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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 19950.o
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
19950.o1 | 19950g5 | \([1, 1, 0, -8887875, -10200416625]\) | \(4969327007303723277361/1123462695162150\) | \(17554104611908593750\) | \([2]\) | \(1474560\) | \(2.6849\) | |
19950.o2 | 19950g3 | \([1, 1, 0, -619125, -120810375]\) | \(1679731262160129361/570261564022500\) | \(8910336937851562500\) | \([2, 2]\) | \(737280\) | \(2.3383\) | |
19950.o3 | 19950g2 | \([1, 1, 0, -254625, 47953125]\) | \(116844823575501841/3760263939600\) | \(58754124056250000\) | \([2, 2]\) | \(368640\) | \(1.9918\) | |
19950.o4 | 19950g1 | \([1, 1, 0, -252625, 48767125]\) | \(114113060120923921/124104960\) | \(1939140000000\) | \([2]\) | \(184320\) | \(1.6452\) | \(\Gamma_0(N)\)-optimal |
19950.o5 | 19950g4 | \([1, 1, 0, 77875, 164660625]\) | \(3342636501165359/751262567039460\) | \(-11738477609991562500\) | \([2]\) | \(737280\) | \(2.3383\) | |
19950.o6 | 19950g6 | \([1, 1, 0, 1817625, -834778125]\) | \(42502666283088696719/43898058864843750\) | \(-685907169763183593750\) | \([2]\) | \(1474560\) | \(2.6849\) |
Rank
sage: E.rank()
The elliptic curves in class 19950.o have rank \(0\).
Complex multiplication
The elliptic curves in class 19950.o do not have complex multiplication.Modular form 19950.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.