Show commands:
SageMath
E = EllipticCurve("dm1")
E.isogeny_class()
Elliptic curves in class 162624dm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 162624.do4 | 162624dm1 | \([0, -1, 0, -263457, 52120257]\) | \(4354703137/1617\) | \(750941408329728\) | \([2]\) | \(1228800\) | \(1.8217\) | \(\Gamma_0(N)\)-optimal |
| 162624.do3 | 162624dm2 | \([0, -1, 0, -302177, 35834625]\) | \(6570725617/2614689\) | \(1214272257269170176\) | \([2, 2]\) | \(2457600\) | \(2.1683\) | |
| 162624.do6 | 162624dm3 | \([0, -1, 0, 975583, 258420417]\) | \(221115865823/190238433\) | \(-88347505748584169472\) | \([2]\) | \(4915200\) | \(2.5149\) | |
| 162624.do2 | 162624dm4 | \([0, -1, 0, -2199457, -1229651135]\) | \(2533811507137/58110129\) | \(26986581391145435136\) | \([2, 2]\) | \(4915200\) | \(2.5149\) | |
| 162624.do5 | 162624dm5 | \([0, -1, 0, 239903, -3810006143]\) | \(3288008303/13504609503\) | \(-6271595843615243108352\) | \([2]\) | \(9830400\) | \(2.8615\) | |
| 162624.do1 | 162624dm6 | \([0, -1, 0, -34995297, -79670741247]\) | \(10206027697760497/5557167\) | \(2580771065741180928\) | \([2]\) | \(9830400\) | \(2.8615\) |
Rank
sage: E.rank()
The elliptic curves in class 162624dm have rank \(0\).
Complex multiplication
The elliptic curves in class 162624dm do not have complex multiplication.Modular form 162624.2.a.dm
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.
