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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 107712.bf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 107712.bf1 | 107712eo6 | \([0, 0, 0, -227487756, -1320642148336]\) | \(6812873765474836663297/74052\) | \(14151557578752\) | \([2]\) | \(6291456\) | \(3.0270\) | |
| 107712.bf2 | 107712eo4 | \([0, 0, 0, -14217996, -20634999280]\) | \(1663303207415737537/5483698704\) | \(1047951141821743104\) | \([2, 2]\) | \(3145728\) | \(2.6804\) | |
| 107712.bf3 | 107712eo5 | \([0, 0, 0, -14022156, -21231057904]\) | \(-1595514095015181697/95635786040388\) | \(-18276283324417402994688\) | \([2]\) | \(6291456\) | \(3.0270\) | |
| 107712.bf4 | 107712eo2 | \([0, 0, 0, -900876, -313074160]\) | \(423108074414017/23284318464\) | \(4449702552602148864\) | \([2, 2]\) | \(1572864\) | \(2.3338\) | |
| 107712.bf5 | 107712eo1 | \([0, 0, 0, -163596, 19291664]\) | \(2533811507137/625016832\) | \(119442576645292032\) | \([2]\) | \(786432\) | \(1.9872\) | \(\Gamma_0(N)\)-optimal |
| 107712.bf6 | 107712eo3 | \([0, 0, 0, 619764, -1262561776]\) | \(137763859017023/3683199928848\) | \(-703870467605841051648\) | \([2]\) | \(3145728\) | \(2.6804\) |
Rank
sage: E.rank()
The elliptic curves in class 107712.bf have rank \(0\).
Complex multiplication
The elliptic curves in class 107712.bf do not have complex multiplication.Modular form 107712.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
