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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 6864.k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6864.k1 | 6864k2 | \([0, -1, 0, -117952, 15612928]\) | \(44308125149913793/61165323648\) | \(250533165662208\) | \([2]\) | \(48384\) | \(1.6672\) | |
| 6864.k2 | 6864k1 | \([0, -1, 0, -5312, 384000]\) | \(-4047806261953/13066420224\) | \(-53520057237504\) | \([2]\) | \(24192\) | \(1.3207\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 6864.k have rank \(0\).
Complex multiplication
The elliptic curves in class 6864.k do not have complex multiplication.Modular form 6864.2.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.
