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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 4774.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4774.i1 | 4774j1 | \([1, 1, 1, -249690, 47919079]\) | \(-1721580238553093926561/2065794891776\) | \(-2065794891776\) | \([5]\) | \(24000\) | \(1.6446\) | \(\Gamma_0(N)\)-optimal |
4774.i2 | 4774j2 | \([1, 1, 1, 596750, 254759559]\) | \(23501790452547877931999/41583720945376050056\) | \(-41583720945376050056\) | \([]\) | \(120000\) | \(2.4494\) |
Rank
sage: E.rank()
The elliptic curves in class 4774.i have rank \(1\).
Complex multiplication
The elliptic curves in class 4774.i do not have complex multiplication.Modular form 4774.2.a.i
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.
\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.