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SageMath
E = EllipticCurve("hl1")
E.isogeny_class()
Elliptic curves in class 169050hl
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
169050.y2 | 169050hl1 | \([1, 1, 0, 1130650, 346583700]\) | \(2173899265153175/1961845235712\) | \(-144255706335175680000\) | \([]\) | \(5256576\) | \(2.5555\) | \(\Gamma_0(N)\)-optimal |
169050.y1 | 169050hl2 | \([1, 1, 0, -25494725, 50120059725]\) | \(-24923353462910020825/341398360424448\) | \(-25103234815984926720000\) | \([]\) | \(15769728\) | \(3.1048\) |
Rank
sage: E.rank()
The elliptic curves in class 169050hl have rank \(0\).
Complex multiplication
The elliptic curves in class 169050hl do not have complex multiplication.Modular form 169050.2.a.hl
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.