Normalized defining polynomial
\( x^{20} - 2 x^{19} + 14 x^{17} - 36 x^{16} - 24 x^{15} + 76 x^{14} - 174 x^{13} - 197 x^{12} + 210 x^{11} + 34 x^{10} - 176 x^{9} + 19 x^{8} - 58 x^{7} - 100 x^{6} + 16 x^{5} + 13 x^{4} + \cdots - 1 \)
Invariants
Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[4, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(49338146756019243307761664\) \(\medspace = 2^{30}\cdot 11^{16}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(19.26\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
Galois root discriminant: | $2^{15/8}11^{4/5}\approx 24.977294240287762$ | ||
Ramified primes: | \(2\), \(11\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{11}a^{13}+\frac{5}{11}a^{12}-\frac{2}{11}a^{11}+\frac{1}{11}a^{10}-\frac{1}{11}a^{8}+\frac{1}{11}a^{7}+\frac{3}{11}a^{5}-\frac{2}{11}a^{4}+\frac{2}{11}a^{3}-\frac{2}{11}a^{2}-\frac{2}{11}a+\frac{2}{11}$, $\frac{1}{11}a^{14}-\frac{5}{11}a^{12}-\frac{5}{11}a^{10}-\frac{1}{11}a^{9}-\frac{5}{11}a^{8}-\frac{5}{11}a^{7}+\frac{3}{11}a^{6}+\frac{5}{11}a^{5}+\frac{1}{11}a^{4}-\frac{1}{11}a^{3}-\frac{3}{11}a^{2}+\frac{1}{11}a+\frac{1}{11}$, $\frac{1}{11}a^{15}+\frac{3}{11}a^{12}-\frac{4}{11}a^{11}+\frac{4}{11}a^{10}-\frac{5}{11}a^{9}+\frac{1}{11}a^{8}-\frac{3}{11}a^{7}+\frac{5}{11}a^{6}+\frac{5}{11}a^{5}-\frac{4}{11}a^{3}+\frac{2}{11}a^{2}+\frac{2}{11}a-\frac{1}{11}$, $\frac{1}{11}a^{16}+\frac{3}{11}a^{12}-\frac{1}{11}a^{11}+\frac{3}{11}a^{10}+\frac{1}{11}a^{9}+\frac{2}{11}a^{7}+\frac{5}{11}a^{6}+\frac{2}{11}a^{5}+\frac{2}{11}a^{4}-\frac{4}{11}a^{3}-\frac{3}{11}a^{2}+\frac{5}{11}a+\frac{5}{11}$, $\frac{1}{11}a^{17}-\frac{5}{11}a^{12}-\frac{2}{11}a^{11}-\frac{2}{11}a^{10}+\frac{5}{11}a^{8}+\frac{2}{11}a^{7}+\frac{2}{11}a^{6}+\frac{4}{11}a^{5}+\frac{2}{11}a^{4}+\frac{2}{11}a^{3}+\frac{5}{11}$, $\frac{1}{121}a^{18}+\frac{2}{121}a^{17}+\frac{4}{121}a^{16}+\frac{3}{121}a^{14}-\frac{1}{121}a^{13}-\frac{6}{121}a^{12}-\frac{29}{121}a^{11}-\frac{36}{121}a^{10}+\frac{39}{121}a^{9}-\frac{7}{121}a^{8}-\frac{52}{121}a^{7}+\frac{26}{121}a^{6}+\frac{56}{121}a^{5}-\frac{13}{121}a^{4}-\frac{18}{121}a^{3}+\frac{26}{121}a^{2}+\frac{20}{121}a-\frac{47}{121}$, $\frac{1}{33972148677841}a^{19}+\frac{117763882196}{33972148677841}a^{18}+\frac{413636325220}{33972148677841}a^{17}+\frac{1222272052372}{33972148677841}a^{16}-\frac{1145141287155}{33972148677841}a^{15}+\frac{401611167424}{33972148677841}a^{14}+\frac{1512310380232}{33972148677841}a^{13}-\frac{2289135694117}{33972148677841}a^{12}+\frac{9116397299993}{33972148677841}a^{11}+\frac{550831379436}{3088377152531}a^{10}+\frac{745377593097}{33972148677841}a^{9}+\frac{633333929291}{3088377152531}a^{8}-\frac{3156794737262}{33972148677841}a^{7}+\frac{13164755328393}{33972148677841}a^{6}+\frac{56376942606}{3088377152531}a^{5}-\frac{958196935655}{3088377152531}a^{4}+\frac{2860033257896}{33972148677841}a^{3}+\frac{12325588528600}{33972148677841}a^{2}+\frac{16534732396230}{33972148677841}a-\frac{16485588049519}{33972148677841}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
