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SageMath
E = EllipticCurve("ce1")
E.isogeny_class()
Elliptic curves in class 105966.ce
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 105966.ce1 | 105966bu4 | \([1, -1, 1, -10172894, -12486088879]\) | \(268498407453697/252\) | \(109273802654268\) | \([2]\) | \(3211264\) | \(2.4221\) | |
| 105966.ce2 | 105966bu6 | \([1, -1, 1, -6918224, 6938387201]\) | \(84448510979617/933897762\) | \(404962538666867241858\) | \([2]\) | \(6422528\) | \(2.7686\) | |
| 105966.ce3 | 105966bu3 | \([1, -1, 1, -787334, -94969807]\) | \(124475734657/63011844\) | \(27323586532291750596\) | \([2, 2]\) | \(3211264\) | \(2.4221\) | |
| 105966.ce4 | 105966bu2 | \([1, -1, 1, -635954, -194880607]\) | \(65597103937/63504\) | \(27536998268875536\) | \([2, 2]\) | \(1605632\) | \(2.0755\) | |
| 105966.ce5 | 105966bu1 | \([1, -1, 1, -30434, -4505119]\) | \(-7189057/16128\) | \(-6993523369873152\) | \([2]\) | \(802816\) | \(1.7289\) | \(\Gamma_0(N)\)-optimal |
| 105966.ce6 | 105966bu5 | \([1, -1, 1, 2921476, -735852175]\) | \(6359387729183/4218578658\) | \(-1829286237126185465922\) | \([2]\) | \(6422528\) | \(2.7686\) |
Rank
sage: E.rank()
The elliptic curves in class 105966.ce have rank \(1\).
Complex multiplication
The elliptic curves in class 105966.ce do not have complex multiplication.Modular form 105966.2.a.ce
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
