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SageMath
E = EllipticCurve("fn1")
E.isogeny_class()
Elliptic curves in class 374850fn
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
374850.fn2 | 374850fn1 | \([1, -1, 0, 19983, 537641]\) | \(658503/476\) | \(-637885524937500\) | \([2]\) | \(1572864\) | \(1.5291\) | \(\Gamma_0(N)\)-optimal |
374850.fn1 | 374850fn2 | \([1, -1, 0, -90267, 4616891]\) | \(60698457/28322\) | \(37954188733781250\) | \([2]\) | \(3145728\) | \(1.8757\) |
Rank
sage: E.rank()
The elliptic curves in class 374850fn have rank \(2\).
Complex multiplication
The elliptic curves in class 374850fn do not have complex multiplication.Modular form 374850.2.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.