Show commands:
SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 13923.i
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 13923.i1 | 13923f5 | \([1, -1, 0, -365211, 84980704]\) | \(7389727131216686257/6115533215337\) | \(4458223713980673\) | \([2]\) | \(98304\) | \(1.9312\) | |
| 13923.i2 | 13923f3 | \([1, -1, 0, -27846, 706927]\) | \(3275619238041697/1605271262049\) | \(1170242750033721\) | \([2, 2]\) | \(49152\) | \(1.5846\) | |
| 13923.i3 | 13923f2 | \([1, -1, 0, -14841, -684608]\) | \(495909170514577/6224736609\) | \(4537832987961\) | \([2, 2]\) | \(24576\) | \(1.2380\) | |
| 13923.i4 | 13923f1 | \([1, -1, 0, -14796, -689045]\) | \(491411892194497/78897\) | \(57515913\) | \([2]\) | \(12288\) | \(0.89147\) | \(\Gamma_0(N)\)-optimal |
| 13923.i5 | 13923f4 | \([1, -1, 0, -2556, -1792715]\) | \(-2533811507137/1904381781393\) | \(-1388294318635497\) | \([2]\) | \(49152\) | \(1.5846\) | |
| 13923.i6 | 13923f6 | \([1, -1, 0, 101439, 5335330]\) | \(158346567380527343/108665074944153\) | \(-79216839634287537\) | \([2]\) | \(98304\) | \(1.9312\) |
Rank
sage: E.rank()
The elliptic curves in class 13923.i have rank \(1\).
Complex multiplication
The elliptic curves in class 13923.i do not have complex multiplication.Modular form 13923.2.a.i
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.
