Normalized defining polynomial
\( x^{20} - 1120 x^{18} - 5285397 x^{16} - 2948764734 x^{14} + 1452248809698 x^{12} + 1562655693365106 x^{10} + 302048559639192771 x^{8} - 33560109862573395248 x^{6} - 10712944772094516747441 x^{4} - 147763063047639772279068 x^{2} + 18412673999053943482324441 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[8, 6]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(74478300488702629539932974793088386039042768856289940340736=2^{40}\cdot 11^{16}\cdot 23^{8}\cdot 199^{4}\cdot 331^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $878.22$ | 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: | $2, 11, 23, 199, 331$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{11} a^{10} + \frac{1}{11} a^{8} - \frac{4}{11} a^{6} - \frac{3}{11} a^{4} + \frac{3}{11} a^{2} + \frac{1}{11}$, $\frac{1}{11} a^{11} + \frac{1}{11} a^{9} - \frac{4}{11} a^{7} - \frac{3}{11} a^{5} + \frac{3}{11} a^{3} + \frac{1}{11} a$, $\frac{1}{11} a^{12} - \frac{5}{11} a^{8} + \frac{1}{11} a^{6} - \frac{5}{11} a^{4} - \frac{2}{11} a^{2} - \frac{1}{11}$, $\frac{1}{11} a^{13} - \frac{5}{11} a^{9} + \frac{1}{11} a^{7} - \frac{5}{11} a^{5} - \frac{2}{11} a^{3} - \frac{1}{11} a$, $\frac{1}{724559} a^{14} - \frac{1120}{724559} a^{12} - \frac{15877}{724559} a^{10} - \frac{138949}{724559} a^{8} - \frac{75610}{724559} a^{6} - \frac{34830}{724559} a^{4} - \frac{111575}{724559} a^{2} - \frac{1}{11}$, $\frac{1}{724559} a^{15} - \frac{1120}{724559} a^{13} - \frac{15877}{724559} a^{11} - \frac{138949}{724559} a^{9} - \frac{75610}{724559} a^{7} - \frac{34830}{724559} a^{5} - \frac{111575}{724559} a^{3} - \frac{1}{11} a$, $\frac{1}{1097697465733} a^{16} - \frac{1120}{1097697465733} a^{14} - \frac{5285397}{1097697465733} a^{12} - \frac{2948764734}{1097697465733} a^{10} + \frac{55179307856}{1097697465733} a^{8} + \frac{233036912235}{1097697465733} a^{6} - \frac{461216693907}{1097697465733} a^{4} + \frac{149367}{724559} a^{2} - \frac{3}{11}$, $\frac{1}{1097697465733} a^{17} - \frac{1120}{1097697465733} a^{15} - \frac{5285397}{1097697465733} a^{13} - \frac{2948764734}{1097697465733} a^{11} + \frac{55179307856}{1097697465733} a^{9} + \frac{233036912235}{1097697465733} a^{7} - \frac{461216693907}{1097697465733} a^{5} + \frac{149367}{724559} a^{3} - \frac{3}{11} a$, $\frac{1}{477295678310647473314508123608136050973622088523313693813225100824272481238605827074818807282782077} a^{18} - \frac{136033622397625575794107056909708207979296503248204096667789751268913811900858267219676}{477295678310647473314508123608136050973622088523313693813225100824272481238605827074818807282782077} a^{16} + \frac{211160217282377630658067590998354889875643377501699280920982196909546624830738101548856588978}{477295678310647473314508123608136050973622088523313693813225100824272481238605827074818807282782077} a^{14} + \frac{20018489634853371990901979831259080447627170357952442639604827188418024271754931551489473714712131}{477295678310647473314508123608136050973622088523313693813225100824272481238605827074818807282782077} a^{12} + \frac{1296999811857907982906585476287173527058282660827089809110367257152407721309897249895155000634722}{477295678310647473314508123608136050973622088523313693813225100824272481238605827074818807282782077} a^{10} + \frac{27580178245235843535440417309955402006534590419751864039088908773599045061348382917730130368023976}{477295678310647473314508123608136050973622088523313693813225100824272481238605827074818807282782077} a^{8} - \frac{25420153096228197297459777663653812084359362192187924523106697914360869102133660104660855683519393}{477295678310647473314508123608136050973622088523313693813225100824272481238605827074818807282782077} a^{6} + \frac{3010051516616800319383853691831238203701498345709582399288982352637008005246478618187589455431}{7246135182113702550737192360718032017696064742493641831714844628342201661458437612151676923633} a^{4} - \frac{817332401568457587113924993215925950163002454845351921134046151878124006435857911145842}{4782968554920736977107521292735866392052251763542289030674748118856598546032697054266259} a^{2} - \frac{12222347232548521214933863816130431675275857149347846692103878086558994503714956494}{72613347020916318406344734134962826095010578019