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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 51842j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
51842.c2 | 51842j1 | \([1, 0, 1, -113482, -14757396]\) | \(-3183010111/8464\) | \(-429770687222128\) | \([2]\) | \(270336\) | \(1.6826\) | \(\Gamma_0(N)\)-optimal |
51842.c1 | 51842j2 | \([1, 0, 1, -1816862, -942758820]\) | \(13062552753151/92\) | \(4671420513284\) | \([2]\) | \(540672\) | \(2.0292\) |
Rank
sage: E.rank()
The elliptic curves in class 51842j have rank \(1\).
Complex multiplication
The elliptic curves in class 51842j do not have complex multiplication.Modular form 51842.2.a.j
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 Cremona 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 Cremona labels.