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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 43758.z
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
43758.z1 | 43758s1 | \([1, -1, 1, -355118, -69665731]\) | \(6793805286030262681/1048227429629952\) | \(764157796200235008\) | \([2]\) | \(1204224\) | \(2.1549\) | \(\Gamma_0(N)\)-optimal |
43758.z2 | 43758s2 | \([1, -1, 1, 618322, -385060291]\) | \(35862531227445945959/108547797844556928\) | \(-79131344628682000512\) | \([2]\) | \(2408448\) | \(2.5015\) |
Rank
sage: E.rank()
The elliptic curves in class 43758.z have rank \(0\).
Complex multiplication
The elliptic curves in class 43758.z do not have complex multiplication.Modular form 43758.2.a.z
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.