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SageMath
E = EllipticCurve("db1")
E.isogeny_class()
Elliptic curves in class 226512.db
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
226512.db1 | 226512bl2 | \([0, 0, 0, -226875, -31286486]\) | \(244140625/61347\) | \(324516599784419328\) | \([2]\) | \(1966080\) | \(2.0700\) | |
226512.db2 | 226512bl1 | \([0, 0, 0, 34485, -3111878]\) | \(857375/1287\) | \(-6808040554917888\) | \([2]\) | \(983040\) | \(1.7235\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 226512.db have rank \(2\).
Complex multiplication
The elliptic curves in class 226512.db do not have complex multiplication.Modular form 226512.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.