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SageMath
E = EllipticCurve("cj1")
E.isogeny_class()
Elliptic curves in class 166782.cj
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
166782.cj1 | 166782e3 | \([1, 0, 0, -81774, 8990058]\) | \(1285429208617/614922\) | \(28929547236282\) | \([2]\) | \(884736\) | \(1.5379\) | |
166782.cj2 | 166782e4 | \([1, 0, 0, -45674, -3698370]\) | \(223980311017/4278582\) | \(201289659620742\) | \([2]\) | \(884736\) | \(1.5379\) | |
166782.cj3 | 166782e2 | \([1, 0, 0, -5964, 89964]\) | \(498677257/213444\) | \(10041661024164\) | \([2, 2]\) | \(442368\) | \(1.1913\) | |
166782.cj4 | 166782e1 | \([1, 0, 0, 1256, 10544]\) | \(4657463/3696\) | \(-173881576176\) | \([2]\) | \(221184\) | \(0.84473\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 166782.cj have rank \(0\).
Complex multiplication
The elliptic curves in class 166782.cj do not have complex multiplication.Modular form 166782.2.a.cj
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.