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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 1752.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1752.a1 | 1752e1 | \([0, -1, 0, -20, 36]\) | \(3631696/657\) | \(168192\) | \([2]\) | \(384\) | \(-0.27835\) | \(\Gamma_0(N)\)-optimal |
1752.a2 | 1752e2 | \([0, -1, 0, 40, 156]\) | \(6740636/15987\) | \(-16370688\) | \([2]\) | \(768\) | \(0.068219\) |
Rank
sage: E.rank()
The elliptic curves in class 1752.a have rank \(2\).
Complex multiplication
The elliptic curves in class 1752.a do not have complex multiplication.Modular form 1752.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.