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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 54777a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
54777.c3 | 54777a1 | \([1, 1, 0, -1461, 17976]\) | \(389017/57\) | \(50587709817\) | \([2]\) | \(43200\) | \(0.77884\) | \(\Gamma_0(N)\)-optimal |
54777.c2 | 54777a2 | \([1, 1, 0, -6266, -175185]\) | \(30664297/3249\) | \(2883499459569\) | \([2, 2]\) | \(86400\) | \(1.1254\) | |
54777.c4 | 54777a3 | \([1, 1, 0, 8149, -846924]\) | \(67419143/390963\) | \(-346981101634803\) | \([2]\) | \(172800\) | \(1.4720\) | |
54777.c1 | 54777a4 | \([1, 1, 0, -97561, -11769650]\) | \(115714886617/1539\) | \(1365868165059\) | \([2]\) | \(172800\) | \(1.4720\) |
Rank
sage: E.rank()
The elliptic curves in class 54777a have rank \(0\).
Complex multiplication
The elliptic curves in class 54777a do not have complex multiplication.Modular form 54777.2.a.a
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.