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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 122694.bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
122694.bc1 | 122694bl1 | \([1, 0, 1, -2126420365, -37741933811320]\) | \(-124352595912593543977/103332962304\) | \(-883598773814213640708096\) | \([]\) | \(62899200\) | \(3.8982\) | \(\Gamma_0(N)\)-optimal |
122694.bc2 | 122694bl2 | \([1, 0, 1, -1650674380, -55074169825270]\) | \(-58169016237585194137/119573538788081664\) | \(-1022471725362366629662987124736\) | \([]\) | \(188697600\) | \(4.4475\) |
Rank
sage: E.rank()
The elliptic curves in class 122694.bc have rank \(1\).
Complex multiplication
The elliptic curves in class 122694.bc do not have complex multiplication.Modular form 122694.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.