Fundamental units: | $\frac{7887087777342}{33972148677841}a^{19}-\frac{23742169075395}{33972148677841}a^{18}+\frac{17502036723136}{33972148677841}a^{17}+\frac{105687996029426}{33972148677841}a^{16}-\frac{390798412906527}{33972148677841}a^{15}+\frac{115809511800072}{33972148677841}a^{14}+\frac{713193674551398}{33972148677841}a^{13}-\frac{19\!\cdots\!79}{33972148677841}a^{12}-\frac{72611012187047}{33972148677841}a^{11}+\frac{28\!\cdots\!77}{33972148677841}a^{10}-\frac{13\!\cdots\!37}{33972148677841}a^{9}-\frac{13\!\cdots\!70}{33972148677841}a^{8}+\frac{99328941754875}{3088377152531}a^{7}-\frac{499401326713761}{33972148677841}a^{6}-\frac{34686834542619}{33972148677841}a^{5}+\frac{532436345100947}{33972148677841}a^{4}-\frac{31260845957416}{33972148677841}a^{3}+\frac{953869028621}{3088377152531}a^{2}+\frac{20318190662028}{33972148677841}a-\frac{32896131180292}{33972148677841}$, $\frac{5023559847238}{33972148677841}a^{19}-\frac{12814763501714}{33972148677841}a^{18}+\frac{10683038619026}{33972148677841}a^{17}+\frac{56560226256602}{33972148677841}a^{16}-\frac{211012779029113}{33972148677841}a^{15}+\frac{46207899271851}{33972148677841}a^{14}+\frac{218992679502350}{33972148677841}a^{13}-\frac{96689973512585}{3088377152531}a^{12}-\frac{113656304506836}{33972148677841}a^{11}+\frac{479361621166475}{33972148677841}a^{10}-\frac{715816565416133}{33972148677841}a^{9}+\frac{375052624422956}{33972148677841}a^{8}+\frac{95941712441398}{33972148677841}a^{7}-\frac{89734391006615}{3088377152531}a^{6}-\frac{35790811842050}{33972148677841}a^{5}-\frac{32657680500839}{33972148677841}a^{4}-\frac{43161925200169}{33972148677841}a^{3}+\frac{143309667130031}{33972148677841}a^{2}+\frac{31538746612081}{33972148677841}a+\frac{8583793255866}{33972148677841}$, $\frac{7240323236098}{33972148677841}a^{19}-\frac{19222403320585}{33972148677841}a^{18}+\frac{8104479961979}{33972148677841}a^{17}+\frac{104645695157606}{33972148677841}a^{16}-\frac{329569294049359}{33972148677841}a^{15}-\frac{17406588853833}{33972148677841}a^{14}+\frac{715656919094928}{33972148677841}a^{13}-\frac{16\!\cdots\!00}{33972148677841}a^{12}-\frac{639023217014436}{33972148677841}a^{11}+\frac{26\!\cdots\!48}{33972148677841}a^{10}-\frac{671379157565289}{33972148677841}a^{9}-\frac{14\!\cdots\!57}{33972148677841}a^{8}+\frac{883638705933455}{33972148677841}a^{7}-\frac{661212219371699}{33972148677841}a^{6}-\frac{79796779275244}{33972148677841}a^{5}+\frac{711607622346914}{33972148677841}a^{4}-\frac{113399380189883}{33972148677841}a^{3}+\frac{81411457522852}{33972148677841}a^{2}+\frac{43907277463986}{33972148677841}a-\frac{51256834104876}{33972148677841}$, $\frac{1143870687911}{33972148677841}a^{19}+\frac{5499019366268}{33972148677841}a^{18}-\frac{20191893381013}{33972148677841}a^{17}+\frac{27530321188659}{33972148677841}a^{16}+\frac{62315475137659}{33972148677841}a^{15}-\frac{369159885335088}{33972148677841}a^{14}+\frac{94401991786442}{33972148677841}a^{13}+\frac{411022900174791}{33972148677841}a^{12}-\frac{19\!