Normalized defining polynomial
\( x^{22} - 11 x^{21} + 67 x^{20} - 285 x^{19} + 929 x^{18} - 2433 x^{17} + 5271 x^{16} - 9630 x^{15} + \cdots + 9 \)
Invariants
Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 11]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-578978183833808423828407471\) \(\medspace = -\,271^{11}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(16.46\) | 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: | $271^{1/2}\approx 16.46207763315433$ | ||
Ramified primes: | \(271\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-271}) \) | ||
$\card{ \Gal(K/\Q) }$: | $22$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{4}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{5}$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{6}$, $\frac{1}{3}a^{15}-\frac{1}{3}a^{7}$, $\frac{1}{3}a^{16}-\frac{1}{3}a^{8}$, $\frac{1}{3}a^{17}-\frac{1}{3}a$, $\frac{1}{27}a^{18}-\frac{1}{9}a^{16}+\frac{1}{9}a^{15}+\frac{1}{9}a^{13}-\frac{2}{27}a^{10}+\frac{1}{9}a^{9}-\frac{2}{9}a^{8}-\frac{1}{9}a^{7}-\frac{1}{9}a^{5}+\frac{10}{27}a^{2}-\frac{1}{9}a-\frac{1}{3}$, $\frac{1}{81}a^{19}+\frac{1}{81}a^{18}+\frac{2}{27}a^{17}-\frac{1}{9}a^{16}-\frac{2}{27}a^{15}+\frac{1}{27}a^{14}+\frac{4}{27}a^{13}+\frac{7}{81}a^{11}+\frac{1}{81}a^{10}-\frac{4}{27}a^{9}-\frac{7}{27}a^{7}+\frac{8}{27}a^{6}+\frac{5}{27}a^{5}+\frac{1}{3}a^{4}-\frac{26}{81}a^{3}+\frac{34}{81}a^{2}-\frac{4}{27}a-\frac{4}{9}$, $\frac{1}{148473}a^{20}-\frac{10}{148473}a^{19}+\frac{2560}{148473}a^{18}-\frac{7585}{49491}a^{17}+\frac{4171}{49491}a^{16}+\frac{6152}{49491}a^{15}-\frac{5677}{49491}a^{14}-\frac{587}{3807}a^{13}-\frac{24644}{148473}a^{12}-\frac{2884}{148473}a^{11}+\frac{4027}{148473}a^{10}-\frac{5464}{49491}a^{9}-\frac{9223}{49491}a^{8}+\frac{9751}{49491}a^{7}+\frac{21301}{49491}a^{6}-\frac{19351}{49491}a^{5}-\frac{515}{11421}a^{4}+\frac{4829}{148473}a^{3}-\frac{22877}{148473}a^{2}-\frac{20704}{49491}a-\frac{5644}{16497}$, $\frac{1}{148473}a^{21}+\frac{209}{49491}a^{19}+\frac{1012}{148473}a^{18}+\frac{2380}{16497}a^{17}+\frac{430}{5499}a^{16}-\frac{6479}{49491}a^{15}-\frac{82}{16497}a^{14}+\frac{21376}{148473}a^{13}-\frac{623}{49491}a^{12}+\frac{3949}{49491}a^{11}+\frac{22045}{148473}a^{10}-\frac{7040}{49491}a^{9}+\frac{5501}{16497}a^{8}-\frac{334}{49491}a^{7}+\frac{4675}{16497}a^{6}-\frac{20828}{148473}a^{5}-\frac{20707}{49491}a^{4}+\frac{2620}{16497}a^{3}-\frac{56258}{148473}a^{2}-\frac{2179}{49491}a+\frac{383}{16497}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $10$ | 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: | $\frac{695605}{49491}a^{21}-\frac{817052}{5499}a^{20}+\frac{4828385}{5499}a^{19}-\frac{179646464}{49491}a^{18}+\frac{21088270}{1833}a^{17}-\frac{161011082}{5499}a^{16}+\frac{1015964059}{16497}a^{15}-\frac{199954799}{1833}a^{14}+\frac{8197221595}{49491}a^{13}-\frac{3621182783}{16497}a^{12}+\frac{4270128809}{16497}a^{11}-\frac{13653529436}{49491}a^{10}+\frac{338312881}{1269}a^{9}-\frac{1265856827}{5499}a^{8}+\frac{2854663187}{16497}a^{7}-\frac{67307030}{611}a^{6}+\frac{2960457967}{49491}a^{5}-\frac{485022211}{16497}a^{4}+\frac{240466393}{16497}a^{3}-\frac{339984797}{49491}a^{2}+\frac{36140630}{16497}a-\frac{1657276}{5499}$, $\frac{987238}{148473}a^{21}-\frac{10616428}{148473}a^{20}+\frac{63466003}{148473}a^{19}-\frac{88361210}{49491}a^{18}+\frac{21747550}{3807}a^{17}-\frac{726083383}{49491}a^{16}+\frac{1540616651}{49491}a^{15}-\frac{2752404185}{49491}a^{14}+\frac{12638647951}{148473}a^{13}-\frac{16872199033}{148