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SageMath
E = EllipticCurve("bn1")
E.isogeny_class()
Elliptic curves in class 342720bn
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
342720.bn2 | 342720bn1 | \([0, 0, 0, 46932, 4672208]\) | \(59822347031/83966400\) | \(-16046228924006400\) | \([2]\) | \(1769472\) | \(1.7951\) | \(\Gamma_0(N)\)-optimal |
342720.bn1 | 342720bn2 | \([0, 0, 0, -298668, 46282448]\) | \(15417797707369/4080067320\) | \(779713007132344320\) | \([2]\) | \(3538944\) | \(2.1417\) |
Rank
sage: E.rank()
The elliptic curves in class 342720bn have rank \(1\).
Complex multiplication
The elliptic curves in class 342720bn do not have complex multiplication.Modular form 342720.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.