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SageMath
E = EllipticCurve("fx1")
E.isogeny_class()
Elliptic curves in class 193200.fx
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 193200.fx1 | 193200y4 | \([0, 1, 0, -18400408, -30386240812]\) | \(10765299591712341649/20708625\) | \(1325352000000000\) | \([2]\) | \(6488064\) | \(2.5833\) | |
| 193200.fx2 | 193200y2 | \([0, 1, 0, -1150408, -474740812]\) | \(2630872462131649/3645140625\) | \(233289000000000000\) | \([2, 2]\) | \(3244032\) | \(2.2368\) | |
| 193200.fx3 | 193200y3 | \([0, 1, 0, -828408, -745864812]\) | \(-982374577874929/3183837890625\) | \(-203765625000000000000\) | \([2]\) | \(6488064\) | \(2.5833\) | |
| 193200.fx4 | 193200y1 | \([0, 1, 0, -92408, -2872812]\) | \(1363569097969/734582625\) | \(47013288000000000\) | \([2]\) | \(1622016\) | \(1.8902\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 193200.fx have rank \(1\).
Complex multiplication
The elliptic curves in class 193200.fx do not have complex multiplication.Modular form 193200.2.a.fx
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.
