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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 20286.k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 20286.k1 | 20286be5 | \([1, -1, 0, -34898103, -79341193359]\) | \(54804145548726848737/637608031452\) | \(54685167576084037692\) | \([2]\) | \(1572864\) | \(2.9382\) | |
| 20286.k2 | 20286be4 | \([1, -1, 0, -7811883, 8405643861]\) | \(614716917569296417/19093020912\) | \(1637534341794122352\) | \([2]\) | \(786432\) | \(2.5917\) | |
| 20286.k3 | 20286be3 | \([1, -1, 0, -2237643, -1171648395]\) | \(14447092394873377/1439452851984\) | \(123456287477054834064\) | \([2, 2]\) | \(786432\) | \(2.5917\) | |
| 20286.k4 | 20286be2 | \([1, -1, 0, -508923, 119705445]\) | \(169967019783457/26337394944\) | \(2258856201591892224\) | \([2, 2]\) | \(393216\) | \(2.2451\) | |
| 20286.k5 | 20286be1 | \([1, -1, 0, 55557, 10309221]\) | \(221115865823/664731648\) | \(-57011454954897408\) | \([2]\) | \(196608\) | \(1.8985\) | \(\Gamma_0(N)\)-optimal |
| 20286.k6 | 20286be6 | \([1, -1, 0, 2763297, -5669493831]\) | \(27207619911317663/177609314617308\) | \(-15232861968195106622268\) | \([2]\) | \(1572864\) | \(2.9382\) |
Rank
sage: E.rank()
The elliptic curves in class 20286.k have rank \(0\).
Complex multiplication
The elliptic curves in class 20286.k do not have complex multiplication.Modular form 20286.2.a.k
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
