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SageMath
E = EllipticCurve("hs1")
E.isogeny_class()
Elliptic curves in class 418950.hs
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
418950.hs1 | 418950hs3 | \([1, -1, 0, -200120517, 1089695673891]\) | \(661397832743623417/443352042\) | \(594134138746391906250\) | \([2]\) | \(62914560\) | \(3.3023\) | \(\Gamma_0(N)\)-optimal* |
418950.hs2 | 418950hs2 | \([1, -1, 0, -12585267, 16806508641]\) | \(164503536215257/4178071044\) | \(5599014792285942562500\) | \([2, 2]\) | \(31457280\) | \(2.9557\) | \(\Gamma_0(N)\)-optimal* |
418950.hs3 | 418950hs1 | \([1, -1, 0, -1780767, -534713859]\) | \(466025146777/177366672\) | \(237688303939364250000\) | \([2]\) | \(15728640\) | \(2.6091\) | \(\Gamma_0(N)\)-optimal* |
418950.hs4 | 418950hs4 | \([1, -1, 0, 2077983, 53625929391]\) | \(740480746823/927484650666\) | \(-1242918137104107602906250\) | \([2]\) | \(62914560\) | \(3.3023\) |
Rank
sage: E.rank()
The elliptic curves in class 418950.hs have rank \(0\).
Complex multiplication
The elliptic curves in class 418950.hs do not have complex multiplication.Modular form 418950.2.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.