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SageMath
E = EllipticCurve("bn1")
E.isogeny_class()
Elliptic curves in class 494190.bn
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
494190.bn1 | 494190bn4 | \([1, -1, 0, -7907094, -8556046862]\) | \(3107086841064961/570\) | \(10029884046570\) | \([2]\) | \(15728640\) | \(2.3304\) | |
494190.bn2 | 494190bn3 | \([1, -1, 0, -572274, -88512170]\) | \(1177918188481/488703750\) | \(8599371834427953750\) | \([2]\) | \(15728640\) | \(2.3304\) | \(\Gamma_0(N)\)-optimal* |
494190.bn3 | 494190bn2 | \([1, -1, 0, -494244, -133566692]\) | \(758800078561/324900\) | \(5717033906544900\) | \([2, 2]\) | \(7864320\) | \(1.9839\) | \(\Gamma_0(N)\)-optimal* |
494190.bn4 | 494190bn1 | \([1, -1, 0, -26064, -2757200]\) | \(-111284641/123120\) | \(-2166454954059120\) | \([2]\) | \(3932160\) | \(1.6373\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 494190.bn have rank \(1\).
Complex multiplication
The elliptic curves in class 494190.bn do not have complex multiplication.Modular form 494190.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.