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SageMath
E = EllipticCurve("bn1")
E.isogeny_class()
Elliptic curves in class 404544bn
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
404544.bn2 | 404544bn1 | \([0, -1, 0, 498559, 302552097]\) | \(444369620591/1540767744\) | \(-47518786339171467264\) | \([]\) | \(12192768\) | \(2.4586\) | \(\Gamma_0(N)\)-optimal |
404544.bn1 | 404544bn2 | \([0, -1, 0, -187849601, -991031767263]\) | \(-23769846831649063249/3261823333284\) | \(-100597826410913285013504\) | \([]\) | \(85349376\) | \(3.4316\) |
Rank
sage: E.rank()
The elliptic curves in class 404544bn have rank \(0\).
Complex multiplication
The elliptic curves in class 404544bn do not have complex multiplication.Modular form 404544.2.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.
\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.