107, 431] [(3 : 3 : 1), (8 : 0 : 1), (10 : 7 : 1)]
[137, 149] []
[137, 233] []
[137, 239] []
[137, 257] []
[137, 293] []
[137, 389] None
[137, 401] []
[137, 431] []
[149, 233] []
[149, 239] []
[149, 257] []
[149, 293] [(0 : 1 : 0), (9 : 3 : 1), (9 : 7 : 1)]
[149, 389] []
[149, 401] []
[149, 431] []
[233, 239] []
[233, 257] None
[233, 293] []
[233, 389] []
[233, 401] []
[233, 431] []
[239, 257] []
[239, 293] []
[239, 389] []
[239, 401] None
[239, 431] [(3 : 3 : 1), (8 : 0 : 1), (10 : 7 : 1)]
[257, 293] []
[257, 389] []
[257, 401] []
[257, 431] []
[293, 389] []
[293, 401] []
[293, 431] []
[389, 401] []
[389, 431] []
[401, 431] [(3 : 3 : 1), (8 : 0 : 1), (10 : 7 : 1)]

def ker(E,p, li):
    """
    computes kernel of the reduction map for rank 3, checks if any are p-divisible
    """
    P, Q, R = E.gens()
    E1 = E.change_ring(GF(li[0]))
    E2 = E.change_ring(GF(li[1]))
    K = []
    for a, b, c in GF(p)**3:
        S = ZZ(a)*P + ZZ(b)*Q + ZZ(c)*R
        S1 = E1(S)
        if len(S1.division_points(p)) > 0:
            S2 = E2(S)
            if len(S2.division_points(p)) > 0:
                K.append(S)
    return K
    
def red_l(E, p, l, li):
    set = ker(E,p, li)
    if len(set) == p:
        P = set[-1]
        return E.change_ring(GF(l))(P).division_points(p)
    else:
        return []

def ell_list(D, p, l, bound):
    """
    computes list of ell_i given D, p, l, and a bound on the ell_i
    """        
    K.<a> = QuadraticField(D)
    L = []
    for q in prime_range(l+1,bound):
        if len(K.factor(q))==1:
            if (gcd(E.ap(q),q+1))%(p) == 0:
                L.append(q)
    return L
                
def ell_pairs(L):
    """
    given list L, returns list of pairs (distinct elements) of L without repetition
    """            
    Lpairs = []
    for i in range(len(L)):
        for j in range(i+1,len(L)):
            Lpairs.append([L[i],L[j]])
    return Lpairs

def tau(E,D, p, l, bound):
    """
    return list of tau given E,D p, l, and a bound on the ell_i
    """
    Lpairs = ell_pairs(ell_list(D,p,l,bound))
    for ells in Lpairs:
        print ells, red_l(E,p,l,ells)

E = EllipticCurve('11642a')
p = 3
l = 5
D = -7
tau(E,D,p,l,300)

[17, 47] []
[17, 59] []
[17, 101] [(0 : 1 : 0), (1 : 1 : 1), (1 : 3 : 1)]
[17, 167] [(0 : 1 : 0), (1 : 1 : 1), (1 : 3 : 1)]
[17, 227] []
[17, 251] []
[17, 269] []
[17, 293] []
[47, 59] []
[47, 101] []
[47, 167] []
[47, 227] []
[47, 251] []
[47, 269] []
[47, 293] []
[59, 101] []
[59, 167] []
[59, 227] []
[59, 251] []
[59, 269] []
[59, 293] [(2 : 0 : 1), (3 : 3 : 1), (4 : 2 : 1)]
[101, 167] [(0 : 1 : 0), (1 : 1 : 1), (1 : 3 : 1)]
[101, 227] []
[101, 251] []
[101, 269] []
[101, 293] []
[167, 227] []
[167, 251] []
[167, 269] []
[167, 293] []
[227, 251] []
[227, 269] []
[227, 293] []
[251, 269] []
[251, 293] []
[269, 293] [(2 : 0 : 1), (3 : 3 : 1), (4 : 2 : 1)]

E = EllipticCurve('11197a')
p = 3
l = 29
D = -19
tau(E,D,p,l,290)

[89, 107] []
[89, 173] []
[89, 257] []
[89, 281] []
[107, 173] []
[107, 257] [(1 : 15 : 1), (2 : 9 : 1), (8 : 4 : 1)]
[107, 281] [(1 : 15 : 1), (2 : 9 : 1), (8 : 4 : 1)]
[173, 257] []
[173, 281] []
[257, 281] []

E = EllipticCurve('11197a')
p = 3
l = 29
D = -43
tau(E,D,p,l,290)

[89, 137] []
[89, 257] []
[89, 263] [(0 : 28 : 1), (11 : 1 : 1), (28 : 2 : 1)]
[137, 257] [(1 : 15 : 1), (2 : 9 : 1), (8 : 4 : 1)]
[137, 263] []
[257, 263] []

E = EllipticCurve('11197a')
p = 3
l = 29
D = -163
tau(E,D,p,l,290)

[89, 107] []
[89, 137] []
[89, 239] []
[89, 257] []
[107, 137] [(1 : 15 : 1), (2 : 9 : 1), (8 : 4 : 1)]
[107, 239] []
[107, 257] [(1 : 15 : 1), (2 : 9 : 1), (8 : 4 : 1)]
[137, 239] []
[137, 257] [(1 : 15 : 1), (2 : 9 : 1), (8 : 4 : 1)]
[239, 257] []

E = EllipticCurve('11197a')
p = 7
l = 13
D = -7
tau(E,D,p,l,400)

[139, 251] []
[139, 349] []
[251, 349] []