473}a^{12}+\frac{20021639923}{148473}a^{11}-\frac{7153237508}{49491}a^{10}+\frac{147952879}{1053}a^{9}-\frac{6049184477}{49491}a^{8}+\frac{4589771410}{49491}a^{7}-\frac{2954170348}{49491}a^{6}+\frac{4851949267}{148473}a^{5}-\frac{2393298787}{148473}a^{4}+\frac{1181839885}{148473}a^{3}-\frac{187988756}{49491}a^{2}+\frac{6904601}{5499}a-\frac{987905}{5499}$, $\frac{1307986}{49491}a^{21}-\frac{4583665}{16497}a^{20}+\frac{80954372}{49491}a^{19}-\frac{111168563}{16497}a^{18}+\frac{27014380}{1269}a^{17}-\frac{296957152}{5499}a^{16}+\frac{69172609}{611}a^{15}-\frac{70154648}{351}a^{14}+\frac{14970526717}{49491}a^{13}-\frac{2197699951}{5499}a^{12}+\frac{23260042046}{49491}a^{11}-\frac{8243165807}{16497}a^{10}+\frac{7945030928}{16497}a^{9}-\frac{2279156447}{5499}a^{8}+\frac{189533336}{611}a^{7}-\frac{3240176960}{16497}a^{6}+\frac{5255733004}{49491}a^{5}-\frac{859641359}{16497}a^{4}+\frac{1281839342}{49491}a^{3}-\frac{200139800}{16497}a^{2}+\frac{20822071}{5499}a-\frac{310185}{611}$, $\frac{987833}{49491}a^{21}-\frac{31462118}{148473}a^{20}+\frac{186488921}{148473}a^{19}-\frac{59467139}{11421}a^{18}+\frac{818854685}{49491}a^{17}-\frac{160713890}{3807}a^{16}+\frac{4405332749}{49491}a^{15}-\frac{7822607692}{49491}a^{14}+\frac{3968590802}{16497}a^{13}-\frac{47439689600}{148473}a^{12}+\frac{4311589952}{11421}a^{11}-\frac{59845497863}{148473}a^{10}+\frac{19313532335}{49491}a^{9}-\frac{16715879449}{49491}a^{8}+\frac{12603503470}{49491}a^{7}-\frac{8053232504}{49491}a^{6}+\frac{4386460132}{49491}a^{5}-\frac{6476442677}{148473}a^{4}+\frac{3206959451}{148473}a^{3}-\frac{116647355}{11421}a^{2}+\frac{163649417}{49491}a-\frac{7705027}{16497}$, $\frac{243190}{148473}a^{21}-\frac{2091038}{148473}a^{20}+\frac{10400150}{148473}a^{19}-\frac{35622727}{148473}a^{18}+\frac{30316835}{49491}a^{17}-\frac{58872899}{49491}a^{16}+\frac{28577749}{16497}a^{15}-\frac{84180361}{49491}a^{14}+\frac{85427923}{148473}a^{13}+\frac{254809444}{148473}a^{12}-\frac{684943330}{148473}a^{11}+\frac{1099522703}{148473}a^{10}-\frac{159919778}{16497}a^{9}+\frac{42403481}{3807}a^{8}-\frac{182470372}{16497}a^{7}+\frac{451992991}{49491}a^{6}-\frac{868671833}{148473}a^{5}+\frac{451903303}{148473}a^{4}-\frac{205677853}{148473}a^{3}+\frac{117792068}{148473}a^{2}-\frac{20578022}{49491}a+\frac{1448092}{16497}$, $\frac{987238}{148473}a^{21}-\frac{10115570}{148473}a^{20}+\frac{58457423}{148473}a^{19}-\frac{18205819}{11421}a^{18}+\frac{245075711}{49491}a^{17}-\frac{47021780}{3807}a^{16}+\frac{419913493}{16497}a^{15}-\frac{2185201768}{49491}a^{14}+\frac{207447626}{3159}a^{13}-\frac{269573572}{3159}a^{12}+\frac{1129063004}{11421}a^{11}-\frac{15391506583}{148473}a^{10}+\frac{1624036862}{16497}a^{9}-\frac{4113184921}{49491}a^{8}+\frac{1000297115}{16497}a^{7}-\frac{1839631640}{49491}a^{6}+\frac{2906012101}{148473}a^{5}-\frac{1411279970}{148473}a^{4}+\frac{711347060}{148473}a^{3}-\frac{24693853}{11421}a^{2}+\frac{28523236}{49491}a-\frac{950198}{16497}$, $\frac{1512778}{148473}a^{21}-\frac{397792}{3807}a^{20}+\frac{9969230}{16497}a^{19}-\frac{363522758}{148473}a^{18}+\frac{125563523}{16497}a^{17}-\frac{313406059}{16497}a^{16}+\frac{1938935743}{49491}a^{15}-\frac{124652066}{1833}a^{14}+\frac{319705526}{3159}a^{13}-\frac{5131832}{39}a^{12}+\frac{7547963051}{49491}a^{11}-\frac{23755880390}{148473}a^{10}+\frac{7523849128}{49491}a^{9}-\frac{235418024}{1833}a^{8}+\frac{4641357125}{49491}a^{7}-\frac{316485653}{5499}a^{6}+\frac{4504818655}{148473}a^{5}-\frac{729970565}{49491}a^{4}+\frac{368215969}{49