\cdots\!13}{33972148677841}a^{11}-\frac{355584704571483}{33972148677841}a^{10}+\frac{21\!\cdots\!96}{33972148677841}a^{9}-\frac{10\!\cdots\!22}{33972148677841}a^{8}-\frac{11\!\cdots\!98}{33972148677841}a^{7}+\frac{733260503788140}{33972148677841}a^{6}-\frac{753896611331625}{33972148677841}a^{5}-\frac{402443067864337}{33972148677841}a^{4}+\frac{294728813424413}{33972148677841}a^{3}-\frac{79601434932171}{33972148677841}a^{2}-\frac{41752107297101}{33972148677841}a+\frac{56944100687612}{33972148677841}$, $\frac{13708459747614}{33972148677841}a^{19}-\frac{28071475717649}{33972148677841}a^{18}+\frac{1708732302495}{33972148677841}a^{17}+\frac{16933763327948}{3088377152531}a^{16}-\frac{491388203447401}{33972148677841}a^{15}-\frac{303042267971239}{33972148677841}a^{14}+\frac{976368487801512}{33972148677841}a^{13}-\frac{22\!\cdots\!17}{33972148677841}a^{12}-\frac{24\!\cdots\!94}{33972148677841}a^{11}+\frac{25\!\cdots\!77}{33972148677841}a^{10}+\frac{11\!\cdots\!88}{33972148677841}a^{9}-\frac{16\!\cdots\!39}{33972148677841}a^{8}-\frac{840829565483151}{33972148677841}a^{7}-\frac{950692264338658}{33972148677841}a^{6}-\frac{443288211770612}{33972148677841}a^{5}+\frac{260596856249530}{33972148677841}a^{4}+\frac{295449525632963}{33972148677841}a^{3}+\frac{174843227129704}{33972148677841}a^{2}+\frac{15484229535595}{33972148677841}a-\frac{2380426760873}{3088377152531}$, $\frac{3930210462336}{33972148677841}a^{19}-\frac{11173744239612}{33972148677841}a^{18}+\frac{11269523314625}{33972148677841}a^{17}+\frac{44947583903407}{33972148677841}a^{16}-\frac{184880468815492}{33972148677841}a^{15}+\frac{7727483134892}{3088377152531}a^{14}+\frac{205081603905633}{33972148677841}a^{13}-\frac{10\!\cdots\!17}{33972148677841}a^{12}+\frac{95375992377058}{33972148677841}a^{11}+\frac{661882113093056}{33972148677841}a^{10}-\frac{12\!\cdots\!54}{33972148677841}a^{9}-\frac{70178316466776}{33972148677841}a^{8}+\frac{777086755469088}{33972148677841}a^{7}-\frac{599426400367342}{33972148677841}a^{6}-\frac{379730328046413}{33972148677841}a^{5}-\frac{120157896018187}{33972148677841}a^{4}-\frac{163017567373647}{33972148677841}a^{3}+\frac{6389163984633}{33972148677841}a^{2}+\frac{11243562986352}{3088377152531}a+\frac{59499480849233}{33972148677841}$, $\frac{674004260629}{33972148677841}a^{19}-\frac{3077693091764}{33972148677841}a^{18}+\frac{2914200976358}{33972148677841}a^{17}+\frac{11561222481887}{33972148677841}a^{16}-\frac{51109057416440}{33972148677841}a^{15}+\frac{40657602639084}{33972148677841}a^{14}+\frac{123447666145924}{33972148677841}a^{13}-\frac{276869602108386