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SageMath
E = EllipticCurve("hq1")
E.isogeny_class()
Elliptic curves in class 286650.hq
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 286650.hq1 | 286650hq3 | \([1, -1, 0, -21570642, 38558117016]\) | \(828279937799497/193444524\) | \(259234163315178187500\) | \([2]\) | \(18874368\) | \(2.9076\) | |
| 286650.hq2 | 286650hq2 | \([1, -1, 0, -1505142, 453732516]\) | \(281397674377/96589584\) | \(129439280448182250000\) | \([2, 2]\) | \(9437184\) | \(2.5611\) | |
| 286650.hq3 | 286650hq1 | \([1, -1, 0, -623142, -183953484]\) | \(19968681097/628992\) | \(842909437188000000\) | \([2]\) | \(4718592\) | \(2.2145\) | \(\Gamma_0(N)\)-optimal |
| 286650.hq4 | 286650hq4 | \([1, -1, 0, 4448358, 3150668016]\) | \(7264187703863/7406095788\) | \(-9924876679549974187500\) | \([2]\) | \(18874368\) | \(2.9076\) |
Rank
sage: E.rank()
The elliptic curves in class 286650.hq have rank \(0\).
Complex multiplication
The elliptic curves in class 286650.hq do not have complex multiplication.Modular form 286650.2.a.hq
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
