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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 32760.o
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
32760.o1 | 32760k4 | \([0, 0, 0, -174963, 28168558]\) | \(396738988420322/2985255\) | \(4456961832960\) | \([2]\) | \(131072\) | \(1.6025\) | |
32760.o2 | 32760k2 | \([0, 0, 0, -11163, 420838]\) | \(206081497444/16769025\) | \(12518010086400\) | \([2, 2]\) | \(65536\) | \(1.2560\) | |
32760.o3 | 32760k1 | \([0, 0, 0, -2343, -36038]\) | \(7622072656/1404585\) | \(262129271040\) | \([2]\) | \(32768\) | \(0.90940\) | \(\Gamma_0(N)\)-optimal |
32760.o4 | 32760k3 | \([0, 0, 0, 11517, 1913182]\) | \(113157757438/1124589375\) | \(-1679002940160000\) | \([2]\) | \(131072\) | \(1.6025\) |
Rank
sage: E.rank()
The elliptic curves in class 32760.o have rank \(0\).
Complex multiplication
The elliptic curves in class 32760.o do not have complex multiplication.Modular form 32760.2.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.