== E1  ordinarity of y^2 = x^3 - x
   p = 1 mod 4 (ordinary, Hasse != 0, matches binomial formula): 37 primes; p = 3 mod 4 control (supersingular): 39 primes; OK: True
== E2  [i] automorphism, Frobenius as a Gaussian integer, point counts
   p=5: a_p=-2, pi=(-1, -2) (1 of 8 norm-p candidates match Frobenius on E(F_p^2)); n=1:#E=8=N(pi^n-1):True,[i]ok:True; n=2:#E=32=N(pi^n-1):True,[i]ok:True; n=3:#E=104=N(pi^n-1):True,[i]ok:True
   p=13: a_p=6, pi=(3, 2) (1 of 8 norm-p candidates match Frobenius on E(F_p^2)); n=1:#E=8=N(pi^n-1):True,[i]ok:True; n=2:#E=160=N(pi^n-1):True,[i]ok:True; n=3:#E=2216=N(pi^n-1):True,[i]ok:True
   p=17: a_p=2, pi=(1, 4) (1 of 8 norm-p candidates match Frobenius on E(F_p^2)); n=1:#E=16=N(pi^n-1):True,[i]ok:True; n=2:#E=320=N(pi^n-1):True,[i]ok:True; n=3:#E=5008=N(pi^n-1):True,[i]ok:True
   p=29: a_p=-10, pi=(-5, -2) (1 of 8 norm-p candidates match Frobenius on E(F_p^2)); n=1:#E=40=N(pi^n-1):True,[i]ok:True; n=2:#E=800=N(pi^n-1):True,[i]ok:True
   p=37: a_p=-2, pi=(-1, -6) (1 of 8 norm-p candidates match Frobenius on E(F_p^2)); n=1:#E=40=N(pi^n-1):True,[i]ok:True; n=2:#E=1440=N(pi^n-1):True,[i]ok:True
   p=41: a_p=10, pi=(5, 4) (1 of 8 norm-p candidates match Frobenius on E(F_p^2)); n=1:#E=32=N(pi^n-1):True,[i]ok:True; n=2:#E=1664=N(pi^n-1):True,[i]ok:True
== E3  cyclic Z[i]-module structure;  E4  M(a) = pi(a) with M from Lemma 2.1
   q=    5 (p=5,n=1): #E=8=N(mu) with mu=pi^n-1=(-2, -2); cyclic: True (G=(3, 2)); Frob=pi on E(F_q): True; all 63 of 63 nonzero points of A checked, failures 0 (lemma 0); 0.0s
   q=   25 (p=5,n=2): #E=32=N(mu) with mu=pi^n-1=(-4, 4); cyclic: True (G=(21, 2)); Frob=pi on E(F_q): True; all 1023 of 1023 nonzero points of A checked, failures 0 (lemma 0); 0.2s
   q=  125 (p=5,n=3): #E=104=N(mu) with mu=pi^n-1=(10, 2); cyclic: True (G=(66, 101)); Frob=pi on E(F_q): True; all 10815 of 10815 nonzero points of A checked, failures 0 (lemma 0); 6.1s
   q=  625 (p=5,n=4): #E=640=N(mu) with mu=pi^n-1=(-8, -24); cyclic: True (G=(138, 278)); Frob=pi on E(F_q): True; sample 6639 of 409599 nonzero points of A checked, failures 0 (lemma 0); 6.2s
   q=   13 (p=13,n=1): #E=8=N(mu) with mu=pi^n-1=(2, 2); cyclic: True (G=(8, 6)); Frob=pi on E(F_q): True; all 63 of 63 nonzero points of A checked, failures 0 (lemma 0); 0.0s
   q=  169 (p=13,n=2): #E=160=N(mu) with mu=pi^n-1=(4, 12); cyclic: True (G=(40, 114)); Frob=pi on E(F_q): True; all 25599 of 25599 nonzero points of A checked, failures 0 (lemma 0); 11.4s
   q= 2197 (p=13,n=3): #E=2216=N(mu) with mu=pi^n-1=(-10, 46); cyclic: True (G=(1219, 2078)); Frob=pi on E(F_q): True; sample 8215 of 4910655 nonzero points of A checked, failures 0 (lemma 0); 8.8s
   q=   17 (p=17,n=1): #E=16=N(mu) with mu=pi^n-1=(0, 4); cyclic: True (G=(12, 4)); Frob=pi on E(F_q): True; all 255 of 255 nonzero points of A checked, failures 0 (lemma 0); 0.0s
   q=  289 (p=17,n=2): #E=320=N(mu) with mu=pi^n-1=(-16, 8); cyclic: True (G=(159, 1)); Frob=pi on E(F_q): True; all 102399 of 102399 nonzero points of A checked, failures 0 (lemma 0); 53.9s
   q= 4913 (p=17,n=3): #E=5008=N(mu) with mu=pi^n-1=(-48, -52); cyclic: True (G=(2454, 4687)); Frob=pi on E(F_q): True; sample 11007 of 25080063 nonzero points of A checked, failures 0 (lemma 0); 9.6s
   q=   29 (p=29,n=1): #E=40=N(mu) with mu=pi^n-1=(-6, -2); cyclic: True (G=(2, 21)); Frob=pi on E(F_q): True; all 1599 of 1599 nonzero points of A checked, failures 0 (lemma 0); 0.2s
