Normalized defining polynomial
\( x^{22} + 55 x^{20} + 1199 x^{18} + 13299 x^{16} + 81114 x^{14} + 279642 x^{12} + 532422 x^{10} + 491194 x^{8} + 80201 x^{6} - 190157 x^{4} - 119053 x^{2} + \cdots - 18225 \)
Invariants
Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(1580153645858805893796455628954412194463744\) \(\medspace = 2^{36}\cdot 7^{10}\cdot 11^{22}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(82.82\) | 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: | not computed | ||
Ramified primes: | \(2\), \(7\), \(11\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{11}a^{8}-\frac{3}{11}a^{6}+\frac{2}{11}a^{4}-\frac{1}{11}a^{2}+\frac{5}{11}$, $\frac{1}{11}a^{9}-\frac{3}{11}a^{7}+\frac{2}{11}a^{5}-\frac{1}{11}a^{3}+\frac{5}{11}a$, $\frac{1}{11}a^{10}+\frac{4}{11}a^{6}+\frac{5}{11}a^{4}+\frac{2}{11}a^{2}+\frac{4}{11}$, $\frac{1}{22}a^{11}-\frac{1}{22}a^{10}-\frac{1}{22}a^{9}-\frac{1}{22}a^{8}-\frac{2}{11}a^{7}+\frac{5}{11}a^{6}-\frac{4}{11}a^{5}+\frac{2}{11}a^{4}+\frac{3}{22}a^{3}-\frac{1}{22}a^{2}-\frac{1}{22}a-\frac{9}{22}$, $\frac{1}{22}a^{12}-\frac{1}{22}a^{8}-\frac{1}{11}a^{6}+\frac{3}{22}a^{4}-\frac{1}{11}a^{2}-\frac{3}{22}$, $\frac{1}{22}a^{13}-\frac{1}{22}a^{9}-\frac{1}{11}a^{7}+\frac{3}{22}a^{5}-\frac{1}{11}a^{3}-\frac{3}{22}a$, $\frac{1}{22}a^{14}-\frac{1}{22}a^{10}-\frac{3}{22}a^{6}+\frac{1}{11}a^{4}-\frac{5}{22}a^{2}+\frac{5}{11}$, $\frac{1}{22}a^{15}-\frac{1}{22}a^{10}-\frac{1}{22}a^{9}-\frac{1}{22}a^{8}-\frac{7}{22}a^{7}+\frac{5}{11}a^{6}-\frac{3}{11}a^{5}+\frac{2}{11}a^{4}-\frac{1}{11}a^{3}-\frac{1}{22}a^{2}+\frac{9}{22}a-\frac{9}{22}$, $\frac{1}{242}a^{16}+\frac{5}{242}a^{14}+\frac{1}{121}a^{12}-\frac{3}{242}a^{10}-\frac{1}{121}a^{8}+\frac{21}{242}a^{6}+\frac{27}{121}a^{4}-\frac{87}{242}a^{2}+\frac{91}{242}$, $\frac{1}{242}a^{17}+\frac{5}{242}a^{15}+\frac{1}{121}a^{13}-\frac{3}{242}a^{11}-\frac{1}{121}a^{9}+\frac{21}{242}a^{7}+\frac{27}{121}a^{5}-\frac{87}{242}a^{3}+\frac{91}{242}a$, $\frac{1}{242}a^{18}-\frac{1}{242}a^{14}-\frac{1}{121}a^{12}-\frac{9}{242}a^{10}-\frac{1}{121}a^{8}-\frac{73}{242}a^{6}-\frac{41}{121}a^{4}-\frac{34}{121}a^{2}+\frac{53}{121}$, $\frac{1}{242}a^{19}-\frac{1}{242}a^{15}-\frac{1}{121}a^{13}+\frac{1}{121}a^{11}-\frac{1}{22}a^{10}+\frac{9}{242}a^{9}-\frac{1}{22}a^{8}+\frac{59}{242}a^{7}+\frac{5}{11}a^{6}+\frac{58}{121}a^{5}+\frac{2}{11}a^{4}-\frac{57}{242}a^{3}-\frac{1}{22}a^{2}-\frac{37}{242}a-\frac{9}{22}$, $\frac{1}{5108398327516}a^{20}-\frac{4086990131}{2554199163758}a^{18}+\frac{6167743089}{5108398327516}a^{16}+\frac{41300152261}{2554199163758}a^{14}+\frac{8127041165}{2554199163758}a^{12}+\frac{15721243864}{1277099581879}a^{10}-\frac{1}{22}a^{9}-\frac{113176767653}{2554199163758}a^{8}-\frac{4}{11}a^{7}+\frac{743420872933}{2554199163758}a^{6}-\frac{1}{11}a^{5}-\frac{1958117383759}{5108398327516}a^{4}-\frac{5}{11}a^{3}-\frac{985605559167}{2554199163758}a^{2}-\frac{5}{22}a-\frac{1210021761347}{5108398327516}$, $\frac{1}{689633774214660}a^{21}+\frac{51308556782}{34481688710733}a^{19}-\frac{732650196841}{689633774214660}a^{17}+\frac{57896978372}{57469481184555}a^{15}-\frac{81322738798}{57469481184555}a^{13}-\frac{2603527405843}{114938962369110}a^{11}-\frac{1}{22}a^{10}+\frac{809343952942}{19156493728185}a^{9}-\frac{23370731790284}{172408443553665}a^{7}-\frac{2}{11}a^{6}+\frac{4801814144641}{62693979474060}a^{5}+\frac{3}{11}a^{4}+\frac{32092329065699}{344816887107330}a^{3}+\frac{9}{22}a^{2}+\frac{7101628718467}{62693979474060}a-\frac{2}{11}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{8}$, which