Sage Reference Manual: Elliptic and Plane Curves - Mirrors

More documents

Recommendations

Info

Sage Reference Manual: Elliptic and Plane Curves, Release 6.1.10sage: i[0](p) / i[1](p)x/yA ValueError is raised if self has no rational pointsage: C = Conic(x^2 + y^2 + 7*z^2)sage: C.parametrization()Traceback (most recent call last):...ValueError: Conic Projective Conic Curve over Rational Field defined by x^2 + y^2 + 7*z^2 haA ValueError is raised if self is not smoothsage: C = Conic(x^2 + y^2)sage: C.parametrization()Traceback (most recent call last):...ValueError: The conic self (=Projective Conic Curve over Rational Field defined by x^2 + y^2point(v, check=True)Constructs a point on self corresponding to the input v.If check is True, then checks if v defines a valid point on self.If no rational point on self is known yet, then also caches the point for use byself.rational_point() and self.parametrization().EXAMPLESsage: c = Conic([1, -1, 1])sage: c.point([15, 17, 8])(15/8 : 17/8 : 1)sage: c.rational_point()(15/8 : 17/8 : 1)sage: d = Conic([1, -1, 1])sage: d.rational_point()(1 : 1 : 0)random_rational_point(*args1, **args2)Return a random rational point of the conic self.ALGORITHM:1.Compute a parametrization f of self using self.parametrization().2.Computes a random point (x : y) on the projective line.3.Output f(x : y).The coordinates x and y are computed using B.random_element, where B is the base field of selfand additional arguments to random_rational_point are passed to random_element.If the base field is a finite field, then the output is uniformly distributed over the points of self.EXAMPLESsage: c = Conic(GF(2), [1,1,1,1,1,0])sage: [c.random_rational_point() for i in range(10)] # output is random[(1 : 0 : 1), (1 : 0 : 1), (1 : 0 : 1), (0 : 1 : 1), (1 : 0 : 1), (0 : 0 : 1), (1 : 0 : 1),sage: d = Conic(QQ, [1, 1, -1])sage: d.random_rational_point(den_bound = 1, num_bound = 5) # output is random24 Chapter 5. Projective plane conics over a field
Sage Reference Manual: Elliptic and Plane Curves, Release 6.1.1(-24/25 : 7/25 : 1)sage: Conic(QQ, [1, 1, 1]).random_rational_point()Traceback (most recent call last):...ValueError: Conic Projective Conic Curve over Rational Field defined by x^2 + y^2 + z^2 hasrational_point(algorithm=’default’, read_cache=True)Return a point on self defined over the base field.Raises ValueError if no rational point exists.See self.has_rational_point for the algorithm used and for the use of the parametersalgorithm and read_cache.EXAMPLES:Examples over Qsage: R. = QQ[]sage: C = Conic(7*x^2 + 2*y*z + z^2)sage: C.rational_point()(0 : 1 : 0)sage: C = Conic(x^2 + 2*y^2 + z^2)sage: C.rational_point()Traceback (most recent call last):...ValueError: Conic Projective Conic Curve over Rational Field defined by x^2 + 2*y^2 + z^2 hasage: C = Conic(x^2 + y^2 + 7*z^2)sage: C.rational_point(algorithm = ’rnfisnorm’)Traceback (most recent call last):...ValueError: Conic Projective Conic Curve over Rational Field defined by x^2 + y^2 + 7*z^2 haExamples over number fieldssage: P. = QQ[]sage: L. = NumberField(x^3-5)sage: C = Conic(L, [3, 2, -5])sage: p = C.rational_point(algorithm = ’rnfisnorm’)sage: p# output is random(60*b^2 - 196*b + 161 : -120*b^2 - 6*b + 361 : 1)sage: C.defining_polynomial()(list(p))0sage: K. = QuadraticField(-1)sage: D = Conic(K, [3, 2, 5])sage: D.rational_point(algorithm = ’rnfisnorm’) # output is random(-3 : 4*i : 1)sage: L. = QuadraticField(2)sage: Conic(QQ, [1, 1, -3]).has_rational_point()Falsesage: E = Conic(L, [1, 1, -3])sage: E.rational_point()# output is random(-1 : -s : 1)Currently Magma is better at solving conics over number fields than Sage, so it helps to use the algorithm25
Page 1: Sage Reference Manual: Elliptic and
Page 4: 22 Kodaira symbols 31123 Isomorphis
Page 14 and 15: Sage Reference Manual: Elliptic and
Page 41 and 42: CHAPTEREIGHTPROJECTIVE PLANE CONICS
Page 43 and 44: CHAPTERNINEPROJECTIVE PLANE CONICS
Page 45 and 46: CHAPTERTENELLIPTIC CURVE CONSTRUCTO
Page 57 and 58: CHAPTERELEVENCONSTRUCT ELLIPTIC CUR
Page 61 and 62: CHAPTERTWELVEELLIPTIC CURVES OVER A
Page 79 and 80:
Sage Reference Manual: Elliptic and
Page 81 and 82:
Page 83 and 84:
CHAPTERTHIRTEENELLIPTIC CURVES OVER
Page 85 and 86:
Page 87 and 88:
Page 89 and 90:
Page 91 and 92:
Page 93 and 94:
Page 95 and 96:
CHAPTERFOURTEENELLIPTIC CURVES OVER
Page 97 and 98:
Page 99 and 100:
Page 101 and 102:
Page 103 and 104:
Page 105 and 106:
Page 107 and 108:
Page 109 and 110:
Page 111 and 112:
Page 113 and 114:
Page 115 and 116:
Page 117 and 118:
Page 119 and 120:
Page 121 and 122:
Page 123 and 124:
Page 125 and 126:
Page 127 and 128:
Page 129 and 130:
Page 131 and 132:
Page 133 and 134:
Page 135 and 136:
Page 137 and 138:
Page 139 and 140:
Page 141 and 142:
Page 143 and 144:
Page 145 and 146:
Page 147 and 148:
Page 149 and 150:
Page 151 and 152:
Page 153 and 154:
Page 155 and 156:
Page 157 and 158:
Page 159 and 160:
Page 161 and 162:
Page 163 and 164:
Page 165 and 166:
Page 167 and 168:
Page 169 and 170:
Page 171 and 172:
CHAPTERFIFTEENHEEGNER POINTS ON ELL
Page 173 and 174:
Page 175 and 176:
Page 177 and 178:
Page 179 and 180:
Page 181 and 182:
Page 183 and 184:
Page 185 and 186:
Page 187 and 188:
Page 189 and 190:
Page 191 and 192:
Page 193 and 194:
Page 195 and 196:
Page 197 and 198:
Page 199 and 200:
Page 201 and 202:
Page 203 and 204:
Page 205 and 206:
Page 207 and 208:
Page 209 and 210:
Page 211 and 212:
Page 213 and 214:
Page 215 and 216:
Page 217 and 218:
Page 219 and 220:
Page 221 and 222:
Page 223 and 224:
CHAPTERSIXTEENTABLES OF ELLIPTIC CU
Page 225 and 226:
CHAPTERSEVENTEENELLIPTIC CURVES OVE
Page 227 and 228:
Page 229 and 230:
Page 231 and 232:
Page 233 and 234:
Page 235 and 236:
Page 237 and 238:
Page 239 and 240:
Page 241 and 242:
Page 243 and 244:
Page 245 and 246:
Page 247 and 248:
Page 249 and 250:
Page 251 and 252:
Page 253 and 254:
CHAPTEREIGHTEENELLIPTIC CURVES OVER
Page 255 and 256:
Page 257 and 258:
Page 259 and 260:
Page 261 and 262:
Page 263 and 264:
Page 265 and 266:
Page 267 and 268:
Page 269 and 270:
Page 271 and 272:
CHAPTERNINETEENPOINTS ON ELLIPTIC C
Page 273 and 274:
Page 275 and 276:
Page 277 and 278:
Page 279 and 280:
Page 281 and 282:
Page 283 and 284:
Page 285 and 286:
Page 287 and 288:
Page 289 and 290:
