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Elliptic divisibility sequences $W_n$ of points on elliptic curves over $\mathbb{Q}$
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Numbers
$N$
$c_4$, $c_6$
$x_P$, $y_P$
$n$ 
$W_n(P)$
37
48, -216
0, 0
1:
1
comment: Cremona label 37a1.
37
48, -216
0, 0
2:
1
comment: Cremona label 37a1.
37
48, -216
0, 0
3:
-1
comment: Cremona label 37a1.
37
48, -216
0, 0
4:
1
comment: Cremona label 37a1.
37
48, -216
0, 0
5:
2
comment: Cremona label 37a1.
37
48, -216
0, 0
6:
-1
comment: Cremona label 37a1.
37
48, -216
0, 0
7:
-3
comment: Cremona label 37a1.
37
48, -216
0, 0
8:
-5
comment: Cremona label 37a1.
37
48, -216
0, 0
9:
7
comment: Cremona label 37a1.
37
48, -216
0, 0
10:
-4
comment: Cremona label 37a1.
37
48, -216
0, 0
11:
-23
comment: Cremona label 37a1.
37
48, -216
0, 0
12:
29
comment: Cremona label 37a1.
37
48, -216
0, 0
13:
59
comment: Cremona label 37a1.
37
48, -216
0, 0
14:
129
comment: Cremona label 37a1.
37
48, -216
0, 0
15:
-314
comment: Cremona label 37a1.
37
48, -216
0, 0
16:
-65
comment: Cremona label 37a1.
37
48, -216
0, 0
17:
1529
comment: Cremona label 37a1.
37
48, -216
0, 0
18:
-3689
comment: Cremona label 37a1.
37
48, -216
0, 0
19:
-8209
comment: Cremona label 37a1.
37
48, -216
0, 0
20:
-16264
comment: Cremona label 37a1.
37
48, -216
0, 0
21:
83313
comment: Cremona label 37a1.
37
48, -216
0, 0
22:
113689
comment: Cremona label 37a1.
37
48, -216
0, 0
23:
-620297
comment: Cremona label 37a1.
37
48, -216
0, 0
24:
2382785
comment: Cremona label 37a1.
37
48, -216
0, 0
25:
7869898
comment: Cremona label 37a1.
37
48, -216
0, 0
26:
7001471
comment: Cremona label 37a1.
37
48, -216
0, 0
27:
-126742987
comment: Cremona label 37a1.
37
48, -216
0, 0
28:
-398035821
comment: Cremona label 37a1.
37
48, -216
0, 0
29:
1687054711
comment: Cremona label 37a1.
37
48, -216
0, 0
30:
-7911171596
comment: Cremona label 37a1.
37
48, -216
0, 0
31:
-47301104551
comment: Cremona label 37a1.
37
48, -216
0, 0
32:
43244638645
comment: Cremona label 37a1.
37
48, -216
0, 0
33:
1123424582771
comment: Cremona label 37a1.
37
48, -216
0, 0
34:
6480598259201
comment: Cremona label 37a1.
37
48, -216
0, 0
35:
-32606721084786
comment: Cremona label 37a1.
37
48, -216
0, 0
36:
124106986093951
comment: Cremona label 37a1.
37
48, -216
0, 0
37:
1662315215971057
comment: Cremona label 37a1.
37
48, -216
0, 0
38:
-5987117709349201
comment: Cremona label 37a1.
37
48, -216
0, 0
39:
-61958046554226593
comment: Cremona label 37a1.
37
48, -216
0, 0
40:
-541051130050800400
comment: Cremona label 37a1.
37
48, -216
0, 0
41:
4257998884448335457
comment: Cremona label 37a1.
37
48, -216
0, 0
42:
-4830209396684261199
comment: Cremona label 37a1.
37
48, -216
0, 0
43:
-334806306946199122193
comment: Cremona label 37a1.
37
48, -216
0, 0
44:
2591758672670554328449
comment: Cremona label 37a1.
37
48, -216
0, 0
45:
23385756731869683322514
comment: Cremona label 37a1.
37
48, -216
0, 0
46:
230318343950087449971199
comment: Cremona label 37a1.
