Show commands:
SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 5082.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 5082.d1 | 5082i3 | \([1, 1, 0, -162626, -25310424]\) | \(268498407453697/252\) | \(446433372\) | \([2]\) | \(20480\) | \(1.3880\) | |
| 5082.d2 | 5082i5 | \([1, 1, 0, -110596, 13974646]\) | \(84448510979617/933897762\) | \(1654456853146482\) | \([2]\) | \(40960\) | \(1.7346\) | |
| 5082.d3 | 5082i4 | \([1, 1, 0, -12586, -197600]\) | \(124475734657/63011844\) | \(111629325368484\) | \([2, 2]\) | \(20480\) | \(1.3880\) | |
| 5082.d4 | 5082i2 | \([1, 1, 0, -10166, -398460]\) | \(65597103937/63504\) | \(112501209744\) | \([2, 2]\) | \(10240\) | \(1.0415\) | |
| 5082.d5 | 5082i1 | \([1, 1, 0, -486, -9324]\) | \(-7189057/16128\) | \(-28571735808\) | \([2]\) | \(5120\) | \(0.69490\) | \(\Gamma_0(N)\)-optimal |
| 5082.d6 | 5082i6 | \([1, 1, 0, 46704, -1466406]\) | \(6359387729183/4218578658\) | \(-7473469425945138\) | \([2]\) | \(40960\) | \(1.7346\) |
Rank
sage: E.rank()
The elliptic curves in class 5082.d have rank \(1\).
Complex multiplication
The elliptic curves in class 5082.d do not have complex multiplication.Modular form 5082.2.a.d
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.
