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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 67280.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
67280.a1 | 67280q3 | \([0, 1, 0, -34761, 2482414]\) | \(488095744/125\) | \(1189646642000\) | \([2]\) | \(145152\) | \(1.3030\) | |
67280.a2 | 67280q4 | \([0, 1, 0, -30556, 3109800]\) | \(-20720464/15625\) | \(-2379293284000000\) | \([2]\) | \(290304\) | \(1.6496\) | |
67280.a3 | 67280q1 | \([0, 1, 0, -1121, -10310]\) | \(16384/5\) | \(47585865680\) | \([2]\) | \(48384\) | \(0.75370\) | \(\Gamma_0(N)\)-optimal |
67280.a4 | 67280q2 | \([0, 1, 0, 3084, -65816]\) | \(21296/25\) | \(-3806869254400\) | \([2]\) | \(96768\) | \(1.1003\) |
Rank
sage: E.rank()
The elliptic curves in class 67280.a have rank \(2\).
Complex multiplication
The elliptic curves in class 67280.a do not have complex multiplication.Modular form 67280.2.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.