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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 10005.k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 10005.k1 | 10005d4 | \([1, 1, 0, -5537, 156294]\) | \(18778886261717401/732035835\) | \(732035835\) | \([4]\) | \(7680\) | \(0.78531\) | |
| 10005.k2 | 10005d3 | \([1, 1, 0, -1667, -24804]\) | \(512787603508921/45649063125\) | \(45649063125\) | \([2]\) | \(7680\) | \(0.78531\) | |
| 10005.k3 | 10005d2 | \([1, 1, 0, -362, 2079]\) | \(5268932332201/900900225\) | \(900900225\) | \([2, 2]\) | \(3840\) | \(0.43874\) | |
| 10005.k4 | 10005d1 | \([1, 1, 0, 43, 216]\) | \(8477185319/21880935\) | \(-21880935\) | \([2]\) | \(1920\) | \(0.092165\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 10005.k have rank \(1\).
Complex multiplication
The elliptic curves in class 10005.k do not have complex multiplication.Modular form 10005.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
