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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 301665.bm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
301665.bm1 | 301665bm3 | \([1, 1, 0, -171538, 21030937]\) | \(115650783909361/27072079335\) | \(130671756182892015\) | \([2]\) | \(3538944\) | \(1.9961\) | |
301665.bm2 | 301665bm2 | \([1, 1, 0, -57463, -5046608]\) | \(4347507044161/258084225\) | \(1245723259988025\) | \([2, 2]\) | \(1769472\) | \(1.6495\) | |
301665.bm3 | 301665bm1 | \([1, 1, 0, -56618, -5209017]\) | \(4158523459441/16065\) | \(77542686585\) | \([2]\) | \(884736\) | \(1.3030\) | \(\Gamma_0(N)\)-optimal |
301665.bm4 | 301665bm4 | \([1, 1, 0, 43092, -20713077]\) | \(1833318007919/39525924375\) | \(-190784087506569375\) | \([2]\) | \(3538944\) | \(1.9961\) |
Rank
sage: E.rank()
The elliptic curves in class 301665.bm have rank \(0\).
Complex multiplication
The elliptic curves in class 301665.bm do not have complex multiplication.Modular form 301665.2.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 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.