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SageMath
E = EllipticCurve("fn1")
E.isogeny_class()
Elliptic curves in class 422370fn
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
422370.fn2 | 422370fn1 | \([1, -1, 1, 12928, -1573729]\) | \(6967871/35100\) | \(-1203805298439900\) | \([2]\) | \(2515968\) | \(1.5759\) | \(\Gamma_0(N)\)-optimal* |
422370.fn1 | 422370fn2 | \([1, -1, 1, -149522, -19898089]\) | \(10779215329/1232010\) | \(42253565975240490\) | \([2]\) | \(5031936\) | \(1.9225\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 422370fn have rank \(0\).
Complex multiplication
The elliptic curves in class 422370fn do not have complex multiplication.Modular form 422370.2.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.