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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 63536.o
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
63536.o1 | 63536u3 | \([0, -1, 0, -45170245, -116834424211]\) | \(-52893159101157376/11\) | \(-2119699214336\) | \([]\) | \(1440000\) | \(2.6621\) | |
63536.o2 | 63536u2 | \([0, -1, 0, -59685, -10145651]\) | \(-122023936/161051\) | \(-31034516197093376\) | \([]\) | \(288000\) | \(1.8574\) | |
63536.o3 | 63536u1 | \([0, -1, 0, -1925, 77869]\) | \(-4096/11\) | \(-2119699214336\) | \([]\) | \(57600\) | \(1.0526\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 63536.o have rank \(1\).
Complex multiplication
The elliptic curves in class 63536.o do not have complex multiplication.Modular form 63536.2.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.
\(\left(\begin{array}{rrr} 1 & 5 & 25 \\ 5 & 1 & 5 \\ 25 & 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.