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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 56784.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
56784.c1 | 56784ci1 | \([0, -1, 0, -31857232, 69219276736]\) | \(-82318551880501/54432\) | \(-2364309954032173056\) | \([]\) | \(3744000\) | \(2.8425\) | \(\Gamma_0(N)\)-optimal |
56784.c2 | 56784ci2 | \([0, -1, 0, 65689568, 357910138624]\) | \(721710134999099/1691848015872\) | \(-73487160211562160325656576\) | \([]\) | \(18720000\) | \(3.6473\) |
Rank
sage: E.rank()
The elliptic curves in class 56784.c have rank \(0\).
Complex multiplication
The elliptic curves in class 56784.c do not have complex multiplication.Modular form 56784.2.a.c
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.
\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.