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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 8664.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 8664.f1 | 8664i3 | \([0, -1, 0, -459312, -116340660]\) | \(111223479026/3518667\) | \(339023439791364096\) | \([2]\) | \(69120\) | \(2.1384\) | |
| 8664.f2 | 8664i2 | \([0, -1, 0, -69432, 4522140]\) | \(768400132/263169\) | \(12678161875854336\) | \([2, 2]\) | \(34560\) | \(1.7918\) | |
| 8664.f3 | 8664i1 | \([0, -1, 0, -62212, 5992132]\) | \(2211014608/513\) | \(6178441459968\) | \([4]\) | \(17280\) | \(1.4453\) | \(\Gamma_0(N)\)-optimal |
| 8664.f4 | 8664i4 | \([0, -1, 0, 204928, 31189932]\) | \(9878111854/10097379\) | \(-972882106052401152\) | \([2]\) | \(69120\) | \(2.1384\) |
Rank
sage: E.rank()
The elliptic curves in class 8664.f have rank \(1\).
Complex multiplication
The elliptic curves in class 8664.f do not have complex multiplication.Modular form 8664.2.a.f
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.
