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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 578.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
578.a1 | 578a4 | \([1, 1, 1, -32663, -1583717]\) | \(159661140625/48275138\) | \(1165244474459522\) | \([2]\) | \(3456\) | \(1.5961\) | |
578.a2 | 578a3 | \([1, 1, 1, -29773, -1989473]\) | \(120920208625/19652\) | \(474351505988\) | \([2]\) | \(1728\) | \(1.2495\) | |
578.a3 | 578a2 | \([1, 1, 1, -12433, 528295]\) | \(8805624625/2312\) | \(55806059528\) | \([2]\) | \(1152\) | \(1.0468\) | |
578.a4 | 578a1 | \([1, 1, 1, -873, 5783]\) | \(3048625/1088\) | \(26261675072\) | \([2]\) | \(576\) | \(0.70021\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 578.a have rank \(0\).
Complex multiplication
The elliptic curves in class 578.a do not have complex multiplication.Modular form 578.2.a.a
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.