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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 364650.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
364650.d1 | 364650d4 | \([1, 1, 0, -221540150, -1269281301750]\) | \(76959285225914579642095969/99697498164371250\) | \(1557773408818300781250\) | \([2]\) | \(108527616\) | \(3.3435\) | |
364650.d2 | 364650d3 | \([1, 1, 0, -34879650, 52529298750]\) | \(300344760859819078829089/97478526510911312910\) | \(1523101976732989264218750\) | \([2]\) | \(108527616\) | \(3.3435\) | |
364650.d3 | 364650d2 | \([1, 1, 0, -13965900, -19476742500]\) | \(19280154901730969757889/675674185355640900\) | \(10557409146181889062500\) | \([2, 2]\) | \(54263808\) | \(2.9970\) | |
364650.d4 | 364650d1 | \([1, 1, 0, 314600, -1069178000]\) | \(220382793629361791/31740865455602160\) | \(-495951022743783750000\) | \([2]\) | \(27131904\) | \(2.6504\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 364650.d have rank \(0\).
Complex multiplication
The elliptic curves in class 364650.d do not have complex multiplication.Modular form 364650.2.a.d
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.