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SageMath
E = EllipticCurve("hs1")
E.isogeny_class()
Elliptic curves in class 92400.hs
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
92400.hs1 | 92400hh4 | \([0, 1, 0, -1239908, -280602312]\) | \(52702650535889104/22020583921875\) | \(88082335687500000000\) | \([2]\) | \(2985984\) | \(2.5241\) | |
92400.hs2 | 92400hh2 | \([0, 1, 0, -1068908, -425718312]\) | \(33766427105425744/9823275\) | \(39293100000000\) | \([2]\) | \(995328\) | \(1.9748\) | |
92400.hs3 | 92400hh1 | \([0, 1, 0, -66533, -6725562]\) | \(-130287139815424/2250652635\) | \(-562663158750000\) | \([2]\) | \(497664\) | \(1.6283\) | \(\Gamma_0(N)\)-optimal |
92400.hs4 | 92400hh3 | \([0, 1, 0, 257467, -32038062]\) | \(7549996227362816/6152409907875\) | \(-1538102476968750000\) | \([2]\) | \(1492992\) | \(2.1776\) |
Rank
sage: E.rank()
The elliptic curves in class 92400.hs have rank \(1\).
Complex multiplication
The elliptic curves in class 92400.hs do not have complex multiplication.Modular form 92400.2.a.hs
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.