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SageMath
E = EllipticCurve("bn1")
E.isogeny_class()
Elliptic curves in class 14700bn
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
14700.ba4 | 14700bn1 | \([0, 1, 0, 8167, 99588]\) | \(2048000/1323\) | \(-38912406750000\) | \([2]\) | \(41472\) | \(1.2966\) | \(\Gamma_0(N)\)-optimal |
14700.ba3 | 14700bn2 | \([0, 1, 0, -34708, 785588]\) | \(9826000/5103\) | \(2401451388000000\) | \([2]\) | \(82944\) | \(1.6432\) | |
14700.ba2 | 14700bn3 | \([0, 1, 0, -138833, 20459088]\) | \(-10061824000/352947\) | \(-10380965400750000\) | \([2]\) | \(124416\) | \(1.8459\) | |
14700.ba1 | 14700bn4 | \([0, 1, 0, -2239708, 1289387588]\) | \(2640279346000/3087\) | \(1452729852000000\) | \([2]\) | \(248832\) | \(2.1925\) |
Rank
sage: E.rank()
The elliptic curves in class 14700bn have rank \(0\).
Complex multiplication
The elliptic curves in class 14700bn do not have complex multiplication.Modular form 14700.2.a.bn
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.