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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 390390.bf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 390390.bf1 | 390390bf3 | \([1, 1, 0, -5575482, 5064858414]\) | \(3971101377248209009/56495958750\) | \(272695202158128750\) | \([2]\) | \(14155776\) | \(2.4853\) | |
| 390390.bf2 | 390390bf2 | \([1, 1, 0, -358452, 74247516]\) | \(1055257664218129/115307784900\) | \(556568653925384100\) | \([2, 2]\) | \(7077888\) | \(2.1387\) | |
| 390390.bf3 | 390390bf1 | \([1, 1, 0, -84672, -8269776]\) | \(13908844989649/1980372240\) | \(9558878551382160\) | \([2]\) | \(3538944\) | \(1.7921\) | \(\Gamma_0(N)\)-optimal |
| 390390.bf4 | 390390bf4 | \([1, 1, 0, 478098, 370218906]\) | \(2503876820718671/13702874328990\) | \(-66141157137037892910\) | \([2]\) | \(14155776\) | \(2.4853\) |
Rank
sage: E.rank()
The elliptic curves in class 390390.bf have rank \(2\).
Complex multiplication
The elliptic curves in class 390390.bf do not have complex multiplication.Modular form 390390.2.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.
