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SageMath
E = EllipticCurve("fn1")
E.isogeny_class()
Elliptic curves in class 405720fn
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
405720.fn2 | 405720fn1 | \([0, 0, 0, 14553, 9501786]\) | \(574992/66125\) | \(-39199920199776000\) | \([2]\) | \(2488320\) | \(1.8633\) | \(\Gamma_0(N)\)-optimal* |
405720.fn1 | 405720fn2 | \([0, 0, 0, -594027, 170532054]\) | \(9776035692/359375\) | \(852172178256000000\) | \([2]\) | \(4976640\) | \(2.2099\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 405720fn have rank \(0\).
Complex multiplication
The elliptic curves in class 405720fn do not have complex multiplication.Modular form 405720.2.a.fn
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.