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SageMath
E = EllipticCurve("cv1")
E.isogeny_class()
Elliptic curves in class 65520cv
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 65520.e5 | 65520cv1 | \([0, 0, 0, -53283, 74582818]\) | \(-5602762882081/801531494400\) | \(-2393360217774489600\) | \([2]\) | \(1179648\) | \(2.2060\) | \(\Gamma_0(N)\)-optimal |
| 65520.e4 | 65520cv2 | \([0, 0, 0, -3002403, 1986202402]\) | \(1002404925316922401/9348917760000\) | \(27915718848675840000\) | \([2, 2]\) | \(2359296\) | \(2.5526\) | |
| 65520.e3 | 65520cv3 | \([0, 0, 0, -5260323, -1405645022]\) | \(5391051390768345121/2833965225000000\) | \(8462174818406400000000\) | \([2, 2]\) | \(4718592\) | \(2.8992\) | |
| 65520.e2 | 65520cv4 | \([0, 0, 0, -47930403, 127721703202]\) | \(4078208988807294650401/359723582400\) | \(1074128861469081600\) | \([2]\) | \(4718592\) | \(2.8992\) | |
| 65520.e6 | 65520cv5 | \([0, 0, 0, 19939677, -10966525022]\) | \(293623352309352854879/187320324116835000\) | \(-559335490687683440640000\) | \([2]\) | \(9437184\) | \(3.2457\) | |
| 65520.e1 | 65520cv6 | \([0, 0, 0, -66587043, -208923000158]\) | \(10934663514379917006241/12996826171875000\) | \(38808315000000000000000\) | \([2]\) | \(9437184\) | \(3.2457\) |
Rank
sage: E.rank()
The elliptic curves in class 65520cv have rank \(0\).
Complex multiplication
The elliptic curves in class 65520cv do not have complex multiplication.Modular form 65520.2.a.cv
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
