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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 34650bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
34650.ba3 | 34650bc1 | \([1, -1, 0, -44644167, 114539415741]\) | \(863913648706111516969/2486234429521920\) | \(28319764048773120000000\) | \([2]\) | \(4816896\) | \(3.1792\) | \(\Gamma_0(N)\)-optimal |
34650.ba2 | 34650bc2 | \([1, -1, 0, -63076167, 10970007741]\) | \(2436531580079063806249/1405478914998681600\) | \(16009283266156857600000000\) | \([2, 2]\) | \(9633792\) | \(3.5258\) | |
34650.ba4 | 34650bc3 | \([1, -1, 0, 251851833, 87497511741]\) | \(155099895405729262880471/90047655797243760000\) | \(-1025699079315479703750000000\) | \([2]\) | \(19267584\) | \(3.8724\) | |
34650.ba1 | 34650bc4 | \([1, -1, 0, -672916167, -6694220792259]\) | \(2958414657792917260183849/12401051653985258880\) | \(141255728996175839430000000\) | \([2]\) | \(19267584\) | \(3.8724\) |
Rank
sage: E.rank()
The elliptic curves in class 34650bc have rank \(1\).
Complex multiplication
The elliptic curves in class 34650bc do not have complex multiplication.Modular form 34650.2.a.bc
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.