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SageMath
E = EllipticCurve("ir1")
E.isogeny_class()
Elliptic curves in class 397488ir
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
397488.ir3 | 397488ir1 | \([0, 1, 0, -930232, -309720748]\) | \(38272753/4368\) | \(10159935049380200448\) | \([2]\) | \(9289728\) | \(2.3794\) | \(\Gamma_0(N)\)-optimal |
397488.ir2 | 397488ir2 | \([0, 1, 0, -3580152, 2277661140]\) | \(2181825073/298116\) | \(693415567120198680576\) | \([2, 2]\) | \(18579456\) | \(2.7260\) | |
397488.ir1 | 397488ir3 | \([0, 1, 0, -55253592, 158062748052]\) | \(8020417344913/187278\) | \(435607215242176094208\) | \([2]\) | \(37158912\) | \(3.0725\) | |
397488.ir4 | 397488ir4 | \([0, 1, 0, 5694568, 12127413780]\) | \(8780064047/32388174\) | \(-75334648399273013796864\) | \([2]\) | \(37158912\) | \(3.0725\) |
Rank
sage: E.rank()
The elliptic curves in class 397488ir have rank \(0\).
Complex multiplication
The elliptic curves in class 397488ir do not have complex multiplication.Modular form 397488.2.a.ir
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.