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SageMath
E = EllipticCurve("ja1")
E.isogeny_class()
Elliptic curves in class 463680.ja
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
463680.ja1 | 463680ja3 | \([0, 0, 0, -2551692, 1563989744]\) | \(9614816895690721/34652610405\) | \(6622216974564065280\) | \([2]\) | \(8388608\) | \(2.4726\) | \(\Gamma_0(N)\)-optimal* |
463680.ja2 | 463680ja2 | \([0, 0, 0, -233292, -466576]\) | \(7347774183121/4251692025\) | \(812510999012966400\) | \([2, 2]\) | \(4194304\) | \(2.1260\) | \(\Gamma_0(N)\)-optimal* |
463680.ja3 | 463680ja1 | \([0, 0, 0, -161292, -24860176]\) | \(2428257525121/8150625\) | \(1557608693760000\) | \([2]\) | \(2097152\) | \(1.7795\) | \(\Gamma_0(N)\)-optimal* |
463680.ja4 | 463680ja4 | \([0, 0, 0, 933108, -3732496]\) | \(470166844956479/272118787605\) | \(-52002710136827412480\) | \([2]\) | \(8388608\) | \(2.4726\) |
Rank
sage: E.rank()
The elliptic curves in class 463680.ja have rank \(0\).
Complex multiplication
The elliptic curves in class 463680.ja do not have complex multiplication.Modular form 463680.2.a.ja
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.