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SageMath
E = EllipticCurve("ij1")
E.isogeny_class()
Elliptic curves in class 221760ij
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
221760.kx1 | 221760ij1 | \([0, 0, 0, -2877132, 1878316144]\) | \(13782741913468081/701662500\) | \(134089791897600000\) | \([2]\) | \(4423680\) | \(2.3564\) | \(\Gamma_0(N)\)-optimal |
221760.kx2 | 221760ij2 | \([0, 0, 0, -2721612, 2090383216]\) | \(-11666347147400401/3126621093750\) | \(-597506595840000000000\) | \([2]\) | \(8847360\) | \(2.7030\) |
Rank
sage: E.rank()
The elliptic curves in class 221760ij have rank \(2\).
Complex multiplication
The elliptic curves in class 221760ij do not have complex multiplication.Modular form 221760.2.a.ij
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.