Normalized defining polynomial
\( x^{20} - x^{19} + 186 x^{18} - 187 x^{17} + 13273 x^{16} - 13461 x^{15} + 465900 x^{14} - 479550 x^{13} + 8563440 x^{12} - 8698080 x^{11} + 81794047 x^{10} - 68039548 x^{9} + 345259493 x^{8} - 12872436 x^{7} + 28194599 x^{6} + 2299307442 x^{5} + 259763990 x^{4} + 2826052345 x^{3} + 10421568750 x^{2} + 7230594575 x + 24518981951 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 10]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(26941966215290889183236469097652013493116161=19^{10}\cdot 41^{19}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $148.43$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $19, 41$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(779=19\cdot 41\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{779}(512,·)$, $\chi_{779}(1,·)$, $\chi_{779}(132,·)$, $\chi_{779}(590,·)$, $\chi_{779}(400,·)$, $\chi_{779}(531,·)$, $\chi_{779}(569,·)$, $\chi_{779}(664,·)$, $\chi_{779}(324,·)$, $\chi_{779}(666,·)$, $\chi_{779}(476,·)$, $\chi_{779}(286,·)$, $\chi_{779}(607,·)$, $\chi_{779}(740,·)$, $\chi_{779}(742,·)$, $\chi_{779}(360,·)$, $\chi_{779}(305,·)$, $\chi_{779}(759,·)$, $\chi_{779}(761,·)$, $\chi_{779}(702,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{166} a^{16} - \frac{9}{83} a^{15} + \frac{33}{166} a^{14} - \frac{15}{83} a^{13} + \frac{21}{83} a^{12} + \frac{27}{83} a^{11} - \frac{75}{166} a^{10} + \frac{69}{166} a^{9} - \frac{4}{83} a^{8} + \frac{77}{166} a^{7} + \frac{22}{83} a^{6} - \frac{32}{83} a^{5} + \frac{77}{166} a^{4} - \frac{1}{166} a^{3} - \frac{15}{83} a^{2} + \frac{11}{83} a + \frac{47}{166}$, $\frac{1}{166} a^{17} + \frac{41}{166} a^{15} + \frac{33}{83} a^{14} - \frac{10}{83} a^{12} + \frac{67}{166} a^{11} + \frac{47}{166} a^{10} + \frac{36}{83} a^{9} - \frac{67}{166} a^{8} - \frac{32}{83} a^{7} + \frac{32}{83} a^{6} - \frac{79}{166} a^{5} + \frac{57}{166} a^{4} - \frac{24}{83} a^{3} - \frac{10}{83} a^{2} - \frac{55}{166} a + \frac{8}{83}$, $\frac{1}{13285478} a^{18} + \frac{11635}{13285478} a^{17} - \frac{18112}{6642739} a^{16} - \frac{2608449}{13285478} a^{15} + \frac{3617249}{13285478} a^{14} - \frac{3017150}{6642739} a^{13} + \frac{5398671}{13285478} a^{12} - \frac{1370555}{6642739} a^{11} - \frac{103296}{6642739} a^{10} + \frac{1841861}{6642739} a^{9} + \frac{4595423}{13285478} a^{8} - \frac{649577}{13285478} a^{7} - \frac{2677431}{13285478} a^{6} + \frac{1029274}{6642739} a^{5} + \frac{2126804}{6642739} a^{4} + \frac{4438177}{13285478} a^{3} + \frac{5617735}{13285478} a^{2} - \frac{3932885}{13285478} a - \frac{1974463}{13285478}$, $\frac{1}{1100727289436683604739351056477011113576546908815116192840547052905279835636705708614057524254} a^{19} - \frac{33323603277225547876775082399036432885285720294030850618712651112785997156397189624623}{1100727289436683604739351056477011113576546908815116192840547052905279835636705708614057524254} a^{18} + \frac{1249332603375606801210488003776997133149508067517263572454361852477615876325417436719876807}{550363644718341802369675528238505556788273454407558096420273526452639917818352854307028762127} a^{17} - \frac{1881873323555614012896134450541631030035409903810978839705913014414162519442126650315882707}{1100727289436683604739351056477011113576546908815116192840547052905279835636705708614057524254} a^{16} - \frac{261131672702682681000182425770297340768399644838081708677960063682077819008642937786134163305}{1100727289436683604739351056477011113576546908815116192840547052905279835636705708614057524254} a^{15} - \frac{101336709870409798369787733637940934797092249038080068403730495989158269751665126087155513001}{550363644718341802369675528238505556788273454407558096420273526452639917818352