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SageMath
E = EllipticCurve("dd1")
E.isogeny_class()
Elliptic curves in class 86640dd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
86640.db2 | 86640dd1 | \([0, 1, 0, -13651696, -988203820]\) | \(212883113611/122880000\) | \(162414036185430097920000\) | \([2]\) | \(14008320\) | \(3.1432\) | \(\Gamma_0(N)\)-optimal |
86640.db1 | 86640dd2 | \([0, 1, 0, -154124016, -734647036716]\) | \(306331959547531/900000000\) | \(1189555929092505600000000\) | \([2]\) | \(28016640\) | \(3.4898\) |
Rank
sage: E.rank()
The elliptic curves in class 86640dd have rank \(1\).
Complex multiplication
The elliptic curves in class 86640dd do not have complex multiplication.Modular form 86640.2.a.dd
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.