Magma V2.30-1
CHECK (0,0) has exact order 13 on the universal curve over the function field of C
j in Q(x): numerator degree 42, denominator [
<t, 13>,
<t + 1, 13>,
<t^3 + 4*t^2 + t - 1, 1>
]
CHECK D(t) = (t^3+t^2+1)^2 + 4(t^2+t) has no real root
CHECK j - 1728 = -D(t) g(t)^2 / den(j), deg g = 18
CHECK class group of Q[t]/(t^3+4t^2+t-1) (disc 169) is trivial (Proof := Full)
#Sel_2(J) = 4, #J(Q)[2] = 4
CHECK rank J(C1)(Q) = 0 (2-Selmer group equals the rational 2-torsion)
C1(Q) = {@ (-1 : 0 : 1), (0 : 0 : 1), (1 : 0 : 0) @}
CHECK C1(Q) consists of t = -1, 0, infinity: cusps of X_1(13) (poles of j)
t = -2: K = Q(sqrt 17), j = -60698457/40960, E0 = [ 1, -1, 0, -139, 965 ], isogeny class degree 13
ALL_CHECKS_PASSED