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SageMath
E = EllipticCurve("bt1")
E.isogeny_class()
Elliptic curves in class 55440bt
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 55440.s4 | 55440bt1 | \([0, 0, 0, 4197, -482998]\) | \(73929353373/954060800\) | \(-105511491993600\) | \([2]\) | \(165888\) | \(1.3716\) | \(\Gamma_0(N)\)-optimal |
| 55440.s2 | 55440bt2 | \([0, 0, 0, -72603, -7041718]\) | \(382704614800227/27778076480\) | \(3072033034076160\) | \([2]\) | \(331776\) | \(1.7182\) | |
| 55440.s3 | 55440bt3 | \([0, 0, 0, -38043, 13571658]\) | \(-75526045083/943250000\) | \(-76046294016000000\) | \([2]\) | \(497664\) | \(1.9209\) | |
| 55440.s1 | 55440bt4 | \([0, 0, 0, -1118043, 453563658]\) | \(1917114236485083/7117764500\) | \(573845334644736000\) | \([2]\) | \(995328\) | \(2.2675\) |
Rank
sage: E.rank()
The elliptic curves in class 55440bt have rank \(0\).
Complex multiplication
The elliptic curves in class 55440bt do not have complex multiplication.Modular form 55440.2.a.bt
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