37
48, -216
0, 0
47:
-3416372868727801226636179
comment: Cremona label 37a1.
37
48, -216
0, 0
48:
-10358960321661880987253845
comment: Cremona label 37a1.
37
48, -216
0, 0
49:
397068399296019032727466599
comment: Cremona label 37a1.
37
48, -216
0, 0
50:
-5423908604123397486016003604
comment: Cremona label 37a1.
43
16, -280
0, 0
1:
1
comment: Cremona label 43a1.
43
16, -280
0, 0
2:
1
comment: Cremona label 43a1.
43
16, -280
0, 0
3:
1
comment: Cremona label 43a1.
43
16, -280
0, 0
4:
-1
comment: Cremona label 43a1.
43
16, -280
0, 0
5:
-2
comment: Cremona label 43a1.
43
16, -280
0, 0
6:
-3
comment: Cremona label 43a1.
43
16, -280
0, 0
7:
-1
comment: Cremona label 43a1.
43
16, -280
0, 0
8:
7
comment: Cremona label 43a1.
43
16, -280
0, 0
9:
11
comment: Cremona label 43a1.
43
16, -280
0, 0
10:
20
comment: Cremona label 43a1.
43
16, -280
0, 0
11:
-19
comment: Cremona label 43a1.
43
16, -280
0, 0
12:
-87
comment: Cremona label 43a1.
43
16, -280
0, 0
13:
-191
comment: Cremona label 43a1.
43
16, -280
0, 0
14:
-197
comment: Cremona label 43a1.
43
16, -280
0, 0
15:
1018
comment: Cremona label 43a1.
43
16, -280
0, 0
16:
2681
comment: Cremona label 43a1.
43
16, -280
0, 0
17:
8191
comment: Cremona label 43a1.
43
16, -280
0, 0
18:
-5841
comment: Cremona label 43a1.
43
16, -280
0, 0
19:
-81289
comment: Cremona label 43a1.
43
16, -280
0, 0
20:
-261080
comment: Cremona label 43a1.
43
16, -280
0, 0
21:
-620551
comment: Cremona label 43a1.
43
16, -280
0, 0
22:
3033521
comment: Cremona label 43a1.
43
16, -280
0, 0
23:
14480129
comment: Cremona label 43a1.
43
16, -280
0, 0
24:
69664119
comment: Cremona label 43a1.
43
16, -280
0, 0
25:
-2664458
comment: Cremona label 43a1.
43
16, -280
0, 0
26:
-1612539083
comment: Cremona label 43a1.
43
16, -280
0, 0
27:
-7758440129
comment: Cremona label 43a1.
43
16, -280
0, 0
28:
-37029252553
comment: Cremona label 43a1.
43
16, -280
0, 0
29:
181003520899
comment: Cremona label 43a1.
43
16, -280
0, 0
30:
1721180313660
comment: Cremona label 43a1.
43
16, -280
0, 0
31:
12437589708389
comment: Cremona label 43a1.
43
16, -280
0, 0
32:
19206818781913
comment: Cremona label 43a1.
43
16, -280
0, 0
33:
-672004824959359
comment: Cremona label 43a1.
43
16, -280
0, 0
34:
-5070370671429517
comment: Cremona label 43a1.
43
16, -280
0, 0
35:
-44138469613743118
comment: Cremona label 43a1.
43
16, -280
0, 0
36:
205791799565838321
comment: Cremona label 43a1.
43
16, -280
0, 0
37:
4451821019236847359
comment: Cremona label 43a1.
43
16, -280
0, 0
38:
47106384726033313759
comment: Cremona label 43a1.
43
16, -280
0, 0
39:
229382731030517041519
comment: Cremona label 43a1.
43
16, -280
0, 0
40:
-5820643101584012912560
comment: Cremona label 43a1.
43
16, -280
0, 0
41:
-73409485506941940842639
comment: Cremona label 43a1.
43
16, -280
0, 0
42:
-1076685351255906765843999
comment: Cremona label 43a1.
43
16, -280
0, 0
43:
3827875777856700313926401
comment: Cremona label 43a1.
43
16, -280
0, 0
44:
247438936198313621573047279
comment: Cremona label 43a1.
