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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 101761.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
101761.f1 | 101761a2 | \([0, 1, 1, -746247, -254833100]\) | \(-32768\) | \(-1402562276304901811\) | \([]\) | \(1108800\) | \(2.2595\) | \(-11\) | |
101761.f2 | 101761a1 | \([0, 1, 1, -6167, 189217]\) | \(-32768\) | \(-791709840251\) | \([]\) | \(100800\) | \(1.0606\) | \(\Gamma_0(N)\)-optimal | \(-11\) |
Rank
sage: E.rank()
The elliptic curves in class 101761.f have rank \(1\).
Complex multiplication
Each elliptic curve in class 101761.f has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-11}) \).Modular form 101761.2.a.f
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.
\(\left(\begin{array}{rr} 1 & 11 \\ 11 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.