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SageMath
E = EllipticCurve("ck1")
E.isogeny_class()
Elliptic curves in class 34650ck
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 34650.co3 | 34650ck1 | \([1, -1, 1, -6605, 983397]\) | \(-75526045083/943250000\) | \(-397933593750000\) | \([2]\) | \(165888\) | \(1.4832\) | \(\Gamma_0(N)\)-optimal |
| 34650.co2 | 34650ck2 | \([1, -1, 1, -194105, 32858397]\) | \(1917114236485083/7117764500\) | \(3002806898437500\) | \([2]\) | \(331776\) | \(1.8298\) | |
| 34650.co4 | 34650ck3 | \([1, -1, 1, 59020, -25485353]\) | \(73929353373/954060800\) | \(-293418417600000000\) | \([2]\) | \(497664\) | \(2.0325\) | |
| 34650.co1 | 34650ck4 | \([1, -1, 1, -1020980, -371085353]\) | \(382704614800227/27778076480\) | \(8543060614935000000\) | \([2]\) | \(995328\) | \(2.3791\) |
Rank
sage: E.rank()
The elliptic curves in class 34650ck have rank \(0\).
Complex multiplication
The elliptic curves in class 34650ck do not have complex multiplication.Modular form 34650.2.a.ck
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.