133264975553722067385242618419849311}$, $\frac{1}{20523714167357841352523849315149850191865749806502488833968679335443716693260050564217208713159629311} a^{19} - \frac{1440479591921462933187067409474035405811728802396101105033630147320713415364321100201383}{20523714167357841352523849315149850191865749806502488833968679335443716693260050564217208713159629311} a^{17} + \frac{10752454188933629856843354776728546234804218963485678498973762306650872511598890070324235262466}{20523714167357841352523849315149850191865749806502488833968679335443716693260050564217208713159629311} a^{15} + \frac{876031095399619347788749412971605268859026365964018863367933880316655010810567436662703626292139190}{20523714167357841352523849315149850191865749806502488833968679335443716693260050564217208713159629311} a^{13} + \frac{618831867448647914108581953431836868762266407061860734652540606224564228122992237423549866380786490}{20523714167357841352523849315149850191865749806502488833968679335443716693260050564217208713159629311} a^{11} + \frac{6388176448313686214777959327304079181965607912026256500540200904067628766627968821889110356278291106}{20523714167357841352523849315149850191865749806502488833968679335443716693260050564217208713159629311} a^{9} + \frac{309472917102700202690741052055266268063273329901866566671012575262278223246949907923958591317633544}{892335398580775710979297796310863051820249991587064731911681710236683334489567415835530813615636057} a^{7} + \frac{144247170073590143774243628559326112834609663338764750953842185843612111532039039066143917985119}{311583812830889209681699271510875376760930783927226598763738319018714671442712817322522107716219} a^{5} - \frac{38170225617117344643732034839635168392665757964382385502793682990728728926314031975505031}{205667647861591690015623415587642254858246825832318428319014169110833737479405973333449137} a^{3} + \frac{1090180284812271949135936190778305200858066554595675357936757175118289688885022755773}{3122373921899401691472823567803401522085454854822730393948810048897565432592053520373} a$
Class group and class number
$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}$, which has order $32$ (assuming GRH)
Unit group
| Rank: | $13$ | 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: | \( 246612959891000000000 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 5120 |
| The 56 conjugacy class representatives for t20n331 are not computed |
| Character table for t20n331 is not computed |
Intermediate fields
| \(\Q(\zeta_{11})^+\), 10.10.2670699013250048.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| $11$ | 11.10.8.5 | $x^{10} - 2321 x^{5} + 2033647$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ |
| 11.10.8.5 | $x^{10} - 2321 x^{5} + 2033647$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ | |
| $23$ | $\Q_{23}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{23}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{23}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{23}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{23}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{23}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 23.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.4.3.2 | $x^{4} - 23$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| 23.4.3.1 | $x^{4} + 46$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| 23.4.2.1 | $x^{4} + 299 x^{2} + 25921$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $199$ | $\Q_{199}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{199}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{199}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{199}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 199.2.0.1 | $x^{2} - x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 199.2.0.1 | $x^{2} - x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 199.2.0.1 | $x^{2} - x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 199.2.0.1 | $x^{2} - x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 199.4.2.2 | $x^{4} - 199 x^{2} + 237606$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 199.4.2.2 | $x^{4} - 199 x^{2} + 237606$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 331 | Data not computed | ||||||