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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 29040o
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 29040.bk5 | 29040o1 | \([0, -1, 0, -1855, 31330]\) | \(24918016/45\) | \(1275523920\) | \([2]\) | \(20480\) | \(0.63923\) | \(\Gamma_0(N)\)-optimal |
| 29040.bk4 | 29040o2 | \([0, -1, 0, -2460, 9792]\) | \(3631696/2025\) | \(918377222400\) | \([2, 2]\) | \(40960\) | \(0.98580\) | |
| 29040.bk6 | 29040o3 | \([0, -1, 0, 9640, 67872]\) | \(54607676/32805\) | \(-59510844011520\) | \([2]\) | \(81920\) | \(1.3324\) | |
| 29040.bk2 | 29040o4 | \([0, -1, 0, -24240, -1436400]\) | \(868327204/5625\) | \(10204191360000\) | \([2, 2]\) | \(81920\) | \(1.3324\) | |
| 29040.bk3 | 29040o5 | \([0, -1, 0, -9720, -3155568]\) | \(-27995042/1171875\) | \(-4251746400000000\) | \([2]\) | \(163840\) | \(1.6789\) | |
| 29040.bk1 | 29040o6 | \([0, -1, 0, -387240, -92622000]\) | \(1770025017602/75\) | \(272111769600\) | \([2]\) | \(163840\) | \(1.6789\) |
Rank
sage: E.rank()
The elliptic curves in class 29040o have rank \(1\).
Complex multiplication
The elliptic curves in class 29040o do not have complex multiplication.Modular form 29040.2.a.o
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
