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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 22218.z
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 22218.z1 | 22218z4 | \([1, 1, 1, -710987, -231045667]\) | \(268498407453697/252\) | \(37305044028\) | \([2]\) | \(180224\) | \(1.7568\) | |
| 22218.z2 | 22218z6 | \([1, 1, 1, -483517, 127965473]\) | \(84448510979617/933897762\) | \(138250385432780418\) | \([2]\) | \(360448\) | \(2.1034\) | |
| 22218.z3 | 22218z3 | \([1, 1, 1, -55027, -1781299]\) | \(124475734657/63011844\) | \(9328014344069316\) | \([2, 2]\) | \(180224\) | \(1.7568\) | |
| 22218.z4 | 22218z2 | \([1, 1, 1, -44447, -3622219]\) | \(65597103937/63504\) | \(9400871095056\) | \([2, 2]\) | \(90112\) | \(1.4103\) | |
| 22218.z5 | 22218z1 | \([1, 1, 1, -2127, -84267]\) | \(-7189057/16128\) | \(-2387522817792\) | \([2]\) | \(45056\) | \(1.0637\) | \(\Gamma_0(N)\)-optimal |
| 22218.z6 | 22218z5 | \([1, 1, 1, 204183, -13497591]\) | \(6359387729183/4218578658\) | \(-624501041953456962\) | \([2]\) | \(360448\) | \(2.1034\) |
Rank
sage: E.rank()
The elliptic curves in class 22218.z have rank \(0\).
Complex multiplication
The elliptic curves in class 22218.z do not have complex multiplication.Modular form 22218.2.a.z
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.
