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SageMath
E = EllipticCurve("hs1")
E.isogeny_class()
Elliptic curves in class 162624.hs
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
162624.hs1 | 162624bh4 | \([0, 1, 0, -108577, -13806817]\) | \(2438569736/21\) | \(1219060727808\) | \([2]\) | \(655360\) | \(1.4869\) | |
162624.hs2 | 162624bh2 | \([0, 1, 0, -6937, -207385]\) | \(5088448/441\) | \(3200034410496\) | \([2, 2]\) | \(327680\) | \(1.1403\) | |
162624.hs3 | 162624bh1 | \([0, 1, 0, -1492, 18038]\) | \(3241792/567\) | \(64286405568\) | \([2]\) | \(163840\) | \(0.79374\) | \(\Gamma_0(N)\)-optimal |
162624.hs4 | 162624bh3 | \([0, 1, 0, 7583, -947905]\) | \(830584/7203\) | \(-418137829638144\) | \([2]\) | \(655360\) | \(1.4869\) |
Rank
sage: E.rank()
The elliptic curves in class 162624.hs have rank \(0\).
Complex multiplication
The elliptic curves in class 162624.hs do not have complex multiplication.Modular form 162624.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.