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SageMath
E = EllipticCurve("cf1")
E.isogeny_class()
Elliptic curves in class 55440.cf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 55440.cf1 | 55440dr6 | \([0, 0, 0, -1908003, -1014368798]\) | \(257260669489908001/14267882475\) | \(42603668784230400\) | \([2]\) | \(1048576\) | \(2.2563\) | |
| 55440.cf2 | 55440dr4 | \([0, 0, 0, -126003, -13953998]\) | \(74093292126001/14707625625\) | \(43916734794240000\) | \([2, 2]\) | \(524288\) | \(1.9097\) | |
| 55440.cf3 | 55440dr2 | \([0, 0, 0, -38883, 2755618]\) | \(2177286259681/161417025\) | \(481988653977600\) | \([2, 2]\) | \(262144\) | \(1.5632\) | |
| 55440.cf4 | 55440dr1 | \([0, 0, 0, -38163, 2869522]\) | \(2058561081361/12705\) | \(37936926720\) | \([2]\) | \(131072\) | \(1.2166\) | \(\Gamma_0(N)\)-optimal |
| 55440.cf5 | 55440dr3 | \([0, 0, 0, 36717, 12175378]\) | \(1833318007919/22507682505\) | \(-67207579837009920\) | \([2]\) | \(524288\) | \(1.9097\) | |
| 55440.cf6 | 55440dr5 | \([0, 0, 0, 262077, -82954622]\) | \(666688497209279/1381398046875\) | \(-4124832465600000000\) | \([2]\) | \(1048576\) | \(2.2563\) |
Rank
sage: E.rank()
The elliptic curves in class 55440.cf have rank \(1\).
Complex multiplication
The elliptic curves in class 55440.cf do not have complex multiplication.Modular form 55440.2.a.cf
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
