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SageMath
E = EllipticCurve("en1")
E.isogeny_class()
Elliptic curves in class 333270.en
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
333270.en1 | 333270en2 | \([1, -1, 1, -2011622, 612666119]\) | \(8341959848041/3327411150\) | \(359088099123237753150\) | \([2]\) | \(16220160\) | \(2.6424\) | |
333270.en2 | 333270en1 | \([1, -1, 1, -916592, -330811729]\) | \(789145184521/17996580\) | \(1942157855340262980\) | \([2]\) | \(8110080\) | \(2.2958\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 333270.en have rank \(1\).
Complex multiplication
The elliptic curves in class 333270.en do not have complex multiplication.Modular form 333270.2.a.en
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.