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SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 55800.bd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 55800.bd1 | 55800bn4 | \([0, 0, 0, -1494075, -702882250]\) | \(15811147933922/1016955\) | \(23723526240000000\) | \([2]\) | \(589824\) | \(2.1996\) | |
| 55800.bd2 | 55800bn3 | \([0, 0, 0, -504075, 129347750]\) | \(607199886722/41558445\) | \(969475404960000000\) | \([2]\) | \(589824\) | \(2.1996\) | |
| 55800.bd3 | 55800bn2 | \([0, 0, 0, -99075, -9567250]\) | \(9220796644/1946025\) | \(22698435600000000\) | \([2, 2]\) | \(294912\) | \(1.8531\) | |
| 55800.bd4 | 55800bn1 | \([0, 0, 0, 13425, -904750]\) | \(91765424/174375\) | \(-508477500000000\) | \([2]\) | \(147456\) | \(1.5065\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 55800.bd have rank \(1\).
Complex multiplication
The elliptic curves in class 55800.bd do not have complex multiplication.Modular form 55800.2.a.bd
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
