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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 60840z
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 60840.ba4 | 60840z1 | \([0, 0, 0, -3491202, 2116794121]\) | \(83587439220736/13990184325\) | \(787645980941339941200\) | \([2]\) | \(2064384\) | \(2.7307\) | \(\Gamma_0(N)\)-optimal |
| 60840.ba2 | 60840z2 | \([0, 0, 0, -53387607, 150139469194]\) | \(18681746265374416/693005625\) | \(624258360142503840000\) | \([2, 2]\) | \(4128768\) | \(3.0773\) | |
| 60840.ba3 | 60840z3 | \([0, 0, 0, -50923587, 164624457166]\) | \(-4053153720264484/903687890625\) | \(-3256162434076640400000000\) | \([2]\) | \(8257536\) | \(3.4239\) | |
| 60840.ba1 | 60840z4 | \([0, 0, 0, -854194107, 9609105685894]\) | \(19129597231400697604/26325\) | \(94854071816524800\) | \([2]\) | \(8257536\) | \(3.4239\) |
Rank
sage: E.rank()
The elliptic curves in class 60840z have rank \(1\).
Complex multiplication
The elliptic curves in class 60840z do not have complex multiplication.Modular form 60840.2.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 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.
