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SageMath
E = EllipticCurve("v1")
E.isogeny_class()
Elliptic curves in class 2112v
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2112.j3 | 2112v1 | \([0, -1, 0, -417, 3393]\) | \(30664297/297\) | \(77856768\) | \([2]\) | \(768\) | \(0.33394\) | \(\Gamma_0(N)\)-optimal |
2112.j2 | 2112v2 | \([0, -1, 0, -737, -2175]\) | \(169112377/88209\) | \(23123460096\) | \([2, 2]\) | \(1536\) | \(0.68052\) | |
2112.j1 | 2112v3 | \([0, -1, 0, -9377, -346047]\) | \(347873904937/395307\) | \(103627358208\) | \([2]\) | \(3072\) | \(1.0271\) | |
2112.j4 | 2112v4 | \([0, -1, 0, 2783, -19775]\) | \(9090072503/5845851\) | \(-1532454764544\) | \([2]\) | \(3072\) | \(1.0271\) |
Rank
sage: E.rank()
The elliptic curves in class 2112v have rank \(1\).
Complex multiplication
The elliptic curves in class 2112v do not have complex multiplication.Modular form 2112.2.a.v
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.