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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 65520.r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 65520.r1 | 65520cq8 | \([0, 0, 0, -5787425523, -169463525207758]\) | \(7179471593960193209684686321/49441793310\) | \(147632403754967040\) | \([2]\) | \(21233664\) | \(3.8302\) | |
| 65520.r2 | 65520cq6 | \([0, 0, 0, -361714323, -2647864079278]\) | \(1752803993935029634719121/4599740941532100\) | \(13734752855559786086400\) | \([2, 2]\) | \(10616832\) | \(3.4837\) | |
| 65520.r3 | 65520cq7 | \([0, 0, 0, -357269043, -2716116019342]\) | \(-1688971789881664420008241/89901485966373558750\) | \(-268444398671815984450560000\) | \([2]\) | \(21233664\) | \(3.8302\) | |
| 65520.r4 | 65520cq5 | \([0, 0, 0, -71482323, -232237338478]\) | \(13527956825588849127121/25701087819771000\) | \(76743037012431089664000\) | \([2]\) | \(7077888\) | \(3.2809\) | |
| 65520.r5 | 65520cq3 | \([0, 0, 0, -22885203, -40302937582]\) | \(443915739051786565201/21894701746029840\) | \(65377229098417165762560\) | \([2]\) | \(5308416\) | \(3.1371\) | |
| 65520.r6 | 65520cq2 | \([0, 0, 0, -5962323, -991050478]\) | \(7850236389974007121/4400862921000000\) | \(13140906268299264000000\) | \([2, 2]\) | \(3538944\) | \(2.9344\) | |
| 65520.r7 | 65520cq1 | \([0, 0, 0, -3704403, 2729550098]\) | \(1882742462388824401/11650189824000\) | \(34787280411426816000\) | \([2]\) | \(1769472\) | \(2.5878\) | \(\Gamma_0(N)\)-optimal |
| 65520.r8 | 65520cq4 | \([0, 0, 0, 23430957, -7863199342]\) | \(476437916651992691759/284661685546875000\) | \(-849995238456000000000000\) | \([2]\) | \(7077888\) | \(3.2809\) |
Rank
sage: E.rank()
The elliptic curves in class 65520.r have rank \(0\).
Complex multiplication
The elliptic curves in class 65520.r do not have complex multiplication.Modular form 65520.2.a.r
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
