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SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 49686.bd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 49686.bd1 | 49686bk4 | \([1, 0, 1, -11129837, -14292541924]\) | \(268498407453697/252\) | \(143103051514332\) | \([2]\) | \(1474560\) | \(2.4445\) | |
| 49686.bd2 | 49686bk6 | \([1, 0, 1, -7569007, 7937471336]\) | \(84448510979617/933897762\) | \(530331823589703832242\) | \([2]\) | \(2949120\) | \(2.7911\) | |
| 49686.bd3 | 49686bk3 | \([1, 0, 1, -861397, -108977620]\) | \(124475734657/63011844\) | \(35782488722004173604\) | \([2, 2]\) | \(1474560\) | \(2.4445\) | |
| 49686.bd4 | 49686bk2 | \([1, 0, 1, -695777, -223255420]\) | \(65597103937/63504\) | \(36061968981611664\) | \([2, 2]\) | \(737280\) | \(2.0980\) | |
| 49686.bd5 | 49686bk1 | \([1, 0, 1, -33297, -5167004]\) | \(-7189057/16128\) | \(-9158595296917248\) | \([2]\) | \(368640\) | \(1.7514\) | \(\Gamma_0(N)\)-optimal |
| 49686.bd6 | 49686bk5 | \([1, 0, 1, 3196293, -840984896]\) | \(6359387729183/4218578658\) | \(-2395601107194585540978\) | \([2]\) | \(2949120\) | \(2.7911\) |
Rank
sage: E.rank()
The elliptic curves in class 49686.bd have rank \(1\).
Complex multiplication
The elliptic curves in class 49686.bd do not have complex multiplication.Modular form 49686.2.a.bd
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.