43
16, -280
0, 0
45:
3828749227226123155576841198
comment: Cremona label 43a1.
43
16, -280
0, 0
46:
43253166457818654521210472557
comment: Cremona label 43a1.
43
16, -280
0, 0
47:
-1033681179609463437347950166401
comment: Cremona label 43a1.
43
16, -280
0, 0
48:
-23555478033514972455736509594393
comment: Cremona label 43a1.
43
16, -280
0, 0
49:
-545176941496421647722644050076549
comment: Cremona label 43a1.
43
16, -280
0, 0
50:
200646549263802764897507144448100
comment: Cremona label 43a1.
53
-15, -297
0, 0
1:
1
comment: Cremona label 53a1.
53
-15, -297
0, 0
2:
1
comment: Cremona label 53a1.
53
-15, -297
0, 0
3:
-1
comment: Cremona label 53a1.
53
-15, -297
0, 0
4:
-2
comment: Cremona label 53a1.
53
-15, -297
0, 0
5:
-1
comment: Cremona label 53a1.
53
-15, -297
0, 0
6:
5
comment: Cremona label 53a1.
53
-15, -297
0, 0
7:
9
comment: Cremona label 53a1.
53
-15, -297
0, 0
8:
-8
comment: Cremona label 53a1.
53
-15, -297
0, 0
9:
-41
comment: Cremona label 53a1.
53
-15, -297
0, 0
10:
-61
comment: Cremona label 53a1.
53
-15, -297
0, 0
11:
241
comment: Cremona label 53a1.
53
-15, -297
0, 0
12:
770
comment: Cremona label 53a1.
53
-15, -297
0, 0
13:
-271
comment: Cremona label 53a1.
53
-15, -297
0, 0
14:
-8649
comment: Cremona label 53a1.
53
-15, -297
0, 0
15:
-27329
comment: Cremona label 53a1.
53
-15, -297
0, 0
16:
106768
comment: Cremona label 53a1.
53
-15, -297
0, 0
17:
651521
comment: Cremona label 53a1.
53
-15, -297
0, 0
18:
740665
comment: Cremona label 53a1.
53
-15, -297
0, 0
19:
-18425809
comment: Cremona label 53a1.
53
-15, -297
0, 0
20:
-107300098
comment: Cremona label 53a1.
53
-15, -297
0, 0
21:
399122991
comment: Cremona label 53a1.
53
-15, -297
0, 0
22:
5615422669
comment: Cremona label 53a1.
53
-15, -297
0, 0
23:
24055184809
comment: Cremona label 53a1.
53
-15, -297
0, 0
24:
-383354254360
comment: Cremona label 53a1.
53
-15, -297
0, 0
25:
-3943757411849
comment: Cremona label 53a1.
53
-15, -297
0, 0
26:
9276714153611
comment: Cremona label 53a1.
53
-15, -297
0, 0
27:
498726356978849
comment: Cremona label 53a1.
53
-15, -297
0, 0
28:
4906162694012418
comment: Cremona label 53a1.
53
-15, -297
0, 0
29:
-74609317288333151
comment: Cremona label 53a1.
53
-15, -297
0, 0
30:
-1416363640207214225
comment: Cremona label 53a1.
53
-15, -297
0, 0
31:
-2771780972000481601
comment: Cremona label 53a1.
53
-15, -297
0, 0
32:
451042247334241856032
comment: Cremona label 53a1.
53
-15, -297
0, 0
33:
8459493968101503667649
comment: Cremona label 53a1.
53
-15, -297
0, 0
34:
-127079825658652223072911
comment: Cremona label 53a1.
53
-15, -297
0, 0
35:
-5139175203350482789079649
comment: Cremona label 53a1.
53
-15, -297
0, 0
36:
-60583104367332966275916290
comment: Cremona label 53a1.
53
-15, -297
0, 0
37:
4032157507318149646832492959
comment: Cremona label 53a1.
53
-15, -297
0, 0
38:
134180632169200713822727506581
comment: Cremona label 53a1.
53
-15, -297
0, 0
39:
-1581813150822504399440199143159
comment: Cremona label 53a1.
53
-15, -297
0, 0
40:
-191907007728904636968948884800552
comment: Cremona label 53a1.
