Show commands:
SageMath
E = EllipticCurve("y1")
E.isogeny_class()
Elliptic curves in class 76050.y
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 76050.y1 | 76050cq4 | \([1, -1, 0, -31485492, -67771509584]\) | \(502270291349/1889568\) | \(12986143771455562500000\) | \([2]\) | \(6144000\) | \(3.1025\) | |
| 76050.y2 | 76050cq2 | \([1, -1, 0, -2016117, 1102222291]\) | \(131872229/18\) | \(123705835347656250\) | \([2]\) | \(1228800\) | \(2.2978\) | |
| 76050.y3 | 76050cq3 | \([1, -1, 0, -1065492, -2033889584]\) | \(-19465109/248832\) | \(-1710109467846000000000\) | \([2]\) | \(3072000\) | \(2.7559\) | |
| 76050.y4 | 76050cq1 | \([1, -1, 0, -114867, 20411041]\) | \(-24389/12\) | \(-82470556898437500\) | \([2]\) | \(614400\) | \(1.9512\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 76050.y have rank \(1\).
Complex multiplication
The elliptic curves in class 76050.y do not have complex multiplication.Modular form 76050.2.a.y
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.
\(\left(\begin{array}{rrrr} 1 & 5 & 2 & 10 \\ 5 & 1 & 10 & 2 \\ 2 & 10 & 1 & 5 \\ 10 & 2 & 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
