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SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 21840.w
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 21840.w1 | 21840bk8 | \([0, -1, 0, -643047280, 6276641208640]\) | \(7179471593960193209684686321/49441793310\) | \(202513585397760\) | \([2]\) | \(2654208\) | \(3.2809\) | |
| 21840.w2 | 21840bk6 | \([0, -1, 0, -40190480, 98082436800]\) | \(1752803993935029634719121/4599740941532100\) | \(18840538896515481600\) | \([2, 2]\) | \(1327104\) | \(2.9344\) | |
| 21840.w3 | 21840bk7 | \([0, -1, 0, -39696560, 100610121792]\) | \(-1688971789881664420008241/89901485966373558750\) | \(-368236486518266096640000\) | \([4]\) | \(2654208\) | \(3.2809\) | |
| 21840.w4 | 21840bk5 | \([0, -1, 0, -7942480, 8604030400]\) | \(13527956825588849127121/25701087819771000\) | \(105271655709782016000\) | \([2]\) | \(884736\) | \(2.7316\) | |
| 21840.w5 | 21840bk3 | \([0, -1, 0, -2542800, 1493548992]\) | \(443915739051786565201/21894701746029840\) | \(89680698351738224640\) | \([2]\) | \(663552\) | \(2.5878\) | |
| 21840.w6 | 21840bk2 | \([0, -1, 0, -662480, 36926400]\) | \(7850236389974007121/4400862921000000\) | \(18025934524416000000\) | \([2, 2]\) | \(442368\) | \(2.3850\) | |
| 21840.w7 | 21840bk1 | \([0, -1, 0, -411600, -100957248]\) | \(1882742462388824401/11650189824000\) | \(47719177519104000\) | \([2]\) | \(221184\) | \(2.0385\) | \(\Gamma_0(N)\)-optimal |
| 21840.w8 | 21840bk4 | \([0, -1, 0, 2603440, 290361792]\) | \(476437916651992691759/284661685546875000\) | \(-1165974264000000000000\) | \([4]\) | \(884736\) | \(2.7316\) |
Rank
sage: E.rank()
The elliptic curves in class 21840.w have rank \(0\).
Complex multiplication
The elliptic curves in class 21840.w do not have complex multiplication.Modular form 21840.2.a.w
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.
