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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 1200.o
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1200.o1 | 1200e5 | \([0, 1, 0, -80008, 8683988]\) | \(1770025017602/75\) | \(2400000000\) | \([2]\) | \(3072\) | \(1.2847\) | |
1200.o2 | 1200e3 | \([0, 1, 0, -5008, 133988]\) | \(868327204/5625\) | \(90000000000\) | \([2, 2]\) | \(1536\) | \(0.93814\) | |
1200.o3 | 1200e6 | \([0, 1, 0, -2008, 295988]\) | \(-27995042/1171875\) | \(-37500000000000\) | \([2]\) | \(3072\) | \(1.2847\) | |
1200.o4 | 1200e2 | \([0, 1, 0, -508, -1012]\) | \(3631696/2025\) | \(8100000000\) | \([2, 2]\) | \(768\) | \(0.59157\) | |
1200.o5 | 1200e1 | \([0, 1, 0, -383, -3012]\) | \(24918016/45\) | \(11250000\) | \([2]\) | \(384\) | \(0.24500\) | \(\Gamma_0(N)\)-optimal |
1200.o6 | 1200e4 | \([0, 1, 0, 1992, -6012]\) | \(54607676/32805\) | \(-524880000000\) | \([4]\) | \(1536\) | \(0.93814\) |
Rank
sage: E.rank()
The elliptic curves in class 1200.o have rank \(0\).
Complex multiplication
The elliptic curves in class 1200.o do not have complex multiplication.Modular form 1200.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.