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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 2006.k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2006.k1 | 2006h2 | \([1, 1, 1, -49719, -786739]\) | \(13592251860742707697/7612392968095424\) | \(7612392968095424\) | \([2]\) | \(14400\) | \(1.7376\) | |
2006.k2 | 2006h1 | \([1, 1, 1, -30839, 2060365]\) | \(3243586268529106417/20244571000832\) | \(20244571000832\) | \([2]\) | \(7200\) | \(1.3910\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 2006.k have rank \(0\).
Complex multiplication
The elliptic curves in class 2006.k do not have complex multiplication.Modular form 2006.2.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.