491}a^{3}-\frac{498362381}{148473}a^{2}+\frac{44661860}{49491}a-\frac{1559542}{16497}$, $\frac{3209632}{148473}a^{21}-\frac{3797698}{16497}a^{20}+\frac{67672595}{49491}a^{19}-\frac{64860086}{11421}a^{18}+\frac{298226368}{16497}a^{17}-\frac{19543669}{423}a^{16}+\frac{4829326522}{49491}a^{15}-\frac{2863085842}{16497}a^{14}+\frac{39276470104}{148473}a^{13}-\frac{17412162776}{49491}a^{12}+\frac{1584285640}{3807}a^{11}-\frac{66034244636}{148473}a^{10}+\frac{21332456896}{49491}a^{9}-\frac{6162310159}{16497}a^{8}+\frac{13960652141}{49491}a^{7}-\frac{2978498975}{16497}a^{6}+\frac{14613941326}{148473}a^{5}-\frac{2397451660}{49491}a^{4}+\frac{131781757}{5499}a^{3}-\frac{129807059}{11421}a^{2}+\frac{182895455}{49491}a-\frac{8585191}{16497}$, $\frac{222094}{148473}a^{21}-\frac{2829142}{148473}a^{20}+\frac{18706591}{148473}a^{19}-\frac{2175782}{3807}a^{18}+\frac{97149121}{49491}a^{17}-\frac{20464633}{3807}a^{16}+\frac{598791392}{49491}a^{15}-\frac{1130384477}{49491}a^{14}+\frac{5455968370}{148473}a^{13}-\frac{7608869122}{148473}a^{12}+\frac{720473947}{11421}a^{11}-\frac{3453284651}{49491}a^{10}+\frac{3465373082}{49491}a^{9}-\frac{3132455810}{49491}a^{8}+\frac{2490590665}{49491}a^{7}-\frac{1690339228}{49491}a^{6}+\frac{2885900968}{148473}a^{5}-\frac{1442303383}{148473}a^{4}+\frac{697855975}{148473}a^{3}-\frac{9001142}{3807}a^{2}+\frac{5043008}{5499}a-\frac{857846}{5499}$, $\frac{711668}{49491}a^{21}-\frac{22686592}{148473}a^{20}+\frac{134530654}{148473}a^{19}-\frac{557854111}{148473}a^{18}+\frac{45461392}{3807}a^{17}-\frac{1508037289}{49491}a^{16}+\frac{3179680780}{49491}a^{15}-\frac{5645450132}{49491}a^{14}+\frac{8589994891}{49491}a^{13}-\frac{34215338854}{148473}a^{12}+\frac{40408725661}{148473}a^{11}-\frac{43125942988}{148473}a^{10}+\frac{13911570586}{49491}a^{9}-\frac{12033538046}{49491}a^{8}+\frac{9064721885}{49491}a^{7}-\frac{5783755645}{49491}a^{6}+\frac{116463631}{1833}a^{5}-\frac{4638579394}{148473}a^{4}+\frac{2297503807}{148473}a^{3}-\frac{1087045504}{148473}a^{2}+\frac{116257519}{49491}a-\frac{5351744}{16497}$ (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: | \( 33482.7947011 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{11}\cdot 33482.7947011 \cdot 1}{2\cdot\sqrt{578978183833808423828407471}}\cr\approx \mathstrut & 0.419217261313 \end{aligned}\] (assuming GRH)
Galois group
A solvable group of order 22 |
The 7 conjugacy class representatives for $D_{11}$ |
Character table for $D_{11}$ |
Intermediate fields
\(\Q(\sqrt{-271}) \), 11.1.1461660310351.1 x11 |
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 | ${\href{/padicField/2.11.0.1}{11} }^{2}$ | ${\href{/padicField/3.2.0.1}{2} }^{11}$ | ${\href{/padicField/5.11.0.1}{11} }^{2}$ | ${\href{/padicField/7.11.0.1}{11} }^{2}$ | ${\href{/padicField/11.11.0.1}{11} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{11}$ | ${\href{/padicField/17.11.0.1}{11} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{11}$ | ${\href{/padicField/23.2.0.1}{2} }^{11}$ | ${\href{/padicField/29.2.0.1}{2} }^{11}$ | ${\href{/padicField/31.11.0.1}{11} }^{2}$ | ${\href{/padicField/37.11.0.1}{11} }^{2}$ | ${\href{/padicField/41.11.0.1}{11} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{11}$ | ${\href{/padicField/47.2.0.1}{2} }^{11}$ | ${\href{/padicField/53.11.0.1}{11} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{11}$ |
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 |
---|---|---|---|---|---|---|---|
\(271\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |