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SageMath
E = EllipticCurve("n1")
E.isogeny_class()
Elliptic curves in class 222768n
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 222768.br4 | 222768n1 | \([0, 0, 0, -14691, -558686]\) | \(117433042273/22801233\) | \(68084116918272\) | \([2]\) | \(655360\) | \(1.3712\) | \(\Gamma_0(N)\)-optimal |
| 222768.br2 | 222768n2 | \([0, 0, 0, -222771, -40468430]\) | \(409460675852593/21538881\) | \(64314754043904\) | \([2, 2]\) | \(1310720\) | \(1.7178\) | |
| 222768.br3 | 222768n3 | \([0, 0, 0, -210531, -45112286]\) | \(-345608484635233/94427721297\) | \(-281959664949301248\) | \([2]\) | \(2621440\) | \(2.0644\) | |
| 222768.br1 | 222768n4 | \([0, 0, 0, -3564291, -2590048190]\) | \(1677087406638588673/4641\) | \(13857951744\) | \([2]\) | \(2621440\) | \(2.0644\) |
Rank
sage: E.rank()
The elliptic curves in class 222768n have rank \(1\).
Complex multiplication
The elliptic curves in class 222768n do not have complex multiplication.Modular form 222768.2.a.n
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.
