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SageMath
E = EllipticCurve("cq1")
E.isogeny_class()
Elliptic curves in class 333270.cq
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 333270.cq1 | 333270cq6 | \([1, -1, 0, -403209189, -3116227718097]\) | \(67176973097223766561/91487391870\) | \(9873151275682013551470\) | \([2]\) | \(69206016\) | \(3.4945\) | |
| 333270.cq2 | 333270cq4 | \([1, -1, 0, -25423839, -47779548327]\) | \(16840406336564161/604708416900\) | \(65259021551467536468900\) | \([2, 2]\) | \(34603008\) | \(3.1479\) | |
| 333270.cq3 | 333270cq2 | \([1, -1, 0, -3999339, 2058123573]\) | \(65553197996161/20996010000\) | \(2265850831230306810000\) | \([2, 2]\) | \(17301504\) | \(2.8013\) | |
| 333270.cq4 | 333270cq1 | \([1, -1, 0, -3618459, 2649782565]\) | \(48551226272641/9273600\) | \(1000789877147961600\) | \([2]\) | \(8650752\) | \(2.4548\) | \(\Gamma_0(N)\)-optimal |
| 333270.cq5 | 333270cq5 | \([1, -1, 0, 9569511, -169199474157]\) | \(898045580910239/115117148363070\) | \(-12423231190465460457818670\) | \([2]\) | \(69206016\) | \(3.4945\) | |
| 333270.cq6 | 333270cq3 | \([1, -1, 0, 11331081, 14025049425]\) | \(1490881681033919/1650501562500\) | \(-178119096787320314062500\) | \([2]\) | \(34603008\) | \(3.1479\) |
Rank
sage: E.rank()
The elliptic curves in class 333270.cq have rank \(1\).
Complex multiplication
The elliptic curves in class 333270.cq do not have complex multiplication.Modular form 333270.2.a.cq
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 8 & 4 & 2 & 1 & 8 & 4 \\ 4 & 2 & 4 & 8 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
