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SageMath
E = EllipticCurve("bn1")
E.isogeny_class()
Elliptic curves in class 59850.bn
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
59850.bn1 | 59850bf4 | \([1, -1, 0, -4084092, -3175788434]\) | \(661397832743623417/443352042\) | \(5050056853406250\) | \([2]\) | \(1310720\) | \(2.3293\) | |
59850.bn2 | 59850bf2 | \([1, -1, 0, -256842, -48925184]\) | \(164503536215257/4178071044\) | \(47590840485562500\) | \([2, 2]\) | \(655360\) | \(1.9828\) | |
59850.bn3 | 59850bf1 | \([1, -1, 0, -36342, 1569316]\) | \(466025146777/177366672\) | \(2020317248250000\) | \([2]\) | \(327680\) | \(1.6362\) | \(\Gamma_0(N)\)-optimal |
59850.bn4 | 59850bf3 | \([1, -1, 0, 42408, -156355934]\) | \(740480746823/927484650666\) | \(-10564629848992406250\) | \([2]\) | \(1310720\) | \(2.3293\) |
Rank
sage: E.rank()
The elliptic curves in class 59850.bn have rank \(0\).
Complex multiplication
The elliptic curves in class 59850.bn do not have complex multiplication.Modular form 59850.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.