Magma V2.21-9 Fri Mar 11 2016 10:34:46 on atkin [Seed = 531199494]
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We will be doing computations with certain modular curves over the field Number
Field with defining polynomial $.1^3 - 13*$.1^2 + 26*$.1 - 13 over the Rational
Field
This is the unique cubic subfield of Q(zeta_{13})
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 K^prime 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 K^prime are [ (2 : 2 : 1), (9 : -33 : 1)
]
These have j-invariants [
16581375,
-3375
]
The modular curve X_0(11) is the elliptic curve Elliptic Curve defined by y^2 +
y = x^3 - x^2 - 10*x - 20 over Rational Field
with Cremona label 11a1
Points on X_0(11) over K^prime are [ (0 : 1 : 0), (5 : -6 : 1), (16 : 60 : 1),
(16 : -61 : 1), (5 : 5 : 1) ]
The cusps of X_0(11) are [
Place at (16 : 60 : 1),
Place at (0 : 1 : 0)
]
The non-cuspidal points on X_0(11) over K^prime are [ (5 : -6 : 1), (16 : -61 :
1), (5 : 5 : 1) ]
These have j-invariants [
-32768,
-121,
-24729001
]
Now we check the claim made in the proof that
every point on X_0(20) over Q(zeta_{13})^+
is in fact defined over Q(sqrt{13})
The modular curve X=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 K^prime are [ (0 : 1 : 0), (4 : 10 : 1), (0 : 2 : 1), (-1
: 0 : 1), (0 : -2 : 1), (4 : -10 : 1) ]
The quadratic twist of E by 13 is Elliptic Curve defined by y^2 = x^3 + 13*x^2 +
676*x + 8788 over Kp
The Mordell--Weil group of this twist over K^prime is
Abelian Group isomorphic to Z/2 + Z
Defined on 2 generators
Relations:
2*A.1 = 0
The generators are (-13 : 0 : 1) (39 : -338 : 1)
Thus if P belongs to X(K) then P+P^sigma and P-P^sigma
both belong to X(Q(sqrt{13}))
Thus 2P belongs to X(Q(sqrt{13}})
As the extension K/Q(sqrt{13}) is cubic, and the equation
2P=R has four solutions, it follows that 2P=2Q
where Q belongs to X(Q(sqrt{13}))
To see that P belongs to X(Q(sqrt{13})), it is
enough to observe that X has full 2-torsion
already over the rationals.
Total time: 13.929 seconds, Total memory usage: 56.06MB