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SageMath
E = EllipticCurve("nz1")
E.isogeny_class()
Elliptic curves in class 388080nz
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 388080.nz6 | 388080nz1 | \([0, 0, 0, 981078, 162126839]\) | \(76102438406144/52315569075\) | \(-71790454839524929200\) | \([2]\) | \(9437184\) | \(2.4986\) | \(\Gamma_0(N)\)-optimal |
| 388080.nz5 | 388080nz2 | \([0, 0, 0, -4313127, 1355440646]\) | \(404151985581136/197735855625\) | \(4341513553810503840000\) | \([2, 2]\) | \(18874368\) | \(2.8452\) | |
| 388080.nz2 | 388080nz3 | \([0, 0, 0, -56606907, 163811297594]\) | \(228410605013945764/187597265625\) | \(16475637537651600000000\) | \([2, 2]\) | \(37748736\) | \(3.1918\) | |
| 388080.nz4 | 388080nz4 | \([0, 0, 0, -36726627, -84728332654]\) | \(62380825826921284/787768887675\) | \(69185414902138325683200\) | \([2]\) | \(37748736\) | \(3.1918\) | |
| 388080.nz1 | 388080nz5 | \([0, 0, 0, -905531907, 10488267362594]\) | \(467508233804095622882/315748125\) | \(55460847399471360000\) | \([2]\) | \(75497472\) | \(3.5383\) | |
| 388080.nz3 | 388080nz6 | \([0, 0, 0, -44382387, 236530077266]\) | \(-55043996611705922/105743408203125\) | \(-18573725579062500000000000\) | \([2]\) | \(75497472\) | \(3.5383\) |
Rank
sage: E.rank()
The elliptic curves in class 388080nz have rank \(1\).
Complex multiplication
The elliptic curves in class 388080nz do not have complex multiplication.Modular form 388080.2.a.nz
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
