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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 5040.bf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 5040.bf1 | 5040s3 | \([0, 0, 0, -26067, 1619714]\) | \(2624033547076/324135\) | \(241965480960\) | \([4]\) | \(12288\) | \(1.2076\) | |
| 5040.bf2 | 5040s2 | \([0, 0, 0, -1767, 20774]\) | \(3269383504/893025\) | \(166659897600\) | \([2, 2]\) | \(6144\) | \(0.86102\) | |
| 5040.bf3 | 5040s1 | \([0, 0, 0, -642, -6001]\) | \(2508888064/118125\) | \(1377810000\) | \([2]\) | \(3072\) | \(0.51444\) | \(\Gamma_0(N)\)-optimal |
| 5040.bf4 | 5040s4 | \([0, 0, 0, 4533, 135434]\) | \(13799183324/18600435\) | \(-13885150325760\) | \([2]\) | \(12288\) | \(1.2076\) |
Rank
sage: E.rank()
The elliptic curves in class 5040.bf have rank \(0\).
Complex multiplication
The elliptic curves in class 5040.bf do not have complex multiplication.Modular form 5040.2.a.bf
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.
