Defining equations for X

accessibility Defining equations for X₁(N)

The table below lists (links to) defining equations f(x,y)=0 that are birationally equivalent to the "raw" equations F(r,s)=0 for X₁(N) listed here. In every case, the map from f(x,y)=0 to F(r,s)=0 is given by

r = (x²y − xy + y − 1) / (x²y − x), s = (xy − y + 1) / (xy),

and the inverse map from F(r,s)=0 to f(x,y)=0 is

x = (s − r) / (rs − 2r + 1), y = (rs − 2r + 1) / (s² − s − r + 1).

The polynomials f(x,y) generally have substantially lower degrees than F(r,s). When N=101, for example, F(r,s) has degree 170 in r and 255 in s, while f(x,y) has degree 140 in x and 134 in y (it also has fewer terms and smaller coefficients). In most cases the equations f(x,y)=0 listed below achieve the same bi-degree as the optimized equations listed here, and in several cases they are actually identical.

Update July 24,2020: Mark van Hoeij has implemented a maple program that can be used both to recreate these polynomials and compute them for larger values of N.

N	deg_x	deg_y	f(x,y)	N	deg_x	deg_y	f(x,y)	N	deg_x	deg_y	f(x,y)	N	deg_x	deg_y	f(x,y)	N	deg_x	deg_y	f(x,y)	N	deg_x	deg_y	f(x,y)
10	1	1	f(x,y)	40	15	16	f(x,y)	70	47	45	f(x,y)	100	98	95	f(x,y)	130	167	158	f(x,y)	160	252	241	f(x,y)
11	2	2	f(x,y)	41	23	22	f(x,y)	71	69	66	f(x,y)	101	140	134	f(x,y)	131	235	225	f(x,y)	161	347	332	f(x,y)
12	1	1	f(x,y)	42	16	15	f(x,y)	72	47	45	f(x,y)	102	95	91	f(x,y)	132	159	150	f(x,y)	162	241	228	f(x,y)
13	3	2	f(x,y)	43	26	24	f(x,y)	73	73	70	f(x,y)	103	145	139	f(x,y)	133	237	226	f(x,y)	163	364	348	f(x,y)
14	2	2	f(x,y)	44	19	19	f(x,y)	74	56	54	f(x,y)	104	110	106	f(x,y)	134	184	176	f(x,y)	164	276	264	f(x,y)
15	2	2	f(x,y)	45	25	23	f(x,y)	75	66	63	f(x,y)	105	126	121	f(x,y)	135	211	204	f(x,y)	165	317	303	f(x,y)
16	2	3	f(x,y)	46	22	21	f(x,y)	76	59	56	f(x,y)	106	115	110	f(x,y)	136	189	182	f(x,y)	166	282	271	f(x,y)
17	4	4	f(x,y)	47	30	29	f(x,y)	77	79	75	f(x,y)	107	157	150	f(x,y)	137	257	246	f(x,y)	167	382	365	f(x,y)
18	4	3	f(x,y)	48	22	19	f(x,y)	78	57	52	f(x,y)	108	107	102	f(x,y)	138	173	166	f(x,y)	168	253	240	f(x,y)
19	5	5	f(x,y)	49	32	31	f(x,y)	79	85	82	f(x,y)	109	163	156	f(x,y)	139	265	253	f(x,y)	169	388	372	f(x,y)
