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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 39039.k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 39039.k1 | 39039z2 | \([1, 0, 0, -185143, -5207944]\) | \(66184391125/37202781\) | \(394516867788356313\) | \([2]\) | \(329472\) | \(2.0665\) | |
| 39039.k2 | 39039z1 | \([1, 0, 0, 45542, -640381]\) | \(985074875/586971\) | \(-6224533601469183\) | \([2]\) | \(164736\) | \(1.7199\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 39039.k have rank \(1\).
Complex multiplication
The elliptic curves in class 39039.k do not have complex multiplication.Modular form 39039.2.a.k
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.
