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SageMath
E = EllipticCurve("db1")
E.isogeny_class()
Elliptic curves in class 42042.db
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
42042.db1 | 42042dk2 | \([1, 0, 0, -2337511, 1374987977]\) | \(28826282175168869972161/9077387406557184\) | \(444791982921302016\) | \([7]\) | \(1119552\) | \(2.3619\) | |
42042.db2 | 42042dk1 | \([1, 0, 0, -72521, -7522971]\) | \(860833894093732321/8282804244\) | \(405857407956\) | \([]\) | \(159936\) | \(1.3890\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 42042.db have rank \(1\).
Complex multiplication
The elliptic curves in class 42042.db do not have complex multiplication.Modular form 42042.2.a.db
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.
\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.