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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 117117.r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 117117.r1 | 117117e2 | \([1, -1, 1, -27407594, 55234236178]\) | \(52652025714902099823/35153041\) | \(1520142321288591\) | \([2]\) | \(4036608\) | \(2.6637\) | |
| 117117.r2 | 117117e1 | \([1, -1, 1, -1712639, 863711398]\) | \(-12846937564867743/10503585169\) | \(-454212320938495119\) | \([2]\) | \(2018304\) | \(2.3171\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 117117.r have rank \(1\).
Complex multiplication
The elliptic curves in class 117117.r do not have complex multiplication.Modular form 117117.2.a.r
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
