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SageMath
E = EllipticCurve("hz1")
E.isogeny_class()
Elliptic curves in class 117600.hz
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
117600.hz1 | 117600dd4 | \([0, 1, 0, -1372408, -619289812]\) | \(303735479048/105\) | \(98825160000000\) | \([2]\) | \(1769472\) | \(2.0413\) | |
117600.hz2 | 117600dd3 | \([0, 1, 0, -178033, 14372063]\) | \(82881856/36015\) | \(271176239040000000\) | \([2]\) | \(1769472\) | \(2.0413\) | |
117600.hz3 | 117600dd1 | \([0, 1, 0, -86158, -9607312]\) | \(601211584/11025\) | \(1297080225000000\) | \([2, 2]\) | \(884736\) | \(1.6947\) | \(\Gamma_0(N)\)-optimal |
117600.hz4 | 117600dd2 | \([0, 1, 0, -408, -27786312]\) | \(-8/354375\) | \(-333534915000000000\) | \([2]\) | \(1769472\) | \(2.0413\) |
Rank
sage: E.rank()
The elliptic curves in class 117600.hz have rank \(1\).
Complex multiplication
The elliptic curves in class 117600.hz do not have complex multiplication.Modular form 117600.2.a.hz
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.