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SageMath
E = EllipticCurve("hh1")
E.isogeny_class()
Elliptic curves in class 284592.hh
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
284592.hh1 | 284592hh3 | \([0, 1, 0, -225493704, 1303240079700]\) | \(2970658109581346/2139291\) | \(913154297698256689152\) | \([2]\) | \(47185920\) | \(3.3338\) | |
284592.hh2 | 284592hh4 | \([0, 1, 0, -32445464, -42260238924]\) | \(8849350367426/3314597517\) | \(1414832749629910300698624\) | \([2]\) | \(47185920\) | \(3.3338\) | |
284592.hh3 | 284592hh2 | \([0, 1, 0, -14184144, 20083907556]\) | \(1478729816932/38900169\) | \(8302249788256599081984\) | \([2, 2]\) | \(23592960\) | \(2.9873\) | |
284592.hh4 | 284592hh1 | \([0, 1, 0, 164036, 1012306700]\) | \(9148592/8301447\) | \(-442933079532208328448\) | \([2]\) | \(11796480\) | \(2.6407\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 284592.hh have rank \(0\).
Complex multiplication
The elliptic curves in class 284592.hh do not have complex multiplication.Modular form 284592.2.a.hh
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.