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SageMath
E = EllipticCurve("pf1")
E.isogeny_class()
Elliptic curves in class 369600.pf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 369600.pf1 | 369600pf4 | \([0, 1, 0, -2977633, -1978343137]\) | \(1425631925916578/270703125\) | \(554400000000000000\) | \([2]\) | \(6291456\) | \(2.4058\) | |
| 369600.pf2 | 369600pf3 | \([0, 1, 0, -1305633, 555616863]\) | \(120186986927618/4332064275\) | \(8872067635200000000\) | \([2]\) | \(6291456\) | \(2.4058\) | |
| 369600.pf3 | 369600pf2 | \([0, 1, 0, -205633, -24083137]\) | \(939083699236/300155625\) | \(307359360000000000\) | \([2, 2]\) | \(3145728\) | \(2.0593\) | |
| 369600.pf4 | 369600pf1 | \([0, 1, 0, 36367, -2545137]\) | \(20777545136/23059575\) | \(-5903251200000000\) | \([2]\) | \(1572864\) | \(1.7127\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 369600.pf have rank \(0\).
Complex multiplication
The elliptic curves in class 369600.pf do not have complex multiplication.Modular form 369600.2.a.pf
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
