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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 328560.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 328560.f1 | 328560f5 | \([0, -1, 0, -4381256, 3531227856]\) | \(1770025017602/75\) | \(394095576422400\) | \([2]\) | \(6635520\) | \(2.2855\) | |
| 328560.f2 | 328560f3 | \([0, -1, 0, -274256, 55063056]\) | \(868327204/5625\) | \(14778584115840000\) | \([2, 2]\) | \(3317760\) | \(1.9389\) | |
| 328560.f3 | 328560f6 | \([0, -1, 0, -109976, 120249360]\) | \(-27995042/1171875\) | \(-6157743381600000000\) | \([2]\) | \(6635520\) | \(2.2855\) | |
| 328560.f4 | 328560f2 | \([0, -1, 0, -27836, -332160]\) | \(3631696/2025\) | \(1330072570425600\) | \([2, 2]\) | \(1658880\) | \(1.5923\) | |
| 328560.f5 | 328560f1 | \([0, -1, 0, -20991, -1161774]\) | \(24918016/45\) | \(1847323014480\) | \([2]\) | \(829440\) | \(1.2457\) | \(\Gamma_0(N)\)-optimal |
| 328560.f6 | 328560f4 | \([0, -1, 0, 109064, -2741600]\) | \(54607676/32805\) | \(-86188702563578880\) | \([2]\) | \(3317760\) | \(1.9389\) |
Rank
sage: E.rank()
The elliptic curves in class 328560.f have rank \(1\).
Complex multiplication
The elliptic curves in class 328560.f do not have complex multiplication.Modular form 328560.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
