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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 45570bm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 45570.bi3 | 45570bm1 | \([1, 0, 1, -5318, 132248]\) | \(141339344329/17141760\) | \(2016710922240\) | \([2]\) | \(110592\) | \(1.0915\) | \(\Gamma_0(N)\)-optimal |
| 45570.bi2 | 45570bm2 | \([1, 0, 1, -20998, -1034344]\) | \(8702409880009/1120910400\) | \(131873987649600\) | \([2, 2]\) | \(221184\) | \(1.4381\) | |
| 45570.bi4 | 45570bm3 | \([1, 0, 1, 31922, -5394952]\) | \(30579142915511/124675335000\) | \(-14667928487415000\) | \([2]\) | \(442368\) | \(1.7847\) | |
| 45570.bi1 | 45570bm4 | \([1, 0, 1, -324798, -71272904]\) | \(32208729120020809/658986840\) | \(77529142739160\) | \([2]\) | \(442368\) | \(1.7847\) |
Rank
sage: E.rank()
The elliptic curves in class 45570bm have rank \(1\).
Complex multiplication
The elliptic curves in class 45570bm do not have complex multiplication.Modular form 45570.2.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 Cremona 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 Cremona labels.
