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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 14586.k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
14586.k1 | 14586j1 | \([1, 1, 1, -84338, -9461761]\) | \(66342819962001390625/4812668669952\) | \(4812668669952\) | \([2]\) | \(59136\) | \(1.4847\) | \(\Gamma_0(N)\)-optimal |
14586.k2 | 14586j2 | \([1, 1, 1, -78898, -10728193]\) | \(-54315282059491182625/17983956399469632\) | \(-17983956399469632\) | \([2]\) | \(118272\) | \(1.8312\) |
Rank
sage: E.rank()
The elliptic curves in class 14586.k have rank \(1\).
Complex multiplication
The elliptic curves in class 14586.k do not have complex multiplication.Modular form 14586.2.a.k
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.