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SageMath
E = EllipticCurve("cf1")
E.isogeny_class()
Elliptic curves in class 38640cf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 38640.bi4 | 38640cf1 | \([0, -1, 0, -12160, -512000]\) | \(48551226272641/9273600\) | \(37984665600\) | \([2]\) | \(49152\) | \(1.0309\) | \(\Gamma_0(N)\)-optimal |
| 38640.bi3 | 38640cf2 | \([0, -1, 0, -13440, -396288]\) | \(65553197996161/20996010000\) | \(85999656960000\) | \([2, 2]\) | \(98304\) | \(1.3774\) | |
| 38640.bi6 | 38640cf3 | \([0, -1, 0, 38080, -2745600]\) | \(1490881681033919/1650501562500\) | \(-6760454400000000\) | \([2]\) | \(196608\) | \(1.7240\) | |
| 38640.bi2 | 38640cf4 | \([0, -1, 0, -85440, 9338112]\) | \(16840406336564161/604708416900\) | \(2476885675622400\) | \([2, 4]\) | \(196608\) | \(1.7240\) | |
| 38640.bi5 | 38640cf5 | \([0, -1, 0, 32160, 32952192]\) | \(898045580910239/115117148363070\) | \(-471519839695134720\) | \([4]\) | \(393216\) | \(2.0706\) | |
| 38640.bi1 | 38640cf6 | \([0, -1, 0, -1355040, 607573632]\) | \(67176973097223766561/91487391870\) | \(374732357099520\) | \([4]\) | \(393216\) | \(2.0706\) |
Rank
sage: E.rank()
The elliptic curves in class 38640cf have rank \(1\).
Complex multiplication
The elliptic curves in class 38640cf do not have complex multiplication.Modular form 38640.2.a.cf
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
