Any more cyclic quintics?
Yes, there are infinitely many cyclic quintics as parameterized by the Emma Lehmer quintic
$$F(y)=y^5 + n^2y^4 - (2n^3 + 6n^2 + 10n + 10)y^3 + (n^4 + 5n^3 + 11n^2 + 15n + 5)y^2 + (n^3 + 4n^2 + 10n + 10)y + 1 = 0$$
This also obeys
$$ y_1 y_2 + y_2 y_3 + y_3 y_4 + y_4 y_5 + y_5 y_1 - (y_1 y_3 + y_3 y_5 + y_5 y_2 + y_2 y_4 + y_4 y_1) = 0$$
Let $p=25 + 25 n + 15 n^2 + 5 n^3 + n^4$. Then the discriminant of $F(y)$ is
$$D = (7 + 10 n + 5 n^2 + n^3)^2\,p^4$$
Also, note that if $m=n+1$, then $n\,p=m^5 + 5m^3 + 5m - 11$. A root is given by $$y = a+b\sum_{k=1}^{(p-1)/5}\,{\zeta_p}^{c^k}$$
with root of unity $\zeta_p = e^{2\pi i/p},\,$ for some integer $a,b,c$. See this MO post for the formulas for $a,b,c$.
P.S. While $p=151$ does not belong to this family, I find that,
$$x^5 + x^4 - 60x^3 - 12x^2 + 784x + 128 = 0$$
with discriminant $d=2^{18}151^4$ has the root $\displaystyle x=\sum_{k=1}^{30}e^{2\pi\, i\, c^k/151}$ for $c=23$. The Mathematica command to find these quintics is,
Table[{c,Recognize[N[Sum[E^(2Pi I c^k/p),{k,1,(p-1)/5}],50],5,x]},{c,p/2}]
for prime $p\equiv1\pmod{10}$. Inspecting the resulting table of candidate quintics, identical ones with small coefficients will stand out and which gives the correct choice of $c$.
There is also an infinite number of cyclic septics, such as the Hashimoto-Hoshi, $$\small x^7 - (a^3 + a^2 + 5a + 6)x^6 + 3(3a^3 + 3a^2 + 8a + 4)x^5 + (a^7 + a^6 + 9a^5 - 5a^4 - 15a^3 - 22a^2 - 36a - 8)x^4 - a(a^7 + 5a^6 + 12a^5 + 24a^4 - 6a^3 + 2a^2 - 20a - 16)x^3 + a^2(2a^6 + 7a^5 + 19a^4 + 14a^3 + 2a^2 + 8a - 8)x^2 - a^4(a^4 + 4a^3 + 8a^2 + 4)x + a^7=0$$
Similar to the Lehmer quintic, the roots of this septic obeys $$ x_1 x_2 + x_2 x_3 + \dots + x_7 x_1 - (x_1 x_3 + x_3 x_5 + \dots + x_6 x_1) = 0$$ For example, let $a=1$ so, $$1 - 17 x + 44 x^2 - 2 x^3 - 75 x^4 + 54 x^5 - 13 x^6 + x^7=0$$ which is the equation involved in $\cos\frac{\pi k}{43}$. See also this post On solvable quintics and septics.
The method of Gauss for this problem is presented in Chapter 9 of Galois Theory by David A. Cox. This was worked out about 30 years before Galois Theory. After doing primes 31, 61, 71 by hand as illustrated there, I was able to write a straightforward program in C++. The input is the prime $p = 10 n + 1$ and a primitive root for that prime. I could have just told the computer to find a primitive root, since I intended primes smaller than 1000 in any case. As I did more of them, I had the machine give a better output; still, for all of these, you will be able to read the quintic polynomial and the collection of exponents of the original $\zeta = e^{2 \pi i / p};$ the sum of these $\zeta^k$ gives one of the five real roots.
