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SageMath
E = EllipticCurve("ch1")
E.isogeny_class()
Elliptic curves in class 17640ch
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 17640.d3 | 17640ch1 | \([0, 0, 0, -139503, -20053838]\) | \(13674725584/945\) | \(20748539992320\) | \([2]\) | \(73728\) | \(1.6090\) | \(\Gamma_0(N)\)-optimal |
| 17640.d2 | 17640ch2 | \([0, 0, 0, -148323, -17374322]\) | \(4108974916/893025\) | \(78429481170969600\) | \([2, 2]\) | \(147456\) | \(1.9556\) | |
| 17640.d1 | 17640ch3 | \([0, 0, 0, -765723, 242798038]\) | \(282678688658/18600435\) | \(3267148101350676480\) | \([2]\) | \(294912\) | \(2.3022\) | |
| 17640.d4 | 17640ch4 | \([0, 0, 0, 327957, -106057658]\) | \(22208984782/40516875\) | \(-7116749217365760000\) | \([2]\) | \(294912\) | \(2.3022\) |
Rank
sage: E.rank()
The elliptic curves in class 17640ch have rank \(0\).
Complex multiplication
The elliptic curves in class 17640ch do not have complex multiplication.Modular form 17640.2.a.ch
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
