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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 83391.o
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
83391.o1 | 83391h4 | \([1, 1, 0, -10865024, 13780073217]\) | \(3015048057243061393/13548843\) | \(637417255465683\) | \([4]\) | \(2488320\) | \(2.4676\) | |
83391.o2 | 83391h3 | \([1, 1, 0, -919474, 49274323]\) | \(1827347754908593/1034850081573\) | \(48685433790523650813\) | \([2]\) | \(2488320\) | \(2.4676\) | |
83391.o3 | 83391h2 | \([1, 1, 0, -679409, 214871160]\) | \(737219801902753/1560329001\) | \(73407052501894881\) | \([2, 2]\) | \(1244160\) | \(2.1210\) | |
83391.o4 | 83391h1 | \([1, 1, 0, -27804, 5705955]\) | \(-50529889873/270937359\) | \(-12746486749968279\) | \([2]\) | \(622080\) | \(1.7745\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 83391.o have rank \(1\).
Complex multiplication
The elliptic curves in class 83391.o do not have complex multiplication.Modular form 83391.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.