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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 272322.x
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
272322.x1 | 272322x4 | \([1, -1, 1, -16296770, -25318080575]\) | \(-189613868625/128\) | \(-323123218936483968\) | \([]\) | \(8482320\) | \(2.6755\) | |
272322.x2 | 272322x3 | \([1, -1, 1, -159170, -49611327]\) | \(-1159088625/2097152\) | \(-806896939346952192\) | \([]\) | \(2827440\) | \(2.1262\) | |
272322.x3 | 272322x1 | \([1, -1, 1, -7880, 284115]\) | \(-140625/8\) | \(-3078067548168\) | \([]\) | \(403920\) | \(1.1532\) | \(\Gamma_0(N)\)-optimal |
272322.x4 | 272322x2 | \([1, -1, 1, 42550, 512731]\) | \(3375/2\) | \(-5048800295882562\) | \([]\) | \(1211760\) | \(1.7025\) |
Rank
sage: E.rank()
The elliptic curves in class 272322.x have rank \(0\).
Complex multiplication
The elliptic curves in class 272322.x do not have complex multiplication.Modular form 272322.2.a.x
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 21 & 7 \\ 3 & 1 & 7 & 21 \\ 21 & 7 & 1 & 3 \\ 7 & 21 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.