}{33972148677841}a^{12}+\frac{132760755193794}{33972148677841}a^{11}+\frac{5108890886919}{280761559321}a^{10}-\frac{423124710111222}{33972148677841}a^{9}-\frac{32056145925594}{3088377152531}a^{8}+\frac{389248605713642}{33972148677841}a^{7}+\frac{2793227257378}{33972148677841}a^{6}-\frac{198048210128}{280761559321}a^{5}+\frac{19301679687769}{3088377152531}a^{4}-\frac{74027986206409}{33972148677841}a^{3}-\frac{100667828648917}{33972148677841}a^{2}+\frac{1321483839209}{33972148677841}a-\frac{26896467726445}{33972148677841}$, $\frac{12715194846701}{33972148677841}a^{19}-\frac{32192342315060}{33972148677841}a^{18}+\frac{14157733937779}{33972148677841}a^{17}+\frac{174482462084031}{33972148677841}a^{16}-\frac{545799903356382}{33972148677841}a^{15}-\frac{56333063446428}{33972148677841}a^{14}+\frac{10\!\cdots\!75}{33972148677841}a^{13}-\frac{26\!\cdots\!50}{33972148677841}a^{12}-\frac{12\!\cdots\!50}{33972148677841}a^{11}+\frac{36\!\cdots\!37}{33972148677841}a^{10}-\frac{557727102303080}{33972148677841}a^{9}-\frac{21\!\cdots\!76}{33972148677841}a^{8}+\frac{417428219564090}{33972148677841}a^{7}-\frac{669207381018069}{33972148677841}a^{6}-\frac{185818185040318}{33972148677841}a^{5}+\frac{403234924270542}{33972148677841}a^{4}-\frac{10801365727555}{33972148677841}a^{3}+\frac{10602102019103}{33972148677841}a^{2}-\frac{65912974617014}{33972148677841}a-\frac{23296280707608}{33972148677841}$, $\frac{8220160487398}{33972148677841}a^{19}-\frac{20576164041306}{33972148677841}a^{18}+\frac{17030784202246}{33972148677841}a^{17}+\frac{93361307515616}{33972148677841}a^{16}-\frac{340577921365874}{33972148677841}a^{15}+\frac{61210681420079}{33972148677841}a^{14}+\frac{359193108876995}{33972148677841}a^{13}-\frac{17\!\cdots\!40}{33972148677841}a^{12}-\frac{349196216834892}{33972148677841}a^{11}+\frac{734518254487518}{33972148677841}a^{10}-\frac{12\!\cdots\!71}{33972148677841}a^{9}+\frac{569387845634}{33972148677841}a^{8}+\frac{97476210857479}{33972148677841}a^{7}-\frac{11\!\cdots\!58}{33972148677841}a^{6}-\frac{338115558591626}{33972148677841}a^{5}-\frac{469713344828328}{33972148677841}a^{4}-\frac{212840465341180}{33972148677841}a^{3}-\frac{24730121411354}{33972148677841}a^{2}-\frac{13412986362258}{33972148677841}a+\frac{9838911706090}{33972148677841}$, $\frac{2885834382345}{33972148677841}a^{19}-\frac{1799134874271}{33972148677841}a^{18}-\frac{12167558197340}{33972148677841}a^{17}+\frac{53333203456118}{33972148677841}a^{16}-\frac{59450364085326}{33972148677841}a^{15}-\frac{265567973765542}{33972148677841}a^{14}+\frac{334960303227771}{33972148677841}a^{13}-\frac{288664864890261}{33972148677841}a^{12}-\frac{15\!\cdots\!00}{33972148677841}a^{11}+\frac{886690435743405}{33972148677841}a^{10}+\frac{826615940332794}{33972148677841}a^{9}-\frac{16\!\cdots\!12}{33972148677841}a^{8}+\frac{252417980030999}{33972148677841}a^{7}+\frac{286075962494679}{33972148677841}a^{6}-\frac{10\!