Page 291 and 292:
Page 293 and 294:
Page 295 and 296:
Page 297 and 298:
Page 299 and 300:
Page 301 and 302:
Page 303 and 304:
CHAPTERTWENTYTORSION SUBGROUPS OF E
Page 305 and 306:
Page 307 and 308:
CHAPTERTWENTYONELOCAL DATA FOR ELLI
Page 309 and 310:
Page 311 and 312:
Page 313 and 314:
Page 315 and 316:
CHAPTERTWENTYTWOKODAIRA SYMBOLSKoda
Page 317 and 318:
CHAPTERTWENTYTHREEISOMORPHISMS BETW
Page 319 and 320:
CHAPTERTWENTYFOURISOGENIESAn isogen
Page 321 and 322:
Page 323 and 324:
Page 325 and 326:
Page 327 and 328:
Page 329 and 330:
Page 331 and 332:
Page 333 and 334:
Page 335 and 336:
Page 337 and 338:
Page 339 and 340:
Page 341 and 342:
Page 343 and 344:
CHAPTERTWENTYFIVEISOGENIES OF SMALL
Page 345 and 346:
Page 347 and 348:
Page 349 and 350:
Page 351 and 352:
Page 353 and 354:
Page 355 and 356:
Page 357 and 358:
Page 359 and 360:
CHAPTERTWENTYSIXWEIERSTRASS ℘ FUN
Page 361 and 362:
Page 363 and 364:
CHAPTERTWENTYSEVENPERIOD LATTICES O
Page 365 and 366:
Page 367 and 368:
Page 369 and 370:
Page 371 and 372:
Page 373 and 374:
Page 375 and 376:
Page 377 and 378:
Page 379 and 380:
CHAPTERTWENTYEIGHTFORMAL GROUPS OF
Page 381 and 382:
Page 383 and 384:
Page 385 and 386:
CHAPTERTWENTYNINETATE’S PARAMETRI
Page 387 and 388:
Page 389 and 390:
Page 391 and 392:
CHAPTERTHIRTYCOMPUTATION OF FROBENI
Page 393 and 394:
Page 395 and 396:
Page 397 and 398:
Page 399 and 400:
Page 401 and 402:
Page 403 and 404:
Page 405 and 406:
Page 407 and 408:
Page 409 and 410:
Page 411 and 412:
Page 413 and 414:
Page 415 and 416:
Page 417 and 418:
CHAPTERTHIRTYONEP-ADIC L-FUNCTIONS
Page 419 and 420:
Page 421 and 422:
Page 423 and 424:
Page 425 and 426:
Page 427 and 428:
Page 429 and 430:
Page 431 and 432:
CHAPTERTHIRTYTWOMODULAR SYMBOLSTo a
Page 433 and 434:
Page 435 and 436:
Page 437 and 438:
CHAPTERTHIRTYTHREEMODULAR PARAMETRI
Page 439 and 440:
CHAPTERTHIRTYFOURGALOIS REPRESENTAT
Page 441 and 442:
Page 443 and 444:
Page 445 and 446:
Page 447 and 448:
Page 449 and 450:
Page 451 and 452:
CHAPTERTHIRTYFIVETATE-SHAFAREVICH G
Page 453 and 454:
Page 455 and 456:
Page 457 and 458:
Page 459 and 460:
Page 461 and 462:
CHAPTERTHIRTYSIXMISCELLANEOUS P-ADI
Page 463 and 464:
Page 465 and 466:
Page 467 and 468:
Page 469 and 470:
Page 471 and 472:
Page 473 and 474:
CHAPTERTHIRTYSEVENCOMPLEX MULTIPLIC
Page 475 and 476:
Page 477 and 478:
Page 479 and 480:
CHAPTERTHIRTYEIGHTHYPERELLIPTIC CUR
Page 481 and 482:
Page 483 and 484:
Page 485 and 486:
Page 487 and 488:
Page 489 and 490:
Page 491 and 492:
Page 493 and 494:
Page 495 and 496:
Page 497 and 498:
Page 499 and 500:
Page 501 and 502:
Page 503 and 504:
Page 505 and 506:
Page 507 and 508:
CHAPTERTHIRTYNINEINDICES AND TABLES
Page 509 and 510:
BIBLIOGRAPHY[WpJacobianVariety] htt
Page 511 and 512:
PYTHON MODULE INDEXisage.interfaces
Page 513 and 514:
INDEXAa1() (sage.schemes.elliptic_c
Page 515 and 516:
Page 517 and 518:
Page 519 and 520:
Page 521 and 522:
Page 523 and 524:
Page 525 and 526:
Page 527 and 528:
Page 529 and 530:
Page 531 and 532:
show all

Sage Reference Manual: Elliptic and Plane Curves - Mirrors

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?