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SageMath
E = EllipticCurve("u1")
E.isogeny_class()
Elliptic curves in class 304920.u
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 304920.u1 | 304920u6 | \([0, 0, 0, -2236109403, -40699369853098]\) | \(467508233804095622882/315748125\) | \(835130551724663040000\) | \([2]\) | \(94371840\) | \(3.7643\) | |
| 304920.u2 | 304920u4 | \([0, 0, 0, -139784403, -635664248098]\) | \(228410605013945764/187597265625\) | \(248090480257712400000000\) | \([2, 2]\) | \(47185920\) | \(3.4178\) | |
| 304920.u3 | 304920u5 | \([0, 0, 0, -109597323, -917847034522]\) | \(-55043996611705922/105743408203125\) | \(-279683532036562500000000000\) | \([2]\) | \(94371840\) | \(3.7643\) | |
| 304920.u4 | 304920u3 | \([0, 0, 0, -90692283, 328785454118]\) | \(62380825826921284/787768887675\) | \(1041795364256789895244800\) | \([2]\) | \(47185920\) | \(3.4178\) | |
| 304920.u5 | 304920u2 | \([0, 0, 0, -10650783, -5259741982]\) | \(404151985581136/197735855625\) | \(65374598108798969760000\) | \([2, 2]\) | \(23592960\) | \(3.0712\) | |
| 304920.u6 | 304920u1 | \([0, 0, 0, 2422662, -629127763]\) | \(76102438406144/52315569075\) | \(-1081022107845911338800\) | \([2]\) | \(11796480\) | \(2.7246\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 304920.u have rank \(0\).
Complex multiplication
The elliptic curves in class 304920.u do not have complex multiplication.Modular form 304920.2.a.u
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
