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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 424536.bm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
424536.bm1 | 424536bm3 | \([0, -1, 0, -10486469632, 413328813423772]\) | \(22501000029889239268/3620708343\) | \(20521227979693303085841408\) | \([2]\) | \(424673280\) | \(4.2593\) | \(\Gamma_0(N)\)-optimal* |
424536.bm2 | 424536bm2 | \([0, -1, 0, -657399892, 6417121071460]\) | \(22174957026242512/278654127129\) | \(394834679335054736588595456\) | \([2, 2]\) | \(212336640\) | \(3.9128\) | \(\Gamma_0(N)\)-optimal* |
424536.bm3 | 424536bm4 | \([0, -1, 0, -112932472, 16723889332060]\) | \(-28104147578308/21301741002339\) | \(-120732697047666938804188867584\) | \([2]\) | \(424673280\) | \(4.2593\) | |
424536.bm4 | 424536bm1 | \([0, -1, 0, -77112247, -102062447528]\) | \(572616640141312/280535480757\) | \(24843777151270658726770128\) | \([2]\) | \(106168320\) | \(3.5662\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 424536.bm have rank \(0\).
Complex multiplication
The elliptic curves in class 424536.bm do not have complex multiplication.Modular form 424536.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.