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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 14586f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
14586.d1 | 14586f1 | \([1, 0, 1, -39458, 2580212]\) | \(6793805286030262681/1048227429629952\) | \(1048227429629952\) | \([2]\) | \(150528\) | \(1.6056\) | \(\Gamma_0(N)\)-optimal |
14586.d2 | 14586f2 | \([1, 0, 1, 68702, 14261492]\) | \(35862531227445945959/108547797844556928\) | \(-108547797844556928\) | \([2]\) | \(301056\) | \(1.9522\) |
Rank
sage: E.rank()
The elliptic curves in class 14586f have rank \(1\).
Complex multiplication
The elliptic curves in class 14586f do not have complex multiplication.Modular form 14586.2.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 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.