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SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 14112.bd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
14112.bd1 | 14112p2 | \([0, 0, 0, -14700, -647584]\) | \(1000000/63\) | \(22131775991808\) | \([2]\) | \(24576\) | \(1.3117\) | |
14112.bd2 | 14112p1 | \([0, 0, 0, 735, -42532]\) | \(8000/147\) | \(-806887666368\) | \([2]\) | \(12288\) | \(0.96514\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 14112.bd have rank \(1\).
Complex multiplication
The elliptic curves in class 14112.bd do not have complex multiplication.Modular form 14112.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.