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SageMath
E = EllipticCurve("cq1")
E.isogeny_class()
Elliptic curves in class 123840.cq
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
123840.cq1 | 123840bk3 | \([0, 0, 0, -19760268, 14686910192]\) | \(4465136636671380769/2096375976562500\) | \(400623687936000000000000\) | \([2]\) | \(13271040\) | \(3.2232\) | |
123840.cq2 | 123840bk1 | \([0, 0, 0, -10118028, -12387131152]\) | \(599437478278595809/33854760000\) | \(6469745387765760000\) | \([2]\) | \(4423680\) | \(2.6739\) | \(\Gamma_0(N)\)-optimal |
123840.cq3 | 123840bk2 | \([0, 0, 0, -9542028, -13859617552]\) | \(-502780379797811809/143268096832200\) | \(-27378959670489592627200\) | \([2]\) | \(8847360\) | \(3.0205\) | |
123840.cq4 | 123840bk4 | \([0, 0, 0, 70239732, 111202910192]\) | \(200541749524551119231/144008551960031250\) | \(-27520462849012604928000000\) | \([2]\) | \(26542080\) | \(3.5698\) |
Rank
sage: E.rank()
The elliptic curves in class 123840.cq have rank \(0\).
Complex multiplication
The elliptic curves in class 123840.cq do not have complex multiplication.Modular form 123840.2.a.cq
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.