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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 53361.bm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
53361.bm1 | 53361bj4 | \([1, -1, 0, -7818498, -8404372971]\) | \(347873904937/395307\) | \(60062912012459053467\) | \([2]\) | \(1658880\) | \(2.7086\) | |
53361.bm2 | 53361bj2 | \([1, -1, 0, -614763, -58125600]\) | \(169112377/88209\) | \(13402467969722268129\) | \([2, 2]\) | \(829440\) | \(2.3620\) | |
53361.bm3 | 53361bj1 | \([1, -1, 0, -347958, 78425199]\) | \(30664297/297\) | \(45126154780209657\) | \([2]\) | \(414720\) | \(2.0154\) | \(\Gamma_0(N)\)-optimal |
53361.bm4 | 53361bj3 | \([1, -1, 0, 2320092, -454331025]\) | \(9090072503/5845851\) | \(-888218104538866678731\) | \([2]\) | \(1658880\) | \(2.7086\) |
Rank
sage: E.rank()
The elliptic curves in class 53361.bm have rank \(0\).
Complex multiplication
The elliptic curves in class 53361.bm do not have complex multiplication.Modular form 53361.2.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.