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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 87120.bf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 87120.bf1 | 87120x6 | \([0, 0, 0, -3485163, 2504279162]\) | \(1770025017602/75\) | \(198369480038400\) | \([2]\) | \(1310720\) | \(2.2283\) | |
| 87120.bf2 | 87120x4 | \([0, 0, 0, -218163, 39000962]\) | \(868327204/5625\) | \(7438855501440000\) | \([2, 2]\) | \(655360\) | \(1.8817\) | |
| 87120.bf3 | 87120x5 | \([0, 0, 0, -87483, 85287818]\) | \(-27995042/1171875\) | \(-3099523125600000000\) | \([2]\) | \(1310720\) | \(2.2283\) | |
| 87120.bf4 | 87120x2 | \([0, 0, 0, -22143, -242242]\) | \(3631696/2025\) | \(669496995129600\) | \([2, 2]\) | \(327680\) | \(1.5351\) | |
| 87120.bf5 | 87120x1 | \([0, 0, 0, -16698, -829213]\) | \(24918016/45\) | \(929856937680\) | \([2]\) | \(163840\) | \(1.1885\) | \(\Gamma_0(N)\)-optimal |
| 87120.bf6 | 87120x3 | \([0, 0, 0, 86757, -1919302]\) | \(54607676/32805\) | \(-43383405284398080\) | \([2]\) | \(655360\) | \(1.8817\) |
Rank
sage: E.rank()
The elliptic curves in class 87120.bf have rank \(1\).
Complex multiplication
The elliptic curves in class 87120.bf do not have complex multiplication.Modular form 87120.2.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.
