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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 21450.h
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
21450.h1 | 21450a4 | \([1, 1, 0, -4010650, -3093105500]\) | \(456612868287073618849/12544848030000\) | \(196013250468750000\) | \([2]\) | \(589824\) | \(2.4215\) | |
21450.h2 | 21450a3 | \([1, 1, 0, -1118650, 411242500]\) | \(9908022260084596129/1047363281250000\) | \(16365051269531250000\) | \([2]\) | \(589824\) | \(2.4215\) | |
21450.h3 | 21450a2 | \([1, 1, 0, -260650, -44355500]\) | \(125337052492018849/18404100000000\) | \(287564062500000000\) | \([2, 2]\) | \(294912\) | \(2.0749\) | |
21450.h4 | 21450a1 | \([1, 1, 0, 27350, -3747500]\) | \(144794100308831/474439680000\) | \(-7413120000000000\) | \([2]\) | \(147456\) | \(1.7283\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 21450.h have rank \(1\).
Complex multiplication
The elliptic curves in class 21450.h do not have complex multiplication.Modular form 21450.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.