Show commands:
SageMath
E = EllipticCurve("cy1")
E.isogeny_class()
Elliptic curves in class 142296cy
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
142296.m4 | 142296cy1 | \([0, -1, 0, 164036, -1012306700]\) | \(9148592/8301447\) | \(-442933079532208328448\) | \([2]\) | \(5898240\) | \(2.6407\) | \(\Gamma_0(N)\)-optimal |
142296.m3 | 142296cy2 | \([0, -1, 0, -14184144, -20083907556]\) | \(1478729816932/38900169\) | \(8302249788256599081984\) | \([2, 2]\) | \(11796480\) | \(2.9873\) | |
142296.m2 | 142296cy3 | \([0, -1, 0, -32445464, 42260238924]\) | \(8849350367426/3314597517\) | \(1414832749629910300698624\) | \([2]\) | \(23592960\) | \(3.3338\) | |
142296.m1 | 142296cy4 | \([0, -1, 0, -225493704, -1303240079700]\) | \(2970658109581346/2139291\) | \(913154297698256689152\) | \([2]\) | \(23592960\) | \(3.3338\) |
Rank
sage: E.rank()
The elliptic curves in class 142296cy have rank \(1\).
Complex multiplication
The elliptic curves in class 142296cy do not have complex multiplication.Modular form 142296.2.a.cy
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.