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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 85680.o
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 85680.o1 | 85680dk4 | \([0, 0, 0, -27165963, -46367463238]\) | \(742525803457216841161/118657634071410000\) | \(354309796815085117440000\) | \([2]\) | \(11796480\) | \(3.2411\) | |
| 85680.o2 | 85680dk2 | \([0, 0, 0, -7570443, 7320342458]\) | \(16069416876629693641/1546622367494400\) | \(4618189643380398489600\) | \([2, 2]\) | \(5898240\) | \(2.8946\) | |
| 85680.o3 | 85680dk1 | \([0, 0, 0, -7386123, 7726251962]\) | \(14924020698027934921/161083883520\) | \(480993898848583680\) | \([2]\) | \(2949120\) | \(2.5480\) | \(\Gamma_0(N)\)-optimal |
| 85680.o4 | 85680dk3 | \([0, 0, 0, 9075957, 35029939898]\) | \(27689398696638536759/193555307298039120\) | \(-577953050707028043694080\) | \([2]\) | \(11796480\) | \(3.2411\) |
Rank
sage: E.rank()
The elliptic curves in class 85680.o have rank \(1\).
Complex multiplication
The elliptic curves in class 85680.o do not have complex multiplication.Modular form 85680.2.a.o
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.
