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SageMath
E = EllipticCurve("cz1")
E.isogeny_class()
Elliptic curves in class 84966cz
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 84966.cv5 | 84966cz1 | \([1, 1, 1, -56939, -11572135]\) | \(-7189057/16128\) | \(-45799663073971968\) | \([2]\) | \(983040\) | \(1.8855\) | \(\Gamma_0(N)\)-optimal |
| 84966.cv4 | 84966cz2 | \([1, 1, 1, -1189819, -499616839]\) | \(65597103937/63504\) | \(180336173353764624\) | \([2, 2]\) | \(1966080\) | \(2.2321\) | |
| 84966.cv3 | 84966cz3 | \([1, 1, 1, -1473039, -244152399]\) | \(124475734657/63011844\) | \(178938568010272948164\) | \([2, 2]\) | \(3932160\) | \(2.5787\) | |
| 84966.cv1 | 84966cz4 | \([1, 1, 1, -19032679, -31967284735]\) | \(268498407453697/252\) | \(715619735530812\) | \([2]\) | \(3932160\) | \(2.5787\) | |
| 84966.cv6 | 84966cz5 | \([1, 1, 1, 5465851, -1878954883]\) | \(6359387729183/4218578658\) | \(-11979754537912253192898\) | \([2]\) | \(7864320\) | \(2.9252\) | |
| 84966.cv2 | 84966cz6 | \([1, 1, 1, -12943449, 17746038645]\) | \(84448510979617/933897762\) | \(2652046307362131781122\) | \([2]\) | \(7864320\) | \(2.9252\) |
Rank
sage: E.rank()
The elliptic curves in class 84966cz have rank \(1\).
Complex multiplication
The elliptic curves in class 84966cz do not have complex multiplication.Modular form 84966.2.a.cz
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.
