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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 1785a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1785.i2 | 1785a1 | \([1, 1, 0, 77, -392]\) | \(49471280711/106269975\) | \(-106269975\) | \([2]\) | \(576\) | \(0.22438\) | \(\Gamma_0(N)\)-optimal |
1785.i1 | 1785a2 | \([1, 1, 0, -598, -4847]\) | \(23711636464489/4590075735\) | \(4590075735\) | \([2]\) | \(1152\) | \(0.57095\) |
Rank
sage: E.rank()
The elliptic curves in class 1785a have rank \(0\).
Complex multiplication
The elliptic curves in class 1785a do not have complex multiplication.Modular form 1785.2.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 Cremona 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 Cremona labels.