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SageMath
E = EllipticCurve("db1")
E.isogeny_class()
Elliptic curves in class 133200db
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 133200.fc3 | 133200db1 | \([0, 0, 0, -270075, -29537750]\) | \(46694890801/18944000\) | \(883851264000000000\) | \([2]\) | \(1327104\) | \(2.1414\) | \(\Gamma_0(N)\)-optimal |
| 133200.fc4 | 133200db2 | \([0, 0, 0, 881925, -215009750]\) | \(1625964918479/1369000000\) | \(-63872064000000000000\) | \([2]\) | \(2654208\) | \(2.4879\) | |
| 133200.fc1 | 133200db3 | \([0, 0, 0, -18990075, -31852097750]\) | \(16232905099479601/4052240\) | \(189061309440000000\) | \([2]\) | \(3981312\) | \(2.6907\) | |
| 133200.fc2 | 133200db4 | \([0, 0, 0, -18918075, -32105609750]\) | \(-16048965315233521/256572640900\) | \(-11970653133830400000000\) | \([2]\) | \(7962624\) | \(3.0373\) |
Rank
sage: E.rank()
The elliptic curves in class 133200db have rank \(1\).
Complex multiplication
The elliptic curves in class 133200db do not have complex multiplication.Modular form 133200.2.a.db
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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
