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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 187200.o
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
187200.o1 | 187200ck4 | \([0, 0, 0, -1490700, -697894000]\) | \(490757540836/2142075\) | \(1599050419200000000\) | \([2]\) | \(4718592\) | \(2.3459\) | |
187200.o2 | 187200ck2 | \([0, 0, 0, -140700, 1406000]\) | \(1650587344/950625\) | \(177409440000000000\) | \([2, 2]\) | \(2359296\) | \(1.9993\) | |
187200.o3 | 187200ck1 | \([0, 0, 0, -100200, 12179000]\) | \(9538484224/26325\) | \(307054800000000\) | \([2]\) | \(1179648\) | \(1.6528\) | \(\Gamma_0(N)\)-optimal |
187200.o4 | 187200ck3 | \([0, 0, 0, 561300, 11234000]\) | \(26198797244/15234375\) | \(-11372400000000000000\) | \([2]\) | \(4718592\) | \(2.3459\) |
Rank
sage: E.rank()
The elliptic curves in class 187200.o have rank \(0\).
Complex multiplication
The elliptic curves in class 187200.o do not have complex multiplication.Modular form 187200.2.a.o
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.