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SageMath
E = EllipticCurve("fq1")
E.isogeny_class()
Elliptic curves in class 121680.fq
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 121680.fq1 | 121680bu4 | \([0, 0, 0, -854194107, -9609105685894]\) | \(19129597231400697604/26325\) | \(94854071816524800\) | \([2]\) | \(16515072\) | \(3.4239\) | |
| 121680.fq2 | 121680bu2 | \([0, 0, 0, -53387607, -150139469194]\) | \(18681746265374416/693005625\) | \(624258360142503840000\) | \([2, 2]\) | \(8257536\) | \(3.0773\) | |
| 121680.fq3 | 121680bu3 | \([0, 0, 0, -50923587, -164624457166]\) | \(-4053153720264484/903687890625\) | \(-3256162434076640400000000\) | \([2]\) | \(16515072\) | \(3.4239\) | |
| 121680.fq4 | 121680bu1 | \([0, 0, 0, -3491202, -2116794121]\) | \(83587439220736/13990184325\) | \(787645980941339941200\) | \([2]\) | \(4128768\) | \(2.7307\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 121680.fq have rank \(1\).
Complex multiplication
The elliptic curves in class 121680.fq do not have complex multiplication.Modular form 121680.2.a.fq
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
