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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 6034.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6034.d1 | 6034c2 | \([1, -1, 0, -4707319, 3932226909]\) | \(11535718620691985847049161/249337512589312\) | \(249337512589312\) | \([2]\) | \(174240\) | \(2.2895\) | |
6034.d2 | 6034c1 | \([1, -1, 0, -293879, 61640029]\) | \(-2806921089205005638601/13094974136516608\) | \(-13094974136516608\) | \([2]\) | \(87120\) | \(1.9429\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 6034.d have rank \(1\).
Complex multiplication
The elliptic curves in class 6034.d do not have complex multiplication.Modular form 6034.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.