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SageMath
E = EllipticCurve("fe1")
E.isogeny_class()
Elliptic curves in class 428736fe
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 428736.fe3 | 428736fe1 | \([0, 1, 0, -3169, 66815]\) | \(13430356633/180873\) | \(47414771712\) | \([2]\) | \(425984\) | \(0.85518\) | \(\Gamma_0(N)\)-optimal |
| 428736.fe2 | 428736fe2 | \([0, 1, 0, -6049, -76609]\) | \(93391282153/44876601\) | \(11764131692544\) | \([2, 2]\) | \(851968\) | \(1.2018\) | |
| 428736.fe4 | 428736fe3 | \([0, 1, 0, 21791, -561025]\) | \(4365111505607/3058314567\) | \(-801718813851648\) | \([2]\) | \(1703936\) | \(1.5483\) | |
| 428736.fe1 | 428736fe4 | \([0, 1, 0, -79969, -8725249]\) | \(215751695207833/163381911\) | \(42829587677184\) | \([2]\) | \(1703936\) | \(1.5483\) |
Rank
sage: E.rank()
The elliptic curves in class 428736fe have rank \(1\).
Complex multiplication
The elliptic curves in class 428736fe do not have complex multiplication.Modular form 428736.2.a.fe
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.
