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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 6930.bm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6930.bm1 | 6930bk4 | \([1, -1, 1, -140837, -15663139]\) | \(423783056881319689/99207416000000\) | \(72322206264000000\) | \([6]\) | \(82944\) | \(1.9468\) | |
6930.bm2 | 6930bk2 | \([1, -1, 1, -131702, -18363571]\) | \(346553430870203929/8300600\) | \(6051137400\) | \([2]\) | \(27648\) | \(1.3975\) | |
6930.bm3 | 6930bk1 | \([1, -1, 1, -8222, -286099]\) | \(-84309998289049/414124480\) | \(-301896745920\) | \([2]\) | \(13824\) | \(1.0509\) | \(\Gamma_0(N)\)-optimal |
6930.bm4 | 6930bk3 | \([1, -1, 1, 20443, -1535011]\) | \(1296134247276791/2137096192000\) | \(-1557943123968000\) | \([6]\) | \(41472\) | \(1.6003\) |
Rank
sage: E.rank()
The elliptic curves in class 6930.bm have rank \(0\).
Complex multiplication
The elliptic curves in class 6930.bm do not have complex multiplication.Modular form 6930.2.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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.