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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 1734.h
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1734.h1 | 1734g1 | \([1, 0, 1, -729, 4744]\) | \(1771561/612\) | \(14772192228\) | \([2]\) | \(2304\) | \(0.65308\) | \(\Gamma_0(N)\)-optimal |
1734.h2 | 1734g2 | \([1, 0, 1, 2161, 33644]\) | \(46268279/46818\) | \(-1130072705442\) | \([2]\) | \(4608\) | \(0.99965\) |
Rank
sage: E.rank()
The elliptic curves in class 1734.h have rank \(0\).
Complex multiplication
The elliptic curves in class 1734.h do not have complex multiplication.Modular form 1734.2.a.h
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.