53
-15, -297
0, 0
41:
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comment: Cremona label 53a1.
53
-15, -297
0, 0
42:
342437683627854778762117208728471635
comment: Cremona label 53a1.
53
-15, -297
0, 0
43:
20529159471813467221779606414723317521
comment: Cremona label 53a1.
53
-15, -297
0, 0
44:
5735569511810198321276797974933889282
comment: Cremona label 53a1.
53
-15, -297
0, 0
45:
-73436531442020355614275036600289880420079
comment: Cremona label 53a1.
53
-15, -297
0, 0
46:
-4402428355142121419015455660268268438938201
comment: Cremona label 53a1.
53
-15, -297
0, 0
47:
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comment: Cremona label 53a1.
53
-15, -297
0, 0
48:
31423521361954194326145067648891170568182924080
comment: Cremona label 53a1.
53
-15, -297
0, 0
49:
952869368085663194754442900484103256253666776641
comment: Cremona label 53a1.
53
-15, -297
0, 0
50:
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comment: Cremona label 53a1.
57
112, -1304
2, 1
1:
1
comment: Cremona label 57a1.
57
112, -1304
2, 1
2:
3
comment: Cremona label 57a1.
57
112, -1304
2, 1
3:
9
comment: Cremona label 57a1.
57
112, -1304
2, 1
4:
81
comment: Cremona label 57a1.
57
112, -1304
2, 1
5:
1458
comment: Cremona label 57a1.
57
112, -1304
2, 1
6:
19683
comment: Cremona label 57a1.
57
112, -1304
2, 1
7:
-531441
comment: Cremona label 57a1.
57
112, -1304
2, 1
8:
-129140163
comment: Cremona label 57a1.
57
112, -1304
2, 1
9:
-17433922005
comment: Cremona label 57a1.
57
112, -1304
2, 1
10:
-3389154437772
comment: Cremona label 57a1.
57
112, -1304
2, 1
11:
-2264802453041139
comment: Cremona label 57a1.
57
112, -1304
2, 1
12:
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comment: Cremona label 57a1.
57
112, -1304
2, 1
13:
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comment: Cremona label 57a1.
57
112, -1304
2, 1
14:
5503884572304203180571909
comment: Cremona label 57a1.
57
112, -1304
2, 1
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comment: Cremona label 57a1.
57
112, -1304
2, 1
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comment: Cremona label 57a1.
57
112, -1304
2, 1
17:
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comment: Cremona label 57a1.
57
112, -1304
2, 1
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comment: Cremona label 57a1.
57
112, -1304
2, 1
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comment: Cremona label 57a1.
57
112, -1304
2, 1
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comment: Cremona label 57a1.
57
112, -1304
2, 1
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comment: Cremona label 57a1.
57
112, -1304
2, 1
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comment: Cremona label 57a1.
57
112, -1304
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comment: Cremona label 57a1.
57
112, -1304
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57
112, -1304
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57
112, -1304
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comment: Cremona label 57a1.
57
112, -1304
2, 1
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comment: Cremona label 57a1.
57
112, -1304
2, 1
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comment: Cremona label 57a1.
57
112, -1304
2, 1
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comment: Cremona label 57a1.
57
112, -1304
2, 1
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comment: Cremona label 57a1.
57
112, -1304
2, 1
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comment: Cremona label 57a1.
57
112, -1304
2, 1
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comment: Cremona label 57a1.
57
112, -1304
2, 1
33:
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comment: Cremona label 57a1.
57
112, -1304
2, 1
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comment: Cremona label 57a1.
57
112, -1304
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Definition
Let $E/\mathbb{Q}$ be the reduced global minimal Weierstrass model recovered from $(c_4,c_6)$ by Formula (1), and let $P=(x_P,y_P)$ be an integral nontorsion point on $E$. The entry is $W_n(P)=\psi_n(P)$, with $\psi_n$ the $n$-th division polynomial.