854307028762127} a^{14} + \frac{518584837692644670690599722383566514539910456343110005396179558373642920759082710102491506421}{1100727289436683604739351056477011113576546908815116192840547052905279835636705708614057524254} a^{13} - \frac{201876288475096232246460572322964533186484693967062439158365570251532611377619976092672022253}{550363644718341802369675528238505556788273454407558096420273526452639917818352854307028762127} a^{12} - \frac{132055814509950518084936272420503255343600025617697576009111395529829523926934154295254020811}{550363644718341802369675528238505556788273454407558096420273526452639917818352854307028762127} a^{11} - \frac{4263900071321703006701654172272316835492830793316492533362497772531970950352394643642251758}{550363644718341802369675528238505556788273454407558096420273526452639917818352854307028762127} a^{10} + \frac{299787227008501572018659135042144656671952681443670243790621867913277907531342174825258354863}{1100727289436683604739351056477011113576546908815116192840547052905279835636705708614057524254} a^{9} + \frac{421218673065096690388370347302837208421273879802870788007456066199721585039772267581796893077}{1100727289436683604739351056477011113576546908815116192840547052905279835636705708614057524254} a^{8} - \frac{34182779955745180300033173598550982520801443246607698935938659559705206172472201442252223969}{1100727289436683604739351056477011113576546908815116192840547052905279835636705708614057524254} a^{7} + \frac{270687303474640564517189868201923872363185810430253591446366168765534701970840541578989061381}{550363644718341802369675528238505556788273454407558096420273526452639917818352854307028762127} a^{6} + \frac{90106430423293696903965900415024848965262625256679121029264395594873436463145245161275763648}{550363644718341802369675528238505556788273454407558096420273526452639917818352854307028762127} a^{5} + \frac{163334706574743260362327937190834879627508221798156006441768911456139384405638617803183705839}{1100727289436683604739351056477011113576546908815116192840547052905279835636705708614057524254} a^{4} - \frac{109886612375451601362587797789521972551811091128101863650047033963785094315825280088579792157}{1100727289436683604739351056477011113576546908815116192840547052905279835636705708614057524254} a^{3} + \frac{27525308032118696169354646706483771911659338489983590078623911255990002977710927747457003237}{1100727289436683604739351056477011113576546908815116192840547052905279835636705708614057524254} a^{2} + \frac{112379394186935860567803561620531508896752581783562729424635375235913785545782233362639134809}{1100727289436683604739351056477011113576546908815116192840547052905279835636705708614057524254} a + \frac{151043333551141782594519406878028497703990094073170979835739192424146368792742048744311409679}{550363644718341802369675528238505556788273454407558096420273526452639917818352854307028762127}$
Class group and class number
$C_{3}\times C_{3}\times C_{3}\times C_{162606}$, which has order $4390362$ (assuming GRH)
Unit group
| Rank: | $9$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 5104264.636551031 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 20 |
| The 20 conjugacy class representatives for $C_{20}$ |
| Character table for $C_{20}$ |
Intermediate fields
| \(\Q(\sqrt{41}) \), 4.0.24880481.1, 5.5.2825761.1, 10.10.327381934393961.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ | $20$ | $20$ | $20$ | $20$ | R | ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ | $20$ | ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ | $20$ | $20$ | ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| 19 | Data not computed | ||||||
| 41 | Data not computed | ||||||