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SageMath
E = EllipticCurve("n1")
E.isogeny_class()
Elliptic curves in class 32490.n
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 32490.n1 | 32490ba7 | \([1, -1, 0, -17328609, -27760411787]\) | \(16778985534208729/81000\) | \(2778012227169000\) | \([2]\) | \(1327104\) | \(2.5857\) | |
| 32490.n2 | 32490ba8 | \([1, -1, 0, -1473489, -93331355]\) | \(10316097499609/5859375000\) | \(200955745599609375000\) | \([2]\) | \(1327104\) | \(2.5857\) | |
| 32490.n3 | 32490ba6 | \([1, -1, 0, -1083609, -433072787]\) | \(4102915888729/9000000\) | \(308668025241000000\) | \([2, 2]\) | \(663552\) | \(2.2391\) | |
| 32490.n4 | 32490ba5 | \([1, -1, 0, -937404, 349562578]\) | \(2656166199049/33750\) | \(1157505094653750\) | \([2]\) | \(442368\) | \(2.0364\) | |
| 32490.n5 | 32490ba4 | \([1, -1, 0, -222624, -34768130]\) | \(35578826569/5314410\) | \(182265382224558090\) | \([2]\) | \(442368\) | \(2.0364\) | |
| 32490.n6 | 32490ba2 | \([1, -1, 0, -60174, 5162080]\) | \(702595369/72900\) | \(2500211004452100\) | \([2, 2]\) | \(221184\) | \(1.6898\) | |
| 32490.n7 | 32490ba3 | \([1, -1, 0, -43929, -11586515]\) | \(-273359449/1536000\) | \(-52679342974464000\) | \([2]\) | \(331776\) | \(1.8925\) | |
| 32490.n8 | 32490ba1 | \([1, -1, 0, 4806, 392548]\) | \(357911/2160\) | \(-74080326057840\) | \([2]\) | \(110592\) | \(1.3432\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 32490.n have rank \(2\).
Complex multiplication
The elliptic curves in class 32490.n do not have complex multiplication.Modular form 32490.2.a.n
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 12 & 3 & 6 & 4 & 12 \\ 4 & 1 & 2 & 3 & 12 & 6 & 4 & 12 \\ 2 & 2 & 1 & 6 & 6 & 3 & 2 & 6 \\ 12 & 3 & 6 & 1 & 4 & 2 & 12 & 4 \\ 3 & 12 & 6 & 4 & 1 & 2 & 12 & 4 \\ 6 & 6 & 3 & 2 & 2 & 1 & 6 & 2 \\ 4 & 4 & 2 & 12 & 12 & 6 & 1 & 3 \\ 12 & 12 & 6 & 4 & 4 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
