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SageMath
E = EllipticCurve("dm1")
E.isogeny_class()
Elliptic curves in class 202800.dm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 202800.dm1 | 202800fw5 | \([0, -1, 0, -609415408, 5790731701312]\) | \(81025909800741361/11088090\) | \(3425285926707840000000\) | \([2]\) | \(49545216\) | \(3.5444\) | |
| 202800.dm2 | 202800fw4 | \([0, -1, 0, -57123408, -166026170688]\) | \(66730743078481/60937500\) | \(18824555100000000000000\) | \([2]\) | \(24772608\) | \(3.1978\) | |
| 202800.dm3 | 202800fw3 | \([0, -1, 0, -38195408, 89956101312]\) | \(19948814692561/231344100\) | \(71465842174521600000000\) | \([2, 2]\) | \(24772608\) | \(3.1978\) | |
| 202800.dm4 | 202800fw6 | \([0, -1, 0, -7775408, 229279701312]\) | \(-168288035761/73415764890\) | \(-22679287981627904640000000\) | \([2]\) | \(49545216\) | \(3.5444\) | |
| 202800.dm5 | 202800fw2 | \([0, -1, 0, -4395408, -1303898688]\) | \(30400540561/15210000\) | \(4698608952960000000000\) | \([2, 2]\) | \(12386304\) | \(2.8513\) | |
| 202800.dm6 | 202800fw1 | \([0, -1, 0, 1012592, -157402688]\) | \(371694959/249600\) | \(-77105377689600000000\) | \([2]\) | \(6193152\) | \(2.5047\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 202800.dm have rank \(0\).
Complex multiplication
The elliptic curves in class 202800.dm do not have complex multiplication.Modular form 202800.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 4 & 8 \\ 8 & 1 & 4 & 8 & 2 & 4 \\ 2 & 4 & 1 & 2 & 2 & 4 \\ 4 & 8 & 2 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
