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SageMath
E = EllipticCurve("n1")
E.isogeny_class()
Elliptic curves in class 491970.n
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
491970.n1 | 491970n4 | \([1, 1, 0, -3506487, -2528714979]\) | \(32208729120020809/658986840\) | \(97553702698700760\) | \([2]\) | \(14192640\) | \(2.3795\) | |
491970.n2 | 491970n2 | \([1, 1, 0, -226687, -36722939]\) | \(8702409880009/1120910400\) | \(165934967553345600\) | \([2, 2]\) | \(7096320\) | \(2.0329\) | |
491970.n3 | 491970n1 | \([1, 1, 0, -57407, 4682949]\) | \(141339344329/17141760\) | \(2537595680624640\) | \([2]\) | \(3548160\) | \(1.6863\) | \(\Gamma_0(N)\)-optimal* |
491970.n4 | 491970n3 | \([1, 1, 0, 344633, -191322131]\) | \(30579142915511/124675335000\) | \(-18456424053097815000\) | \([2]\) | \(14192640\) | \(2.3795\) |
Rank
sage: E.rank()
The elliptic curves in class 491970.n have rank \(1\).
Complex multiplication
The elliptic curves in class 491970.n do not have complex multiplication.Modular form 491970.2.a.n
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.