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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 116886.bc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 116886.bc1 | 116886bh4 | \([1, 1, 1, -743124, 235277505]\) | \(25618370387495257/1291301719092\) | \(2287619764776342612\) | \([2]\) | \(2949120\) | \(2.2813\) | |
| 116886.bc2 | 116886bh2 | \([1, 1, 1, -130864, -13544959]\) | \(139903436105497/36583447824\) | \(64809809410533264\) | \([2, 2]\) | \(1474560\) | \(1.9347\) | |
| 116886.bc3 | 116886bh1 | \([1, 1, 1, -121184, -16286335]\) | \(111097343765017/12241152\) | \(21685947478272\) | \([2]\) | \(737280\) | \(1.5882\) | \(\Gamma_0(N)\)-optimal |
| 116886.bc4 | 116886bh3 | \([1, 1, 1, 326516, -86725759]\) | \(2173106048486183/3097446973236\) | \(-5487316257352941396\) | \([2]\) | \(2949120\) | \(2.2813\) |
Rank
sage: E.rank()
The elliptic curves in class 116886.bc have rank \(1\).
Complex multiplication
The elliptic curves in class 116886.bc do not have complex multiplication.Modular form 116886.2.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 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.
