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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 244398.bm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
244398.bm1 | 244398bm4 | \([1, 0, 0, -7476368, 7867182714]\) | \(312196988566716625/25367712678\) | \(3755331898184300742\) | \([2]\) | \(7299072\) | \(2.6093\) | |
244398.bm2 | 244398bm3 | \([1, 0, 0, -435378, 140400288]\) | \(-61653281712625/21875235228\) | \(-3238319894061097692\) | \([2]\) | \(3649536\) | \(2.2627\) | |
244398.bm3 | 244398bm2 | \([1, 0, 0, -192038, -16262004]\) | \(5290763640625/2291573592\) | \(339235133900643288\) | \([2]\) | \(2433024\) | \(2.0600\) | |
244398.bm4 | 244398bm1 | \([1, 0, 0, 40722, -1877436]\) | \(50447927375/39517632\) | \(-5850027784294848\) | \([2]\) | \(1216512\) | \(1.7134\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 244398.bm have rank \(1\).
Complex multiplication
The elliptic curves in class 244398.bm do not have complex multiplication.Modular form 244398.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.