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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 182070.j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
182070.j1 | 182070dh3 | \([1, -1, 0, -289363905, 1891622040525]\) | \(152277495831664137649/282362258900400\) | \(4968527571751912194020400\) | \([2]\) | \(56623104\) | \(3.6296\) | |
182070.j2 | 182070dh4 | \([1, -1, 0, -238176225, -1406989214739]\) | \(84917632843343402929/537144431250000\) | \(9451748002979458181250000\) | \([2]\) | \(56623104\) | \(3.6296\) | |
182070.j3 | 182070dh2 | \([1, -1, 0, -24061905, 8349263325]\) | \(87557366190249649/48960807840000\) | \(861528465722097159840000\) | \([2, 2]\) | \(28311552\) | \(3.2830\) | |
182070.j4 | 182070dh1 | \([1, -1, 0, 5901615, 1032171741]\) | \(1291859362462031/773834342400\) | \(-13616611799167977062400\) | \([2]\) | \(14155776\) | \(2.9364\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 182070.j have rank \(0\).
Complex multiplication
The elliptic curves in class 182070.j do not have complex multiplication.Modular form 182070.2.a.j
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.