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SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 101430.bp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 101430.bp1 | 101430co4 | \([1, -1, 0, -48686409, 130767600865]\) | \(148809678420065817601/20700\) | \(1775358704700\) | \([2]\) | \(4718592\) | \(2.6753\) | |
| 101430.bp2 | 101430co6 | \([1, -1, 0, -11391039, -12672659777]\) | \(1905890658841300321/293666194803750\) | \(25186610397147993753750\) | \([2]\) | \(9437184\) | \(3.0219\) | |
| 101430.bp3 | 101430co3 | \([1, -1, 0, -3122289, 1931606473]\) | \(39248884582600321/3935264062500\) | \(337512333751326562500\) | \([2, 2]\) | \(4718592\) | \(2.6753\) | |
| 101430.bp4 | 101430co2 | \([1, -1, 0, -3042909, 2043802165]\) | \(36330796409313601/428490000\) | \(36749925187290000\) | \([2, 2]\) | \(2359296\) | \(2.3287\) | |
| 101430.bp5 | 101430co1 | \([1, -1, 0, -185229, 33710053]\) | \(-8194759433281/965779200\) | \(-82831135726483200\) | \([2]\) | \(1179648\) | \(1.9821\) | \(\Gamma_0(N)\)-optimal |
| 101430.bp6 | 101430co5 | \([1, -1, 0, 3876381, 9354395875]\) | \(75108181893694559/484313964843750\) | \(-41537730110778808593750\) | \([2]\) | \(9437184\) | \(3.0219\) |
Rank
sage: E.rank()
The elliptic curves in class 101430.bp have rank \(1\).
Complex multiplication
The elliptic curves in class 101430.bp do not have complex multiplication.Modular form 101430.2.a.bp
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.
