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SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 422331.bd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
422331.bd1 | 422331bd1 | \([0, -1, 1, -19959, -1383523]\) | \(-44226936832/16395939\) | \(-325995224832459\) | \([]\) | \(1410048\) | \(1.4950\) | \(\Gamma_0(N)\)-optimal* |
422331.bd2 | 422331bd2 | \([0, -1, 1, 152031, 14009582]\) | \(19545301188608/15606257499\) | \(-310294239456474819\) | \([]\) | \(4230144\) | \(2.0443\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 422331.bd have rank \(1\).
Complex multiplication
The elliptic curves in class 422331.bd do not have complex multiplication.Modular form 422331.2.a.bd
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.