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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 4650.bm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4650.bm1 | 4650bx2 | \([1, 0, 0, -1297763, -563641983]\) | \(123759873855465821/1378809464832\) | \(2692987236000000000\) | \([2]\) | \(126720\) | \(2.3506\) | |
4650.bm2 | 4650bx1 | \([1, 0, 0, -17763, -22201983]\) | \(-317354125661/108829605888\) | \(-212557824000000000\) | \([2]\) | \(63360\) | \(2.0040\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 4650.bm have rank \(1\).
Complex multiplication
The elliptic curves in class 4650.bm do not have complex multiplication.Modular form 4650.2.a.bm
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.