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SageMath
E = EllipticCurve("ch1")
E.isogeny_class()
Elliptic curves in class 162450.ch
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 162450.ch1 | 162450ei3 | \([1, -1, 0, -34765992, -78891216084]\) | \(8671983378625/82308\) | \(44107374690167062500\) | \([2]\) | \(14929920\) | \(2.9317\) | |
| 162450.ch2 | 162450ei4 | \([1, -1, 0, -33953742, -82753464834]\) | \(-8078253774625/846825858\) | \(-453798724499783822531250\) | \([2]\) | \(29859840\) | \(3.2783\) | |
| 162450.ch3 | 162450ei1 | \([1, -1, 0, -651492, 15622416]\) | \(57066625/32832\) | \(17594077438737000000\) | \([2]\) | \(4976640\) | \(2.3824\) | \(\Gamma_0(N)\)-optimal |
| 162450.ch4 | 162450ei2 | \([1, -1, 0, 2597508, 122839416]\) | \(3616805375/2105352\) | \(-1128220215759010125000\) | \([2]\) | \(9953280\) | \(2.7290\) |
Rank
sage: E.rank()
The elliptic curves in class 162450.ch have rank \(1\).
Complex multiplication
The elliptic curves in class 162450.ch do not have complex multiplication.Modular form 162450.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
