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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 59850.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 59850.e1 | 59850bg4 | \([1, -1, 0, -1276917, 555701741]\) | \(20214562937713929/665000\) | \(7574765625000\) | \([2]\) | \(737280\) | \(1.9720\) | |
| 59850.e2 | 59850bg2 | \([1, -1, 0, -79917, 8672741]\) | \(4955605568649/28302400\) | \(322382025000000\) | \([2, 2]\) | \(368640\) | \(1.6254\) | |
| 59850.e3 | 59850bg3 | \([1, -1, 0, -34917, 18347741]\) | \(-413327139849/12516028840\) | \(-142565391005625000\) | \([2]\) | \(737280\) | \(1.9720\) | |
| 59850.e4 | 59850bg1 | \([1, -1, 0, -7917, -39259]\) | \(4818245769/2723840\) | \(31026240000000\) | \([2]\) | \(184320\) | \(1.2788\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 59850.e have rank \(2\).
Complex multiplication
The elliptic curves in class 59850.e do not have complex multiplication.Modular form 59850.2.a.e
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
