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SageMath
E = EllipticCurve("ce1")
E.isogeny_class()
Elliptic curves in class 40656.ce
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 40656.ce1 | 40656dk6 | \([0, 1, 0, -8748824, -9963217068]\) | \(10206027697760497/5557167\) | \(40324547902205952\) | \([2]\) | \(1228800\) | \(2.5149\) | |
| 40656.ce2 | 40656dk4 | \([0, 1, 0, -549864, -153981324]\) | \(2533811507137/58110129\) | \(421665334236647424\) | \([2, 2]\) | \(614400\) | \(2.1683\) | |
| 40656.ce3 | 40656dk2 | \([0, 1, 0, -75544, 4441556]\) | \(6570725617/2614689\) | \(18973004019830784\) | \([2, 2]\) | \(307200\) | \(1.8217\) | |
| 40656.ce4 | 40656dk1 | \([0, 1, 0, -65864, 6482100]\) | \(4354703137/1617\) | \(11733459505152\) | \([2]\) | \(153600\) | \(1.4752\) | \(\Gamma_0(N)\)-optimal |
| 40656.ce5 | 40656dk5 | \([0, 1, 0, 59976, -476220780]\) | \(3288008303/13504609503\) | \(-97993685056488173568\) | \([2]\) | \(1228800\) | \(2.5149\) | |
| 40656.ce6 | 40656dk3 | \([0, 1, 0, 243896, 32424500]\) | \(221115865823/190238433\) | \(-1380429777321627648\) | \([2]\) | \(614400\) | \(2.1683\) |
Rank
sage: E.rank()
The elliptic curves in class 40656.ce have rank \(1\).
Complex multiplication
The elliptic curves in class 40656.ce do not have complex multiplication.Modular form 40656.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 8 & 4 & 2 & 1 & 8 & 4 \\ 4 & 2 & 4 & 8 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
