Magma V2.21-9 Fri Mar 11 2016 08:17:38 on lehner [Seed = 647087714]
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The modular curve X_0(20) is the elliptic curve Elliptic Curve defined by y^2 =
x^3 + x^2 + 4*x + 4 over Rational Field
with Cremona label 20a1
Points on X_0(20) over Q(zeta_{11})^+ are [ (0 : 1 : 0), (4 : 10 : 1), (0 : 2 :
1), (-1 : 0 : 1), (0 : -2 : 1), (4 : -10 : 1) ]
The cusps of X_0(20) are [
Place at (4 : 10 : 1),
Place at (4 : -10 : 1),
Place at (-1 : 0 : 1),
Place at (0 : -2 : 1),
Place at (0 : 2 : 1),
Place at (0 : 1 : 0)
]
Thus every Q(zeta_{11})^+ point on X_0(20) is a cusp.
The modular curve X_0(14) is the elliptic curve Elliptic Curve defined by y^2 +
x*y + y = x^3 + 4*x - 6 over Rational Field
with Cremona label 14a1
Points on X_0(14) over Q(zeta_{11})^+ are [ (0 : 1 : 0), (9 : 23 : 1), (2 : 2 :
1), (1 : -1 : 1), (2 : -5 : 1), (9 : -33 : 1) ]
The cusps of X_0(14) are [
Place at (9 : 23 : 1),
Place at (2 : -5 : 1),
Place at (1 : -1 : 1),
Place at (0 : 1 : 0)
]
The non-cuspidal points on X_0(14) over Q(zeta_{11})^+ are [ (2 : 2 : 1), (9 :
-33 : 1) ]
These have j-invariants [
16581375,
-3375
]
The modular curve X_0(19) is the elliptic curve Elliptic Curve defined by y^2 +
y = x^3 + x^2 - 9*x - 15 over Rational Field
with Cremona label 19a1
WARNING
Class group unconditional proof is expensive.
Hint: to obtain class group proven under GRH, input
> SetClassGroupBounds("GRH");
before starting the current calculation.
Points on X_0(19) over Q(zeta_{11})^+ are [ (0 : 1 : 0), (5 : -10 : 1), (5 : 9 :
1) ]
The cusps of X_0(19) are [
Place at (5 : 9 : 1),
Place at (0 : 1 : 0)
]
The non-cuspidal points on X_0(19) over Q(zeta_{11})^+ are [ (5 : -10 : 1) ]
These have j-invariants [
-884736
]
Total time: 1082.170 seconds, Total memory usage: 98.22MB