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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 3648.o
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3648.o1 | 3648g3 | \([0, -1, 0, -6497, 203745]\) | \(115714886617/1539\) | \(403439616\) | \([2]\) | \(3072\) | \(0.79471\) | |
3648.o2 | 3648g2 | \([0, -1, 0, -417, 3105]\) | \(30664297/3249\) | \(851705856\) | \([2, 2]\) | \(1536\) | \(0.44814\) | |
3648.o3 | 3648g1 | \([0, -1, 0, -97, -287]\) | \(389017/57\) | \(14942208\) | \([2]\) | \(768\) | \(0.10156\) | \(\Gamma_0(N)\)-optimal |
3648.o4 | 3648g4 | \([0, -1, 0, 543, 14433]\) | \(67419143/390963\) | \(-102488604672\) | \([2]\) | \(3072\) | \(0.79471\) |
Rank
sage: E.rank()
The elliptic curves in class 3648.o have rank \(0\).
Complex multiplication
The elliptic curves in class 3648.o do not have complex multiplication.Modular form 3648.2.a.o
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.