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SageMath
E = EllipticCurve("mp1")
E.isogeny_class()
Elliptic curves in class 463680.mp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
463680.mp1 | 463680mp4 | \([0, 0, 0, -2551692, -1563989744]\) | \(9614816895690721/34652610405\) | \(6622216974564065280\) | \([2]\) | \(8388608\) | \(2.4726\) | |
463680.mp2 | 463680mp2 | \([0, 0, 0, -233292, 466576]\) | \(7347774183121/4251692025\) | \(812510999012966400\) | \([2, 2]\) | \(4194304\) | \(2.1260\) | |
463680.mp3 | 463680mp1 | \([0, 0, 0, -161292, 24860176]\) | \(2428257525121/8150625\) | \(1557608693760000\) | \([2]\) | \(2097152\) | \(1.7795\) | \(\Gamma_0(N)\)-optimal* |
463680.mp4 | 463680mp3 | \([0, 0, 0, 933108, 3732496]\) | \(470166844956479/272118787605\) | \(-52002710136827412480\) | \([2]\) | \(8388608\) | \(2.4726\) |
Rank
sage: E.rank()
The elliptic curves in class 463680.mp have rank \(1\).
Complex multiplication
The elliptic curves in class 463680.mp do not have complex multiplication.Modular form 463680.2.a.mp
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.