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SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 359310.bp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 359310.bp1 | 359310bp3 | \([1, 0, 0, -480276670, -3941497920448]\) | \(12251751642270795504999416813281/378975018052733327169964740\) | \(378975018052733327169964740\) | \([2]\) | \(184000000\) | \(3.8755\) | |
| 359310.bp2 | 359310bp1 | \([1, 0, 0, -64482970, 199297549412]\) | \(29652328953610455812191224481/157579868246774400000\) | \(157579868246774400000\) | \([10]\) | \(36800000\) | \(3.0707\) | \(\Gamma_0(N)\)-optimal |
| 359310.bp3 | 359310bp2 | \([1, 0, 0, -63355450, 206603202500]\) | \(-28123906954810971599002864801/2165402912639419687500000\) | \(-2165402912639419687500000\) | \([10]\) | \(73600000\) | \(3.4173\) | |
| 359310.bp4 | 359310bp4 | \([1, 0, 0, 135057800, -13341716484850]\) | \(272447436027978082902346243199/77054400342590877150376155150\) | \(-77054400342590877150376155150\) | \([2]\) | \(368000000\) | \(4.2220\) |
Rank
sage: E.rank()
The elliptic curves in class 359310.bp have rank \(1\).
Complex multiplication
The elliptic curves in class 359310.bp do not have complex multiplication.Modular form 359310.2.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.
\(\left(\begin{array}{rrrr} 1 & 5 & 10 & 2 \\ 5 & 1 & 2 & 10 \\ 10 & 2 & 1 & 5 \\ 2 & 10 & 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
