Show commands:
SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 112710.q
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 112710.q1 | 112710o4 | \([1, 1, 0, -88920537, -322758546111]\) | \(3221338935539503699129/200350631681460\) | \(4835977196404826770740\) | \([2]\) | \(14745600\) | \(3.2202\) | |
| 112710.q2 | 112710o3 | \([1, 1, 0, -29768017, 58615358521]\) | \(120859257477573578809/8424459021127500\) | \(203345960910137489047500\) | \([4]\) | \(14745600\) | \(3.2202\) | |
| 112710.q3 | 112710o2 | \([1, 1, 0, -5890837, -4406070371]\) | \(936615448738871929/194959225328400\) | \(4705841753550802659600\) | \([2, 2]\) | \(7372800\) | \(2.8736\) | |
| 112710.q4 | 112710o1 | \([1, 1, 0, 790843, -414434739]\) | \(2266209994236551/4390344840960\) | \(-105972251532466026240\) | \([2]\) | \(3686400\) | \(2.5270\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 112710.q have rank \(1\).
Complex multiplication
The elliptic curves in class 112710.q do not have complex multiplication.Modular form 112710.2.a.q
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