   q=  841 (p=29,n=2): #E=800=N(mu) with mu=pi^n-1=(20, 20); cyclic: True (G=(620, 357)); Frob=pi on E(F_q): True; sample 6799 of 639999 nonzero points of A checked, failures 0 (lemma 0); 4.5s
   q=   37 (p=37,n=1): #E=40=N(mu) with mu=pi^n-1=(-2, -6); cyclic: True (G=(5, 34)); Frob=pi on E(F_q): True; all 1599 of 1599 nonzero points of A checked, failures 0 (lemma 0); 0.3s
   q= 1369 (p=37,n=2): #E=1440=N(mu) with mu=pi^n-1=(-36, 12); cyclic: True (G=(1139, 1066)); Frob=pi on E(F_q): True; sample 7439 of 2073599 nonzero points of A checked, failures 0 (lemma 0); 5.8s
   q=   41 (p=41,n=1): #E=32=N(mu) with mu=pi^n-1=(4, 4); cyclic: True (G=(34, 22)); Frob=pi on E(F_q): True; all 1023 of 1023 nonzero points of A checked, failures 0 (lemma 0); 0.1s
   q= 1681 (p=41,n=2): #E=1664=N(mu) with mu=pi^n-1=(8, 40); cyclic: True (G=(1526, 858)); Frob=pi on E(F_q): True; sample 7663 of 2768895 nonzero points of A checked, failures 0 (lemma 0); 4.7s
   q= 2809 (p=53,n=2): #E=2720=N(mu) with mu=pi^n-1=(44, 28); cyclic: True (G=(2458, 1462)); Frob=pi on E(F_q): True; sample 8719 of 7398399 nonzero points of A checked, failures 0 (lemma 0); 6.6s
   q= 3721 (p=61,n=2): #E=3744=N(mu) with mu=pi^n-1=(-12, -60); cyclic: True (G=(3199, 3085)); Frob=pi on E(F_q): True; sample 9743 of 14017535 nonzero points of A checked, failures 0 (lemma 0); 7.4s
   total nonzero points of A(F_q) over these fields: 57443182; checked with EC arithmetic: 210662 (incl. 19007 points of the curve C); OK: True
== E5  log-free orbits of SL_2(Z[i]) and GL_2(Z[i]) on E(F_q)^2 (BFS on points)
   q=   5: #E=8, orbits SL2=4 GL2=4, ideals containing mu=4; pi(a) outside orbit of a: 0/64; negative control (1+i)a separated from a for 63 points; 0.0s
   q=  25: #E=32, orbits SL2=6 GL2=6, ideals containing mu=6; pi(a) outside orbit of a: 0/1024; negative control (1+i)a separated from a for 1023 points; 0.1s
   q= 125: #E=104, orbits SL2=8 GL2=8, ideals containing mu=8; pi(a) outside orbit of a: 0/10816; negative control (1+i)a separated from a for 10647 points; 0.8s
   q=  13: #E=8, orbits SL2=4 GL2=4, ideals containing mu=4; pi(a) outside orbit of a: 0/64; negative control (1+i)a separated from a for 63 points; 0.0s
   q= 169: #E=160, orbits SL2=12 GL2=12, ideals containing mu=12; pi(a) outside orbit of a: 0/25600; negative control (1+i)a separated from a for 25575 points; 1.6s
   q=  17: #E=16, orbits SL2=5 GL2=5, ideals containing mu=5; pi(a) outside orbit of a: 0/256; negative control (1+i)a separated from a for 255 points; 0.0s
   q= 289: #E=320, orbits SL2=14 GL2=14, ideals containing mu=14; pi(a) outside orbit of a: 0/102400; negative control (1+i)a separated from a for 102375 points; 6.7s
   q=  29: #E=40, orbits SL2=8 GL2=8, ideals containing mu=8; pi(a) outside orbit of a: 0/1600; negative control (1+i)a separated from a for 1575 points; 0.1s
   q=  37: #E=40, orbits SL2=8 GL2=8, ideals containing mu=8; pi(a) outside orbit of a: 0/1600; negative control (1+i)a separated from a for 1575 points; 0.0s
   q=  41: #E=32, orbits SL2=6 GL2=6, ideals containing mu=6; pi(a) outside orbit of a: 0/1024; negative control (1+i)a separated from a for 1023 points; 0.0s
== E6  Remark 5.5(1): a fixed alpha serves finitely many points
   288 matrices alpha in GL_2(Z[i]) (entries in {0,+-1,+-i}) x 4 fields; max(count - N(det(alpha - pi I))) = 0; OK: True
== SUMMARY {'E1': True, 'E2': True, 'E3_E4': True, 'E5': True, 'E6': True} total 143.8s
