Normalized defining polynomial
\( x^{20} - 6 x^{19} + 34 x^{18} - 132 x^{17} + 502 x^{16} - 1450 x^{15} + 3398 x^{14} - 7814 x^{13} + 13793 x^{12} - 30364 x^{11} + 73744 x^{10} - 254376 x^{9} + 822868 x^{8} - 2139784 x^{7} + 4734462 x^{6} - 7831492 x^{5} + 11001528 x^{4} - 10686038 x^{3} + 8803206 x^{2} - 3761584 x + 1294987 \)
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: | \(7303816416639774784171798530359296=2^{30}\cdot 11^{16}\cdot 23^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $49.34$ | 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: | $2, 11, 23$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{23} a^{17} - \frac{1}{23} a^{16} - \frac{8}{23} a^{15} - \frac{7}{23} a^{14} - \frac{9}{23} a^{13} - \frac{6}{23} a^{12} - \frac{1}{23} a^{11} - \frac{9}{23} a^{10} - \frac{1}{23} a^{9} - \frac{11}{23} a^{8} + \frac{9}{23} a^{7} + \frac{6}{23} a^{6} + \frac{11}{23} a^{5} - \frac{6}{23} a^{4} - \frac{10}{23} a^{3} + \frac{1}{23} a^{2} + \frac{5}{23} a + \frac{11}{23}$, $\frac{1}{23} a^{18} - \frac{9}{23} a^{16} + \frac{8}{23} a^{15} + \frac{7}{23} a^{14} + \frac{8}{23} a^{13} - \frac{7}{23} a^{12} - \frac{10}{23} a^{11} - \frac{10}{23} a^{10} + \frac{11}{23} a^{9} - \frac{2}{23} a^{8} - \frac{8}{23} a^{7} - \frac{6}{23} a^{6} + \frac{5}{23} a^{5} + \frac{7}{23} a^{4} - \frac{9}{23} a^{3} + \frac{6}{23} a^{2} - \frac{7}{23} a + \frac{11}{23}$, $\frac{1}{102645251009738911346243283297425942815871962086542776921153} a^{19} - \frac{758734246940755277562999326761754178428835308375631926138}{102645251009738911346243283297425942815871962086542776921153} a^{18} + \frac{1300263162722492333314503311275908225019363975089528403222}{102645251009738911346243283297425942815871962086542776921153} a^{17} - \frac{16995227964311612869003730451717231678083309927012263906502}{102645251009738911346243283297425942815871962086542776921153} a^{16} + \frac{7833166464238532218396883527180123079876140668504722533046}{102645251009738911346243283297425942815871962086542776921153} a^{15} + \frac{20636847673493551063633348414658761239148820079548580481539}{102645251009738911346243283297425942815871962086542776921153} a^{14} + \frac{33144979789395208336395423973171174139561938126561566232424}{102645251009738911346243283297425942815871962086542776921153} a^{13} - \frac{48653923721351365148566210243661509234722562913644846620240}{102645251009738911346243283297425942815871962086542776921153} a^{12} + \frac{4584281850074682599903751935767936813813434837713832325409}{102645251009738911346243283297425942815871962086542776921153} a^{11} - \frac{7514921053990420294984479590088721638291489365074199700349}{102645251009738911346243283297425942815871962086542776921153} a^{10} + \frac{15971063483769253694685846820070731315620406417874799578210}{102645251009738911346243283297425942815871962086542776921153} a^{9} - \frac{41374089164207038538664696420919861202707365831099057582887}{102645251009738911346243283297425942815871962086542776921153} a^{8} - \frac{22411187823622959498922791763572539757984872900671509244514}{102645251009738911346243283297425942815871962086542776921153} a^{7} + \frac{36133344573199681446081729588380074000805327091216349405618}{102645251009738911346243283297425942815871962086542776921153} a^{6} + \frac{21650376954693819051305324902883117603993588171356281488727}{102645251009738911346243283297425942815871962086542776921153} a^{5} - \frac{46915498964371322627685036261157935883852220684827615477418}{102645251009738911346243283297425942815871962086542776921153} a^{4} - \frac{11509029397859921724364514850944586088281256581956957980136}{102645251009738911346243283297425942815871962086542776921153} a^{3} - \frac{38875124881130069157832156185562800637596260181621063019047}{102645251009738911346243283297425942815871962086542776921153} a^{2} + \frac{15173098876209810286550556821829166450939881577182111713374}{102645251009738911346243283297425942815871962086542776921153} a - \frac{40942773582178947981566135046427146143957880376159174714463}{102645251009738911346243283297425942815871962086542776921153}$
Class group and class number
$C_{2}\times C_{4}\times C_{20}$, which has order $160$ (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: | \( 2417625.83378 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times C_2^4:C_5$ (as 20T40):
| A solvable group of order 160 |
| The 16 conjugacy class representatives for $C_2\times C_2^4:C_5$ |
| Character table for $C_2\times C_2^4:C_5$ |
Intermediate fields
| \(\Q(\zeta_{11})^+\), 10.0.2670699013250048.1, 10.0.5048580365312.1, 10.10.116117348402176.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 10 siblings: | data not computed |
| Degree 20 siblings: | data not computed |
| Degree 32 sibling: | data not computed |
| Degree 40 siblings: | data not computed |
| Arithmetically equvalently siblings: | data not computed |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ | 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 | ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ | ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$ |
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 | $x^{10} - 2321 x^{5} + 2033647$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ | |
| $23$ | 23.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 23.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.1.1 | $x^{2} - 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.1.1 | $x^{2} - 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.1.1 | $x^{2} - 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.1.1 | $x^{2} - 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.4.2.1 | $x^{4} + 299 x^{2} + 25921$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |