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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 390775.bm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 390775.bm1 | 390775bm4 | \([1, -1, 0, -65360717, -203348417434]\) | \(16798320881842096017/2132227789307\) | \(3919601049752800671875\) | \([2]\) | \(33030144\) | \(3.1652\) | |
| 390775.bm2 | 390775bm3 | \([1, -1, 0, -25927967, 48746574316]\) | \(1048626554636928177/48569076788309\) | \(89282864297933836578125\) | \([2]\) | \(33030144\) | \(3.1652\) | |
| 390775.bm3 | 390775bm2 | \([1, -1, 0, -4435342, -2599306809]\) | \(5249244962308257/1448621666569\) | \(2662951413284004390625\) | \([2, 2]\) | \(16515072\) | \(2.8186\) | |
| 390775.bm4 | 390775bm1 | \([1, -1, 0, 715783, -265847184]\) | \(22062729659823/29354283343\) | \(-53960970015946984375\) | \([2]\) | \(8257536\) | \(2.4720\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 390775.bm have rank \(1\).
Complex multiplication
The elliptic curves in class 390775.bm do not have complex multiplication.Modular form 390775.2.a.bm
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.
