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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 51714.k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 51714.k1 | 51714l2 | \([1, -1, 0, -8683011, -8525445291]\) | \(45204035637810785581/6545053349462016\) | \(10482642530191907831808\) | \([2]\) | \(3612672\) | \(2.9506\) | |
| 51714.k2 | 51714l1 | \([1, -1, 0, 901629, -721631403]\) | \(50611530622079699/169662750916608\) | \(-271734067483801288704\) | \([2]\) | \(1806336\) | \(2.6040\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 51714.k have rank \(1\).
Complex multiplication
The elliptic curves in class 51714.k do not have complex multiplication.Modular form 51714.2.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.
