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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 36225.bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
36225.bc1 | 36225be1 | \([0, 0, 1, -4800, -395969]\) | \(-1073741824/5325075\) | \(-60655932421875\) | \([]\) | \(82944\) | \(1.3291\) | \(\Gamma_0(N)\)-optimal |
36225.bc2 | 36225be2 | \([0, 0, 1, 42450, 9691906]\) | \(742692847616/3992296875\) | \(-45474756591796875\) | \([]\) | \(248832\) | \(1.8784\) |
Rank
sage: E.rank()
The elliptic curves in class 36225.bc have rank \(0\).
Complex multiplication
The elliptic curves in class 36225.bc do not have complex multiplication.Modular form 36225.2.a.bc
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.