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SageMath
E = EllipticCurve("ce1")
E.isogeny_class()
Elliptic curves in class 32448.ce
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 32448.ce1 | 32448dm4 | \([0, 1, 0, -767898369, -8190629925633]\) | \(18013780041269221/9216\) | \(25619612623586721792\) | \([2]\) | \(5990400\) | \(3.4922\) | |
| 32448.ce2 | 32448dm3 | \([0, 1, 0, -47985409, -128036721409]\) | \(-4395631034341/3145728\) | \(-8744827775517601038336\) | \([2]\) | \(2995200\) | \(3.1456\) | |
| 32448.ce3 | 32448dm2 | \([0, 1, 0, -2287809, 503315775]\) | \(476379541/236196\) | \(656602650091220631552\) | \([2]\) | \(1198080\) | \(2.6875\) | |
| 32448.ce4 | 32448dm1 | \([0, 1, 0, 524351, 60681791]\) | \(5735339/3888\) | \(-10808274075575648256\) | \([2]\) | \(599040\) | \(2.3409\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 32448.ce have rank \(0\).
Complex multiplication
The elliptic curves in class 32448.ce do not have complex multiplication.Modular form 32448.2.a.ce
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
