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SageMath
E = EllipticCurve("dm1")
E.isogeny_class()
Elliptic curves in class 217800.dm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 217800.dm1 | 217800fp3 | \([0, 0, 0, -1437489075, 20977558582750]\) | \(15897679904620804/2475\) | \(51142131572400000000\) | \([2]\) | \(47185920\) | \(3.6284\) | |
| 217800.dm2 | 217800fp6 | \([0, 0, 0, -762309075, -7945856707250]\) | \(1185450336504002/26043266205\) | \(1076289411580726004640000000\) | \([2]\) | \(94371840\) | \(3.9749\) | |
| 217800.dm3 | 217800fp4 | \([0, 0, 0, -103464075, 221844757750]\) | \(5927735656804/2401490025\) | \(49623159122568147600000000\) | \([2, 2]\) | \(47185920\) | \(3.6284\) | |
| 217800.dm4 | 217800fp2 | \([0, 0, 0, -89851575, 327709170250]\) | \(15529488955216/6125625\) | \(31644193910422500000000\) | \([2, 2]\) | \(23592960\) | \(3.2818\) | |
| 217800.dm5 | 217800fp1 | \([0, 0, 0, -4773450, 6709404625]\) | \(-37256083456/38671875\) | \(-12485871965917968750000\) | \([2]\) | \(11796480\) | \(2.9352\) | \(\Gamma_0(N)\)-optimal |
| 217800.dm6 | 217800fp5 | \([0, 0, 0, 337580925, 1614223822750]\) | \(102949393183198/86815346805\) | \(-3587815667721087649440000000\) | \([2]\) | \(94371840\) | \(3.9749\) |
Rank
sage: E.rank()
The elliptic curves in class 217800.dm have rank \(1\).
Complex multiplication
The elliptic curves in class 217800.dm do not have complex multiplication.Modular form 217800.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
