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SageMath
E = EllipticCurve("bn1")
E.isogeny_class()
Elliptic curves in class 455175bn
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
455175.bn2 | 455175bn1 | \([1, -1, 1, 25033270, 385346259272]\) | \(1284365503/48234375\) | \(-65154512888185603271484375\) | \([2]\) | \(90243072\) | \(3.6331\) | \(\Gamma_0(N)\)-optimal* |
455175.bn1 | 455175bn2 | \([1, -1, 1, -665857355, 6325623853022]\) | \(24170156844497/1191196125\) | \(1609055850286631658392578125\) | \([2]\) | \(180486144\) | \(3.9797\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 455175bn have rank \(0\).
Complex multiplication
The elliptic curves in class 455175bn do not have complex multiplication.Modular form 455175.2.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.