Magma V2.30-1
model: Hyperelliptic Curve defined by y^2 + (-x^3 - 1)*y = -2*x^5 + 2*x^4 - 5*x^3 + 2*x^2 - 2*x over Rational Field
CHECK model: #X(F_p) = p+1-tr T_p for 3 <= p < 100, p <> 13
CHECK w_2 is the moduli Atkin-Lehner involution (Phi_2 test at 5 quadratic points)
X_0(26)/w_2 is isomorphic to [ 1, 0, 1, -5, -8 ]
26a1
26a1: rank 0 (proved), torsion Z/3
rational points of the quotient: [ (0 : 1 : 0), (4 : -9 : 1), (4 : 4 : 1) ]
 images: [ (0 : 1 : 0), (4 : -9 : 1), (4 : 4 : 1) ]
H_{-52} = t^2 - 6896880000*t - 567663552000000
fibre over a rational point of X_0(26)/w_2 (reduced degree 2):
   (-1 : r1 : 1) over a field of degree 2: j has minimal polynomial Y^2 - 6896880000*Y - 567663552000000
   (-1 : -r1 : 1) over a field of degree 2: j has minimal polynomial Y^2 - 6896880000*Y - 567663552000000
fibre over a rational point of X_0(26)/w_2 (reduced degree 2):
   (0 : 1 : 1) : rational cusp
   (1 : 1 : 0) : rational cusp
fibre over a rational point of X_0(26)/w_2 (reduced degree 2):
   (0 : 0 : 1) : rational cusp
   (1 : 0 : 0) : rational cusp
CHECK fibres: 4 rational cusps and one conjugate pair over Q(sqrt 13) with j a root of H_{-52}
CHECK X_0(26)(Q) consists of the 4 rational cusps
ALL_CHECKS_PASSED