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SageMath
E = EllipticCurve("cz1")
E.isogeny_class()
Elliptic curves in class 18480.cz
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 18480.cz1 | 18480df5 | \([0, 1, 0, -212000, 37498548]\) | \(257260669489908001/14267882475\) | \(58441246617600\) | \([4]\) | \(131072\) | \(1.7070\) | |
| 18480.cz2 | 18480df3 | \([0, 1, 0, -14000, 512148]\) | \(74093292126001/14707625625\) | \(60242434560000\) | \([2, 4]\) | \(65536\) | \(1.3604\) | |
| 18480.cz3 | 18480df2 | \([0, 1, 0, -4320, -103500]\) | \(2177286259681/161417025\) | \(661164134400\) | \([2, 2]\) | \(32768\) | \(1.0139\) | |
| 18480.cz4 | 18480df1 | \([0, 1, 0, -4240, -107692]\) | \(2058561081361/12705\) | \(52039680\) | \([2]\) | \(16384\) | \(0.66729\) | \(\Gamma_0(N)\)-optimal |
| 18480.cz5 | 18480df4 | \([0, 1, 0, 4080, -449580]\) | \(1833318007919/22507682505\) | \(-92191467540480\) | \([2]\) | \(65536\) | \(1.3604\) | |
| 18480.cz6 | 18480df6 | \([0, 1, 0, 29120, 3082100]\) | \(666688497209279/1381398046875\) | \(-5658206400000000\) | \([4]\) | \(131072\) | \(1.7070\) |
Rank
sage: E.rank()
The elliptic curves in class 18480.cz have rank \(1\).
Complex multiplication
The elliptic curves in class 18480.cz do not have complex multiplication.Modular form 18480.2.a.cz
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.
