Show commands:
SageMath
E = EllipticCurve("dq1")
E.isogeny_class()
Elliptic curves in class 363888dq
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 363888.dq3 | 363888dq1 | \([0, 0, 0, -8396499, 5479229842]\) | \(466025146777/177366672\) | \(24916159324072695103488\) | \([2]\) | \(22118400\) | \(2.9968\) | \(\Gamma_0(N)\)-optimal |
| 363888.dq2 | 363888dq2 | \([0, 0, 0, -59340819, -172041347630]\) | \(164503536215257/4178071044\) | \(586928099996140986187776\) | \([2, 2]\) | \(44236800\) | \(3.3434\) | |
| 363888.dq4 | 363888dq3 | \([0, 0, 0, 9797901, -549054787790]\) | \(740480746823/927484650666\) | \(-130291418709293718396248064\) | \([2]\) | \(88473600\) | \(3.6900\) | |
| 363888.dq1 | 363888dq4 | \([0, 0, 0, -943588659, -11156344865678]\) | \(661397832743623417/443352042\) | \(62281318077191915347968\) | \([2]\) | \(88473600\) | \(3.6900\) |
Rank
sage: E.rank()
The elliptic curves in class 363888dq have rank \(0\).
Complex multiplication
The elliptic curves in class 363888dq do not have complex multiplication.Modular form 363888.2.a.dq
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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
