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SageMath
E = EllipticCurve("bn1")
E.isogeny_class()
Elliptic curves in class 411840bn
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
411840.bn2 | 411840bn1 | \([0, 0, 0, -1051788, 420802288]\) | \(-673350049820449/10617750000\) | \(-2029083623424000000\) | \([2]\) | \(7077888\) | \(2.3142\) | \(\Gamma_0(N)\)-optimal* |
411840.bn1 | 411840bn2 | \([0, 0, 0, -16891788, 26721538288]\) | \(2789222297765780449/677605500\) | \(129492427603968000\) | \([2]\) | \(14155776\) | \(2.6608\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 411840bn have rank \(2\).
Complex multiplication
The elliptic curves in class 411840bn do not have complex multiplication.Modular form 411840.2.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.