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SageMath
E = EllipticCurve("cl1")
E.isogeny_class()
Elliptic curves in class 40425.cl
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 40425.cl1 | 40425k6 | \([1, 1, 0, -5535800, 5010934125]\) | \(10206027697760497/5557167\) | \(10215549068484375\) | \([2]\) | \(983040\) | \(2.4005\) | |
| 40425.cl2 | 40425k4 | \([1, 1, 0, -347925, 77265000]\) | \(2533811507137/58110129\) | \(106821852605015625\) | \([2, 2]\) | \(491520\) | \(2.0539\) | |
| 40425.cl3 | 40425k2 | \([1, 1, 0, -47800, -2268125]\) | \(6570725617/2614689\) | \(4806492908765625\) | \([2, 2]\) | \(245760\) | \(1.7073\) | |
| 40425.cl4 | 40425k1 | \([1, 1, 0, -41675, -3291000]\) | \(4354703137/1617\) | \(2972475515625\) | \([2]\) | \(122880\) | \(1.3607\) | \(\Gamma_0(N)\)-optimal |
| 40425.cl5 | 40425k5 | \([1, 1, 0, 37950, 239718375]\) | \(3288008303/13504609503\) | \(-24825059428413234375\) | \([2]\) | \(983040\) | \(2.4005\) | |
| 40425.cl6 | 40425k3 | \([1, 1, 0, 154325, -16214750]\) | \(221115865823/190238433\) | \(-349708771937765625\) | \([2]\) | \(491520\) | \(2.0539\) |
Rank
sage: E.rank()
The elliptic curves in class 40425.cl have rank \(0\).
Complex multiplication
The elliptic curves in class 40425.cl do not have complex multiplication.Modular form 40425.2.a.cl
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.
