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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 67270h
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
67270.i3 | 67270h1 | \([1, 1, 0, -565568, -163945922]\) | \(22542871522249/2170\) | \(1925882987770\) | \([]\) | \(691200\) | \(1.7911\) | \(\Gamma_0(N)\)-optimal |
67270.i2 | 67270h2 | \([1, 1, 0, -628033, -125583763]\) | \(30867540216409/10218313000\) | \(9068790401110153000\) | \([]\) | \(2073600\) | \(2.3404\) | |
67270.i1 | 67270h3 | \([1, 1, 0, -20439048, 35551551808]\) | \(1063985165884855369/217000000000\) | \(192588298777000000000\) | \([]\) | \(6220800\) | \(2.8897\) |
Rank
sage: E.rank()
The elliptic curves in class 67270h have rank \(1\).
Complex multiplication
The elliptic curves in class 67270h do not have complex multiplication.Modular form 67270.2.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 Cremona numbering.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.