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sofar 1 1 1 1 1 1
prime was 11
primitive root used was 2
smallest generator is 10
actual value of the constant a, usually 2 but not always, was 2
polynomial is x^5 + x^4 - 4 x^3 - 3 x^2 + 3 x + 1
list of the 2 exponents
1 10
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sofar 185 185 185 185 185 185
prime was 31
primitive root used was 3
smallest generator is 6
actual value of the constant a, usually 2 but not always, was 2
polynomial is x^5 + x^4 - 12 x^3 - 21 x^2 + 1 x + 5
list of the 6 exponents
1 5 6 25 26 30
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sofar 711 711 711 711 711 711
prime was 41
primitive root used was 6
smallest generator is 3
actual value of the constant a, usually 2 but not always, was 2
polynomial is x^5 + x^4 - 16 x^3 + 5 x^2 + 21 x - 9
list of the 8 exponents
1 3 9 14 27 32 38 40
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
sofar 3707 3707 3707 3707 3707 3707
prime was 61
primitive root used was 2
smallest generator is 21
actual value of the constant a, usually 2 but not always, was 2
polynomial is x^5 + x^4 - 24 x^3 - 17 x^2 + 41 x - 13
list of the 12 exponents
1 11 13 14 21 29 32 40 47 48
50 60
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sofar 7141 7141 7141 7141 7141 7141
prime was 71
primitive root used was 7
smallest generator is 23
actual value of the constant a, usually 2 but not always, was 2
polynomial is x^5 + x^4 - 28 x^3 + 37 x^2 + 25 x + 1
list of the 14 exponents
1 20 23 26 30 32 34 37 39 41
45 48 51 70
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sofar 30463 30463 30463 30463 30463 30463
prime was 101
primitive root used was 2
smallest generator is 32
actual value of the constant a, usually 2 but not always, was 2
polynomial is x^5 + x^4 - 40 x^3 + 93 x^2 - 21 x - 17
list of the 20 exponents
1 6 10 14 17 32 36 39 41 44
57 60 62 65 69 84 87 91 95 100
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sofar 86773 86773 86773 86773 86773 86773
prime was 131
primitive root used was 2
smallest generator is 18
actual value of the constant a, usually 2 but not always, was 2
polynomial is x^5 + x^4 - 52 x^3 - 89 x^2 + 109 x + 193
list of the 26 exponents
1 18 19 24 32 39 45 47 51 52
60 62 63 68 69 71 79 80 84 86
92 99 107 112 113 130
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sofar 155648 155648 155648 155648 155648 155648
prime was 151
primitive root used was 6
smallest generator is 23
actual value of the constant a, usually 2 but not always, was 3
polynomial is x^5 + x^4 - 60 x^3 - 12 x^2 + 784 x + 128
list of the 30 exponents
1 2 4 8 16 19 23 32 33 38
46 59 64 66 75 76 85 87 92 105
113 118 119 128 132 135 143 147 149 150
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sofar 323951 323951 323951 323951 323951 323951
prime was 181
primitive root used was 2
smallest generator is 17
actual value of the constant a, usually 2 but not always, was 2
polynomial is x^5 + x^4 - 72 x^3 - 123 x^2 + 223 x - 49
list of the 36 exponents
1 7 17 19 26 32 39 43 48 49
61 62 65 72 73 80 88 89 92 93
101 108 109 116 119 120 132 133 138 142
149 155 162 164 174 180
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sofar 401125 401125 401125 401125 401125 401125
prime was 191
primitive root used was 19
smallest generator is 11
actual value of the constant a, usually 2 but not always, was 2
polynomial is x^5 + x^4 - 76 x^3 - 359 x^2 - 437 x - 155
list of the 38 exponents
1 5 6 11 14 25 30 31 32 36
37 38 41 52 55 66 69 70 84 107
