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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 412224bc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 412224.bc1 | 412224bc1 | \([0, -1, 0, -8417, -294015]\) | \(251598106297/412224\) | \(108062048256\) | \([2]\) | \(479232\) | \(1.0136\) | \(\Gamma_0(N)\)-optimal |
| 412224.bc2 | 412224bc2 | \([0, -1, 0, -5857, -478847]\) | \(-84778086457/331891848\) | \(-87003456602112\) | \([2]\) | \(958464\) | \(1.3602\) |
Rank
sage: E.rank()
The elliptic curves in class 412224bc have rank \(2\).
Complex multiplication
The elliptic curves in class 412224bc do not have complex multiplication.Modular form 412224.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
