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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 43758.r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 43758.r1 | 43758u1 | \([1, -1, 1, -27185, 1575969]\) | \(3047678972871625/304559880768\) | \(222024153079872\) | \([2]\) | \(172032\) | \(1.4892\) | \(\Gamma_0(N)\)-optimal |
| 43758.r2 | 43758u2 | \([1, -1, 1, 33655, 7586961]\) | \(5783051584712375/37533175779528\) | \(-27361685143275912\) | \([2]\) | \(344064\) | \(1.8358\) |
Rank
sage: E.rank()
The elliptic curves in class 43758.r have rank \(1\).
Complex multiplication
The elliptic curves in class 43758.r do not have complex multiplication.Modular form 43758.2.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.