121 122 125 136 139 150 153 154 155 159
160 161 166 177 180 185 186 190
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sofar 604481 604481 604481 604481 604481 604481
prime was 211
primitive root used was 2
smallest generator is 26
actual value of the constant a, usually 2 but not always, was 2
polynomial is x^5 + x^4 - 84 x^3 - 59 x^2 + 1661 x + 269
list of the 42 exponents
1 12 14 15 26 31 32 33 34 38
40 43 50 54 58 63 67 73 88 94
101 110 117 123 138 144 148 153 157 161
168 171 173 177 178 179 180 185 196 197
199 210
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sofar 1033472 1033472 1033472 1033472 1033472 1033472
prime was 241
primitive root used was 7
smallest generator is 11
actual value of the constant a, usually 2 but not always, was 3
polynomial is x^5 + x^4 - 96 x^3 - 212 x^2 + 1232 x + 512
list of the 48 exponents
1 2 4 8 11 15 16 19 22 30
32 38 44 60 63 64 65 76 88 89
111 113 115 120 121 126 128 130 152 153
165 176 177 178 181 197 203 209 211 219
222 225 226 230 233 237 239 240
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sofar 1220224 1220224 1220224 1220224 1220224 1220224
prime was 251
primitive root used was 6
smallest generator is 2
actual value of the constant a, usually 2 but not always, was 3
polynomial is x^5 + x^4 - 100 x^3 - 20 x^2 + 1504 x + 1024
list of the 50 exponents
1 2 4 5 8 10 16 20 25 32
40 47 50 51 63 64 69 80 91 94
100 102 113 123 125 126 128 138 149 151
157 160 171 182 187 188 200 201 204 211
219 226 231 235 241 243 246 247 249 250
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sofar 1658645 1658645 1658645 1658645 1658645 1658645
prime was 271
primitive root used was 6
smallest generator is 12
actual value of the constant a, usually 2 but not always, was 2
polynomial is x^5 + x^4 - 108 x^3 - 401 x^2 - 13 x + 845
list of the 54 exponents
1 5 12 13 23 25 28 29 32 33
54 60 65 77 83 88 93 102 106 111
113 114 115 125 126 127 131 140 144 145
146 156 157 158 160 165 169 178 183 188
194 206 211 217 238 239 242 243 246 248
258 259 266 270
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sofar 1923223 1923223 1923223 1923223 1923223 1923223
prime was 281
primitive root used was 3
smallest generator is 6
actual value of the constant a, usually 2 but not always, was 2
polynomial is x^5 + x^4 - 112 x^3 - 191 x^2 + 2257 x + 967
list of the 56 exponents
1 6 10 28 32 34 36 37 38 39
47 53 59 60 65 73 77 79 88 89
92 99 100 109 113 116 124 134 147 157
165 168 172 181 182 189 192 193 202 204
208 216 221 222 228 234 242 243 244 245
247 249 253 271 275 280
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sofar 2904781 2904781 2904781 2904781 2904781 2904781
prime was 311
primitive root used was 17
smallest generator is 11
actual value of the constant a, usually 2 but not always, was 2
polynomial is x^5 + x^4 - 124 x^3 + 535 x^2 - 413 x - 539
list of the 62 exponents
1 7 11 13 15 18 20 24 32 41
46 47 49 51 61 68 77 83 86 87
89 91 105 113 116 121 126 140 142 143
146 165 168 169 171 185 190 195 198 206
220 222 224 225 228 234 243 250 260 262
264 265 270 279 287 291 293 296 298 300
304 310
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sofar 3714113 3714113 3714113 3714113 3714113 3714113
prime was 331
primitive root used was 3
smallest generator is 13
actual value of the constant a, usually 2 but not always, was 2
polynomial is x^5 + x^4 - 132 x^3 - 887 x^2 - 1843 x - 1027
list of the 66 exponents
1 13 23 31 32 34 38 47 48 51
57 61 72 74 79 80 85 88 89 95
108 111 112 119 120 131 132 133 146 151
162 163 164 167 168 169 180 185 198 199
