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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 70644.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
70644.d1 | 70644d4 | \([0, -1, 0, -1537628, -733366920]\) | \(2640279346000/3087\) | \(470072215533312\) | \([2]\) | \(870912\) | \(2.0985\) | |
70644.d2 | 70644d3 | \([0, -1, 0, -95313, -11632494]\) | \(-10061824000/352947\) | \(-3359057706831792\) | \([2]\) | \(435456\) | \(1.7519\) | |
70644.d3 | 70644d2 | \([0, -1, 0, -23828, -445512]\) | \(9826000/5103\) | \(777058152208128\) | \([2]\) | \(290304\) | \(1.5492\) | |
70644.d4 | 70644d1 | \([0, -1, 0, 5607, -56970]\) | \(2048000/1323\) | \(-12591220058928\) | \([2]\) | \(145152\) | \(1.2026\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 70644.d have rank \(1\).
Complex multiplication
The elliptic curves in class 70644.d do not have complex multiplication.Modular form 70644.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.