\cdots\!13}{33972148677841}a^{5}+\frac{107116654351323}{33972148677841}a^{4}+\frac{4667927554256}{3088377152531}a^{3}-\frac{137926595309680}{33972148677841}a^{2}+\frac{81365383790832}{33972148677841}a+\frac{36030337921958}{33972148677841}$, $\frac{11053159271431}{33972148677841}a^{19}-\frac{23194992208676}{33972148677841}a^{18}+\frac{776762352813}{33972148677841}a^{17}+\frac{157688633364799}{33972148677841}a^{16}-\frac{413620068987647}{33972148677841}a^{15}-\frac{244264259443873}{33972148677841}a^{14}+\frac{915007596820619}{33972148677841}a^{13}-\frac{179658469345770}{3088377152531}a^{12}-\frac{20\!\cdots\!61}{33972148677841}a^{11}+\frac{27\!\cdots\!28}{33972148677841}a^{10}+\frac{421824643650899}{33972148677841}a^{9}-\frac{22\!\cdots\!48}{33972148677841}a^{8}+\frac{142573475533699}{33972148677841}a^{7}-\frac{31983513899166}{3088377152531}a^{6}-\frac{814345337600310}{33972148677841}a^{5}+\frac{169302247665560}{33972148677841}a^{4}+\frac{191853711723504}{33972148677841}a^{3}+\frac{14646167684626}{33972148677841}a^{2}+\frac{86685318107295}{33972148677841}a+\frac{49775173052788}{33972148677841}$ (assuming GRH) | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 90919.8145635 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 90919.8145635 \cdot 1}{2\cdot\sqrt{49338146756019243307761664}}\cr\approx \mathstrut & 0.251533630318 \end{aligned}\] (assuming GRH)
Galois group
$C_2^4:C_5$ (as 20T23):
A solvable group of order 80 |
The 8 conjugacy class representatives for $C_2^4:C_5$ |
Character table for $C_2^4:C_5$ |
Intermediate fields
\(\Q(\zeta_{11})^+\), 10.6.219503494144.1, 10.2.219503494144.1, 10.6.219503494144.2 |
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 16 sibling: | data not computed |
Degree 20 siblings: | data not computed |
Degree 40 siblings: | data not computed |
Minimal sibling: | 10.2.219503494144.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.5.0.1}{5} }^{4}$ | ${\href{/padicField/5.5.0.1}{5} }^{4}$ | ${\href{/padicField/7.5.0.1}{5} }^{4}$ | R | ${\href{/padicField/13.5.0.1}{5} }^{4}$ | ${\href{/padicField/17.5.0.1}{5} }^{4}$ | ${\href{/padicField/19.5.0.1}{5} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ | ${\href{/padicField/29.5.0.1}{5} }^{4}$ | ${\href{/padicField/31.5.0.1}{5} }^{4}$ | ${\href{/padicField/37.5.0.1}{5} }^{4}$ | ${\href{/padicField/41.5.0.1}{5} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.5.0.1}{5} }^{4}$ | ${\href{/padicField/53.5.0.1}{5} }^{4}$ | ${\href{/padicField/59.5.0.1}{5} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | Deg $20$ | $4$ | $5$ | $30$ | |||
\(11\) | 11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ |
11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ | |
11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ | |
11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ |