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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 2142e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2142.d2 | 2142e1 | \([1, -1, 0, -162, 832]\) | \(647214625/3332\) | \(2429028\) | \([2]\) | \(384\) | \(0.071129\) | \(\Gamma_0(N)\)-optimal |
2142.d1 | 2142e2 | \([1, -1, 0, -252, -122]\) | \(2433138625/1387778\) | \(1011690162\) | \([2]\) | \(768\) | \(0.41770\) |
Rank
sage: E.rank()
The elliptic curves in class 2142e have rank \(1\).
Complex multiplication
The elliptic curves in class 2142e do not have complex multiplication.Modular form 2142.2.a.e
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.