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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 2112r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2112.r4 | 2112r1 | \([0, 1, 0, -129, -33]\) | \(912673/528\) | \(138412032\) | \([2]\) | \(768\) | \(0.25162\) | \(\Gamma_0(N)\)-optimal |
2112.r2 | 2112r2 | \([0, 1, 0, -1409, -20769]\) | \(1180932193/4356\) | \(1141899264\) | \([2, 2]\) | \(1536\) | \(0.59819\) | |
2112.r1 | 2112r3 | \([0, 1, 0, -22529, -1309089]\) | \(4824238966273/66\) | \(17301504\) | \([2]\) | \(3072\) | \(0.94477\) | |
2112.r3 | 2112r4 | \([0, 1, 0, -769, -39073]\) | \(-192100033/2371842\) | \(-621764149248\) | \([4]\) | \(3072\) | \(0.94477\) |
Rank
sage: E.rank()
The elliptic curves in class 2112r have rank \(1\).
Complex multiplication
The elliptic curves in class 2112r do not have complex multiplication.Modular form 2112.2.a.r
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.