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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 122034.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 122034.e1 | 122034e4 | \([1, 0, 1, -650887, 202064240]\) | \(4824238966273/66\) | \(417209961234\) | \([2]\) | \(1290240\) | \(1.7856\) | |
| 122034.e2 | 122034e2 | \([1, 0, 1, -40717, 3148820]\) | \(1180932193/4356\) | \(27535857441444\) | \([2, 2]\) | \(645120\) | \(1.4391\) | |
| 122034.e3 | 122034e3 | \([1, 0, 1, -22227, 6025864]\) | \(-192100033/2371842\) | \(-14993274376866258\) | \([2]\) | \(1290240\) | \(1.7856\) | |
| 122034.e4 | 122034e1 | \([1, 0, 1, -3737, -1876]\) | \(912673/528\) | \(3337679689872\) | \([2]\) | \(322560\) | \(1.0925\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 122034.e have rank \(1\).
Complex multiplication
The elliptic curves in class 122034.e do not have complex multiplication.Modular form 122034.2.a.e
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
