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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 1850.k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1850.k1 | 1850j3 | \([1, -1, 1, -9880, -375503]\) | \(6825481747209/46250\) | \(722656250\) | \([2]\) | \(1536\) | \(0.88123\) | |
| 1850.k2 | 1850j2 | \([1, -1, 1, -630, -5503]\) | \(1767172329/136900\) | \(2139062500\) | \([2, 2]\) | \(768\) | \(0.53466\) | |
| 1850.k3 | 1850j1 | \([1, -1, 1, -130, 497]\) | \(15438249/2960\) | \(46250000\) | \([4]\) | \(384\) | \(0.18808\) | \(\Gamma_0(N)\)-optimal |
| 1850.k4 | 1850j4 | \([1, -1, 1, 620, -25503]\) | \(1689410871/18741610\) | \(-292837656250\) | \([2]\) | \(1536\) | \(0.88123\) |
Rank
sage: E.rank()
The elliptic curves in class 1850.k have rank \(1\).
Complex multiplication
The elliptic curves in class 1850.k do not have complex multiplication.Modular form 1850.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
