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SageMath
E = EllipticCurve("gy1")
E.isogeny_class()
Elliptic curves in class 277200.gy
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
277200.gy1 | 277200gy6 | \([0, 0, 0, -16268475, -25256209750]\) | \(10206027697760497/5557167\) | \(259275183552000000\) | \([2]\) | \(10485760\) | \(2.6700\) | |
277200.gy2 | 277200gy4 | \([0, 0, 0, -1022475, -389983750]\) | \(2533811507137/58110129\) | \(2711186178624000000\) | \([2, 2]\) | \(5242880\) | \(2.3234\) | |
277200.gy3 | 277200gy2 | \([0, 0, 0, -140475, 11326250]\) | \(6570725617/2614689\) | \(121990929984000000\) | \([2, 2]\) | \(2621440\) | \(1.9768\) | |
277200.gy4 | 277200gy1 | \([0, 0, 0, -122475, 16492250]\) | \(4354703137/1617\) | \(75442752000000\) | \([2]\) | \(1310720\) | \(1.6302\) | \(\Gamma_0(N)\)-optimal |
277200.gy5 | 277200gy5 | \([0, 0, 0, 111525, -1207597750]\) | \(3288008303/13504609503\) | \(-630071060971968000000\) | \([2]\) | \(10485760\) | \(2.6700\) | |
277200.gy6 | 277200gy3 | \([0, 0, 0, 453525, 82012250]\) | \(221115865823/190238433\) | \(-8875764330048000000\) | \([2]\) | \(5242880\) | \(2.3234\) |
Rank
sage: E.rank()
The elliptic curves in class 277200.gy have rank \(2\).
Complex multiplication
The elliptic curves in class 277200.gy do not have complex multiplication.Modular form 277200.2.a.gy
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 8 & 4 & 2 & 1 & 8 & 4 \\ 4 & 2 & 4 & 8 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.