Parameters
$N$
—   conductor of $E$ ($N\geq1$)
$c_4$
—   invariant $c_4$ of the reduced global minimal model of $E$
$c_6$
—   invariant $c_6$ of the reduced global minimal model of $E$
$x_P$
—   $x$-coordinate of $P$
$y_P$
—   $y$-coordinate of $P$
$n$
—   sequence index ($n\geq1$)
Formulas
(1)
For the reduced models in this table, $b_2$ is the unique element of $\{-4,-3,0,1,4,5\}$ with $b_2\equiv -c_6\pmod{12}$, $b_4=(b_2^2-c_4)/24$, and $b_6=(-c_6-b_2^3+36b_2b_4)/216$. The model has $a_1\equiv b_2\pmod 2$, $a_3\equiv b_6\pmod 2$ with $a_1,a_3\in\{0,1\}$ and $a_2\in\{-1,0,1\}$, and $a_2=(b_2-a_1^2)/4$, $a_4=(b_4-a_1a_3)/2$ and $a_6=(b_6-a_3^2)/4$.
(2)
With $W_0=0$ and $W_1=1$, the terms are a divisibility sequence: $W_m(P)\mid W_n(P)$ whenever $m\mid n$. They satisfy $W_{2m+1}=W_{m+2}W_m^3-W_{m-1}W_{m+1}^3$ for $m\geq2$, and $W_{2m}W_2=W_m(W_{m+2}W_{m-1}^2-W_{m-2}W_{m+1}^2)$ for $m\geq3$.
(3)
For $n\geq1$, $x(nP)=x(P)-W_{n-1}(P)W_{n+1}(P)/W_n(P)^2$.
Comments
(4)
This table uses the division-polynomial convention $W_n(P)=\psi_n(P)$ for elliptic divisibility sequences [1] [2]. Here $\psi_n\in\mathbb{Z}[x,y]$ is the full division polynomial, so $\psi_2=2y+a_1x+a_3$. In Sage [3], evaluate this convention with two_torsion_multiplicity=1, since the default uses two_torsion_multiplicity=2. The table does not store the denominator sequence obtained from writing $x(nP)=A_n/D_n^2$.
(5)
The curve is identified by $N$, $c_4$ and $c_6$; in Cremona's labelling [4], the five curves here are 37a1, 43a1, 53a1, 57a1 and 58a1. LMFDB [5] numbers curves within an isogeny class differently from Cremona's tables, so a label here is not an LMFDB label.
(6)
In the computed range, $P$ is the first Mordell-Weil generator returned by Sage for a rank-one curve, replaced by $-P$ when necessary so that $W_2(P)>0$.
Programs
(P1)
Sage
E = EllipticCurve('37a1')
P = E(0, 0)
[E.division_polynomial(n, P, two_torsion_multiplicity=1) for n in [1..10]]
References
[1]
Katherine E. Stange, "Integral points on elliptic curves and explicit valuations of division polynomials", Canadian Journal of Mathematics 68 (2016), 978-1008 (arXiv) (doi)
Links
Similar tables
Division polynomials $\psi_n$ of elliptic curves over $\mathbb{Q}$ —   gives the polynomials whose values at $P$ are the terms $W_n(P)$
Highest known ranks of elliptic curves over $\mathbb{Q}$ with prescribed torsion subgroup —   records rank records by rational torsion subgroup, while this table records divisibility sequences from individual nontorsion points
Regulators of elliptic curves over $\mathbb{Q}$ of rank $1$ —   uses the same $(N,c_4,c_6)$ curve index for rank-one curves and records their regulators
Data properties
Entries are of type: integer
How well the digits are known: exact
Table is complete: no (it holds $W_n(P)$ for $1\leq n\leq50$, for every rank-one curve in Sage's mini Cremona database [4] with conductor $N\leq60$, taking $P$ to be the first Mordell-Weil generator returned by Sage, negated where necessary so that $W_2(P)>0$)
Sources of data: [4]
How they were obtained:

The generator reads exact Weierstrass coefficients from Sage's mini Cremona database [4] and evaluates the exact division polynomials by the same coordinate-ring recurrences used for the division-polynomial table.

more

It verifies the elliptic divisibility sequence recurrences in Formula (2), verifies Formula (3) against exact point multiplication in Sage, and checks the first terms for 37a1, 43a1 and 53a1 against OEIS A006769 [6], A050512 [7] and A178627 [8].



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