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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 1710.o
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1710.o1 | 1710o4 | \([1, -1, 1, -67298, 6505697]\) | \(46237740924063961/1806561830400\) | \(1316983574361600\) | \([6]\) | \(6912\) | \(1.6684\) | |
| 1710.o2 | 1710o2 | \([1, -1, 1, -9923, -375253]\) | \(148212258825961/1218375000\) | \(888195375000\) | \([2]\) | \(2304\) | \(1.1191\) | |
| 1710.o3 | 1710o1 | \([1, -1, 1, -203, -13669]\) | \(-1263214441/110808000\) | \(-80779032000\) | \([2]\) | \(1152\) | \(0.77253\) | \(\Gamma_0(N)\)-optimal |
| 1710.o4 | 1710o3 | \([1, -1, 1, 1822, 367841]\) | \(918046641959/80912056320\) | \(-58984889057280\) | \([6]\) | \(3456\) | \(1.3218\) |
Rank
sage: E.rank()
The elliptic curves in class 1710.o have rank \(0\).
Complex multiplication
The elliptic curves in class 1710.o do not have complex multiplication.Modular form 1710.2.a.o
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
