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SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 212940bp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
212940.bn2 | 212940bp1 | \([0, 0, 0, -18252, -533871]\) | \(442368/175\) | \(266017028331600\) | \([2]\) | \(677376\) | \(1.4663\) | \(\Gamma_0(N)\)-optimal |
212940.bn1 | 212940bp2 | \([0, 0, 0, -132327, 18151614]\) | \(10536048/245\) | \(5958781434627840\) | \([2]\) | \(1354752\) | \(1.8129\) |
Rank
sage: E.rank()
The elliptic curves in class 212940bp have rank \(0\).
Complex multiplication
The elliptic curves in class 212940bp do not have complex multiplication.Modular form 212940.2.a.bp
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.