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SageMath
E = EllipticCurve("u1")
E.isogeny_class()
Elliptic curves in class 2112u
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2112.e4 | 2112u1 | \([0, -1, 0, -129, 33]\) | \(912673/528\) | \(138412032\) | \([2]\) | \(768\) | \(0.25162\) | \(\Gamma_0(N)\)-optimal |
2112.e2 | 2112u2 | \([0, -1, 0, -1409, 20769]\) | \(1180932193/4356\) | \(1141899264\) | \([2, 2]\) | \(1536\) | \(0.59819\) | |
2112.e1 | 2112u3 | \([0, -1, 0, -22529, 1309089]\) | \(4824238966273/66\) | \(17301504\) | \([4]\) | \(3072\) | \(0.94477\) | |
2112.e3 | 2112u4 | \([0, -1, 0, -769, 39073]\) | \(-192100033/2371842\) | \(-621764149248\) | \([2]\) | \(3072\) | \(0.94477\) |
Rank
sage: E.rank()
The elliptic curves in class 2112u have rank \(0\).
Complex multiplication
The elliptic curves in class 2112u do not have complex multiplication.Modular form 2112.2.a.u
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.