20	4	3	f(x,y)	50	25	23	f(x,y)	80	64	60	f(x,y)	110	118	114	f(x,y)	140	189	182	f(x,y)	170	283	272	f(x,y)
21	5	5	f(x,y)	51	31	30	f(x,y)	81	79	77	f(x,y)	111	149	143	f(x,y)	141	242	231	f(x,y)	171	356	340	f(x,y)
22	5	4	f(x,y)	52	28	26	f(x,y)	82	69	66	f(x,y)	112	127	120	f(x,y)	142	207	198	f(x,y)	172	304	290	f(x,y)
23	7	7	f(x,y)	53	39	37	f(x,y)	83	95	90	f(x,y)	113	175	167	f(x,y)	143	275	264	f(x,y)	173	410	392	f(x,y)
24	5	6	f(x,y)	54	25	26	f(x,y)	84	63	61	f(x,y)	114	118	114	f(x,y)	144	189	182	f(x,y)	174	277	264	f(x,y)
25	8	8	f(x,y)	55	39	37	f(x,y)	85	95	90	f(x,y)	115	174	166	f(x,y)	145	275	264	f(x,y)	175	394	377	f(x,y)
26	6	7	f(x,y)	56	31	31	f(x,y)	86	75	73	f(x,y)	116	138	132	f(x,y)	146	219	209	f(x,y)	176	316	301	f(x,y)
27	10	8	f(x,y)	57	39	37	f(x,y)	87	91	88	f(x,y)	117	167	158	f(x,y)	147	258	246	f(x,y)	177	380	364	f(x,y)
28	8	7	f(x,y)	58	34	33	f(x,y)	88	79	76	f(x,y)	118	143	136	f(x,y)	148	225	214	f(x,y)	178	326	311	f(x,y)
29	12	11	f(x,y)	59	48	46	f(x,y)	89	108	104	f(x,y)	119	189	181	f(x,y)	149	304	291	f(x,y)	179	439	420	f(x,y)
30	8	8	f(x,y)	60	32	31	f(x,y)	90	70	67	f(x,y)	120	127	119	f(x,y)	150	197	189	f(x,y)	180	283	272	f(x,y)
31	13	13	f(x,y)	61	51	49	f(x,y)	91	110	106	f(x,y)	121	198	190	f(x,y)	151	312	299	f(x,y)	181	449	429	f(x,y)
32	11	10	f(x,y)	62	40	37	f(x,y)	92	87	82	f(x,y)	122	153	146	f(x,y)	152	236	227	f(x,y)	182	332	316	f(x,y)
33	12	12	f(x,y)	63	48	45	f(x,y)	93	105	100	f(x,y)	123	184	176	f(x,y)	153	285	271	f(x,y)	183	407	389	f(x,y)
34	12	11	f(x,y)	64	42	40	f(x,y)	94	91	87	f(x,y)	124	157	151	f(x,y)	154	236	227	f(x,y)	184	346	333	f(x,y)
35	16	15	f(x,y)	65	54	53	f(x,y)	95	118	113	f(x,y)	125	206	196	f(x,y)	155	316	301	f(x,y)	185	449	430	f(x,y)
36	12	11	f(x,y)	66	40	39	f(x,y)	96	83	81	f(x,y)	126	141	136	f(x,y)	156	220	212	f(x,y)	186	315	303	f(x,y)
37	19	18	f(x,y)	67	62	59	f(x,y)	97	129	123	f(x,y)	127	221	211	f(x,y)	157	338	323	f(x,y)	187	473	452	f(x,y)
38	15	14	f(x,y)	68	47	45	f(x,y)	98	97	92	f(x,y)	128	168	161	f(x,y)	158	257	245	f(x,y)	188	363	346	f(x,y)
39	17	18	f(x,y)	69	58	55	f(x,y)	99	120	113	f(x,y)	129	201	194	f(x,y)	159	306	294	f(x,y)	189	425	408	f(x,y)