200 211 212 219 220 223 236 242 243 246
251 252 257 259 270 274 280 283 284 293
297 299 300 308 318 330
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sofar 8075491 8075491 8075491 8075491 8075491 8075491
prime was 401
primitive root used was 3
smallest generator is 26
actual value of the constant a, usually 2 but not always, was 2
polynomial is x^5 + x^4 - 160 x^3 + 369 x^2 + 879 x - 29
list of the 80 exponents
1 20 22 26 29 30 32 33 35 39
45 48 56 68 72 76 83 84 98 102
108 114 119 126 133 142 147 148 151 153
155 157 158 162 164 171 179 188 189 199
202 212 213 222 230 237 239 243 244 246
248 250 253 254 259 268 275 282 287 293
299 303 317 318 325 329 333 345 353 356
362 366 368 369 371 372 375 379 381 400
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sofar 9819947 9819947 9819947 9819947 9819947 9819947
prime was 421
primitive root used was 2
smallest generator is 32
actual value of the constant a, usually 2 but not always, was 2
polynomial is x^5 + x^4 - 168 x^3 + 219 x^2 + 3853 x - 3517
list of the 84 exponents
1 6 20 21 29 32 33 36 51 52
70 75 86 93 95 109 110 111 112 115
120 122 126 135 137 149 152 159 170 171
174 176 178 182 184 188 192 195 198 202
205 207 214 216 219 223 226 229 233 237
239 243 245 247 250 251 262 269 272 284
286 295 299 301 306 309 310 311 312 326
328 335 346 351 369 370 385 388 389 392
400 401 415 420
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sofar 14139931 14139931 14139931 14139931 14139931 14139931
prime was 461
primitive root used was 2
smallest generator is 13
actual value of the constant a, usually 2 but not always, was 2
polynomial is x^5 + x^4 - 184 x^3 - 129 x^2 + 4551 x + 5419
list of the 92 exponents
1 13 14 20 21 22 23 30 32 33
37 38 41 45 48 57 61 68 71 72
86 102 108 113 124 129 134 139 145 153
162 163 167 169 175 179 181 182 186 188
196 199 201 211 218 229 232 243 250 260
262 265 273 275 279 280 282 286 292 294
298 299 308 316 322 327 332 337 348 353
359 375 389 390 393 400 404 413 416 420
423 424 428 429 431 438 439 440 441 447
448 460
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sofar 18223497 18223497 18223497 18223497 18223497 18223497
prime was 491
primitive root used was 2
smallest generator is 32
actual value of the constant a, usually 2 but not always, was 2
polynomial is x^5 + x^4 - 196 x^3 + 59 x^2 + 2019 x + 1377
list of the 98 exponents
1 3 9 14 17 27 32 35 37 42
43 46 51 53 77 80 81 96 97 104
105 109 111 113 115 118 126 129 137 138
146 152 153 158 159 164 176 178 179 196
200 202 203 223 229 231 238 240 243 248
251 253 260 262 268 288 289 291 295 312
313 315 327 332 333 338 339 345 353 354
362 365 373 376 378 380 382 386 387 394
395 410 411 414 438 440 445 448 449 454
456 459 464 474 477 482 488 490
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sofar 23112547 23112547 23112547 23112547 23112547 23112547
prime was 521
primitive root used was 3
smallest generator is 24
actual value of the constant a, usually 2 but not always, was 2
polynomial is x^5 + x^4 - 208 x^3 - 771 x^2 + 4143 x + 2083
list of the 104 exponents
1 10 18 24 29 32 34 39 42 43
46 52 55 56 61 62 74 75 89 91
98 99 100 101 106 114 131 132 135 152
175 176 180 181 187 197 201 205 206 213
214 219 226 229 231 235 237 240 243 247
253 255 266 268 274 278 281 284 286 290
292 295 302 307 308 315 316 320 324 334
340 341 345 346 369 386 389 390 407 415
420 421 422 423 430 432 446 447 459 460
465 466 469 475 478 479 482 487 489 492
497 503 511 520
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