The table below lists an alternative set of defining equations f(x,y)=0 for X₁(N). For these equations, the birational map map from f(x,y)=0 to F(r,s)=0 is given by

r = (x²(x + 1) + x + y) / (−xy(x + 1) + x + y), s = (x² + x + y) / (−xy + x + y),
and the map from F(r,s)=0 to f(r,s)=0 is defined by

x = −(rs − 2r+1) / (s − r), y = (rs− 2r+1)(s² − s − r+1) / ((s − r)(rs² + s² − 3rs + r)).

N	deg_x	deg_y	f(x,y)	N	deg_x	deg_y	f(x,y)	N	deg_x	deg_y	f(x,y)	N	deg_x	deg_y	f(x,y)	N	deg_x	deg_y	f(x,y)	N	deg_x	deg_y	f(x,y)
				40	17	16	f(x,y)	70	49	45	f(x,y)	100	103	95	f(x,y)	130	173	158	f(x,y)	160	263	241	f(x,y)
11	2	2	f(x,y)	41	24	22	f(x,y)	71	72	66	f(x,y)	101	146	134	f(x,y)	131	245	225	f(x,y)	161	361	332	f(x,y)
12	0	1	f(x,y)	42	17	15	f(x,y)	72	48	45	f(x,y)	102	100	91	f(x,y)	132	165	150	f(x,y)	162	250	228	f(x,y)
13	3	2	f(x,y)	43	27	24	f(x,y)	73	77	70	f(x,y)	103	152	139	f(x,y)	133	247	226	f(x,y)	163	380	348	f(x,y)
14	2	2	f(x,y)	44	20	19	f(x,y)	74	59	54	f(x,y)	104	116	106	f(x,y)	134	192	176	f(x,y)	164	288	264	f(x,y)
15	2	2	f(x,y)	45	26	23	f(x,y)	75	68	63	f(x,y)	105	133	121	f(x,y)	135	221	204	f(x,y)	165	330	303	f(x,y)
16	3	3	f(x,y)	46	23	21	f(x,y)	76	61	56	f(x,y)	106	120	110	f(x,y)	136	199	182	f(x,y)	166	295	271	f(x,y)
17	5	4	f(x,y)	47	32	29	f(x,y)	77	82	75	f(x,y)	107	164	150	f(x,y)	137	269	246	f(x,y)	167	399	365	f(x,y)
18	4	3	f(x,y)	48	21	19	f(x,y)	78	58	52	f(x,y)	108	111	102	f(x,y)	138	181	166	f(x,y)	168	261	240	f(x,y)
19	5	5	f(x,y)	49	33	31	f(x,y)	79	89	82	f(x,y)	109	170	156	f(x,y)	139	276	253	f(x,y)	169	405	372	f(x,y)
20	3	3	f(x,y)	50	25	23	f(x,y)	80	66	60	f(x,y)	110	123	114	f(x,y)	140	199	182	f(x,y)	170	296	272	f(x,y)
21	4	5	f(x,y)	51	31	30	f(x,y)	81	83	77	f(x,y)	111	155	143	f(x,y)	141	251	231	f(x,y)	171	371	340	f(x,y)
22	5	4	f(x,y)	52	28	26	f(x,y)	82	72	66	f(x,y)	112	131	120	f(x,y)	142	216	198	f(x,y)	172	316	290	f(x,y)
23	8	7	f(x,y)	53	41	37	f(x,y)	83	99	90	f(x,y)	113	183	167	f(x,y)	143	288	264	f(x,y)	173	428	392	f(x,y)
24	7	6	f(x,y)	54	27	26	f(x,y)	84	68	61	f(x,y)	114	123	114	f(x,y)	144	199	182	f(x,y)	174	288	264	f(x,y)
25	9	8	f(x,y)	55	41	37	f(x,y)	85	98	90	f(x,y)	115	181	166	f(x,y)	145	288	264	f(x,y)	175	411	377	f(x,y)
26	7	7	f(x,y)	56	34	31	f(x,y)	86	79	73	f(x,y)	116	144	132	f(x,y)	146	228	209	f(x,y)	176	329	301	f(x,y)
27	10	8	f(x,y)	57	41	37	f(x,y)	87	96	88	f(x,y)	117	174	158	f(x,y)	147	270	246	f(x,y)	177	397	364	f(x,y)
28	7	7	f(x,y)	58	36	33	f(x,y)	88	83	76	f(x,y)	118	149	136	f(x,y)	148	233	214	f(x,y)	178	340	311	f(x,y)
29	12	11	f(x,y)	59	50	46	f(x,y)	89	113	104	f(x,y)	119	196	181	f(x,y)	149	317	291	f(x,y)	179	458	420	f(x,y)
30	8	8	f(x,y)	60	35	31	f(x,y)	90	73	67	f(x,y)	120	130	119	f(x,y)	150	207	189	f(x,y)	180	295	272	f(x,y)
31	14	13	f(x,y)	61	53	49	f(x,y)	91	114	106	f(x,y)	121	207	190	f(x,y)	151	326	299	f(x,y)	181	468	429	f(x,y)
32	11	10	f(x,y)	62	41	37	f(x,y)	92	89	82	f(x,y)	122	160	146	f(x,y)	152	247	227	f(x,y)	182	346	316	f(x,y)
33	13	12	f(x,y)	63	51	45	f(x,y)	93	109	100	f(x,y)	123	192	176	f(x,y)	153	298	271	f(x,y)	183	425	389	f(x,y)
34	12	11	f(x,y)	64	44	40	f(x,y)	94	95	87	f(x,y)	124	164	151	f(x,y)	154	247	227	f(x,y)	184	363	333	f(x,y)
35	16	15	f(x,y)	65	57	53	f(x,y)	95	124	113	f(x,y)	125	214	196	f(x,y)	155	329	301	f(x,y)	185	469	430	f(x,y)
36	13	11	f(x,y)	66	42	39	f(x,y)	96	88	81	f(x,y)	126	147	136	f(x,y)	156	232	212	f(x,y)
37	20	18	f(x,y)	67	65	59	f(x,y)	97	135	123	f(x,y)	127	231	211	f(x,y)	157	353	323	f(x,y)
38	16	14	f(x,y)	68	48	45	f(x,y)	98	101	92	f(x,y)	128	175	161	f(x,y)	158	268	245	f(x,y)
39	18	18	f(x,y)	69	59	55	f(x,y)	99	124	113	f(x,y)	129	210	194	f(x,y)	159	319	294	f(x,y)