has order $8$ (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: | $\frac{20145560366}{172408443553665}a^{21}+\frac{451322338697}{68963377421466}a^{19}+\frac{4596468339343}{31346989737030}a^{17}+\frac{194999982699701}{114938962369110}a^{15}+\frac{12\!\cdots\!11}{114938962369110}a^{13}+\frac{47\!\cdots\!83}{114938962369110}a^{11}+\frac{33\!\cdots\!01}{38312987456370}a^{9}+\frac{29\!\cdots\!43}{344816887107330}a^{7}+\frac{32\!\cdots\!57}{344816887107330}a^{5}-\frac{69\!\cdots\!97}{172408443553665}a^{3}-\frac{30\!\cdots\!68}{172408443553665}a$, $\frac{15780158843}{344816887107330}a^{21}+\frac{192450086857}{68963377421466}a^{19}+\frac{23390933213377}{344816887107330}a^{17}+\frac{47830536579662}{57469481184555}a^{15}+\frac{617087346076819}{114938962369110}a^{13}+\frac{990796877456821}{57469481184555}a^{11}+\frac{901464748999829}{38312987456370}a^{9}+\frac{811532457787291}{172408443553665}a^{7}-\frac{22\!\cdots\!11}{172408443553665}a^{5}-\frac{18\!\cdots\!71}{344816887107330}a^{3}+\frac{179542921510478}{172408443553665}a$, $\frac{18539009987}{172408443553665}a^{21}+\frac{27337759903}{6269397947406}a^{19}+\frac{71785472848}{1424863169865}a^{17}-\frac{9368686148233}{114938962369110}a^{15}-\frac{587191019272333}{114938962369110}a^{13}-\frac{37\!\cdots\!69}{114938962369110}a^{11}-\frac{30\!\cdots\!53}{38312987456370}a^{9}-\frac{26\!\cdots\!09}{344816887107330}a^{7}+\frac{278759710099919}{344816887107330}a^{5}+\frac{60\!\cdots\!96}{172408443553665}a^{3}+\frac{27\!\cdots\!83}{344816887107330}a$, $\frac{2462346191}{1277099581879}a^{20}+\frac{134296548389}{1277099581879}a^{18}+\frac{5774848785117}{2554199163758}a^{16}+\frac{62522862444153}{2554199163758}a^{14}+\frac{182403678779331}{1277099581879}a^{12}+\frac{11\!\cdots\!37}{2554199163758}a^{10}+\frac{938446841006152}{1277099581879}a^{8}+\frac{105755858786319}{232199923978}a^{6}-\frac{185251424712533}{1277099581879}a^{4}-\frac{58552913438431}{232199923978}a^{2}-\frac{165071819201789}{2554199163758}$, $\frac{4574155358}{1277099581879}a^{20}+\frac{493274019943}{2554199163758}a^{18}+\frac{10437444860397}{2554199163758}a^{16}+\frac{55232551020327}{1277099581879}a^{14}+\frac{312118667859806}{1277099581879}a^{12}+\frac{949073864560498}{1277099581879}a^{10}+\frac{131217737474293}{116099961989}a^{8}+\frac{762492931186490}{1277099581879}a^{6}-\frac{391117818732171}{1277099581879}a^{4}-\frac{909531831674701}{2554199163758}a^{2}-\frac{165847536930431}{2554199163758}$, $\frac{39769560431}{31346989737030}a^{21}+\frac{2346566288434}{34481688710733}a^{19}+\frac{246593794830307}{172408443553665}a^{17}+\frac{862822129613584}{57469481184555}a^{15}+\frac{48\!\cdots\!44}{57469481184555}a^{13}+\frac{14\!\cdots\!57}{57469481184555}a^{11}+\frac{77\!\cdots\!74}{19156493728185}a^{9}+\frac{40\!\cdots\!57}{15673494868515}a^{7}-\frac{22\!\cdots\!69}{344816887107330}a^{5}-\frac{24\!\cdots\!41}{172408443553665}a^{3}-\frac{69\!\cdots\!69}{172408443553665}a$, $\frac{273263248597}{344816887107330}a^{21}+\frac{2758200846317}{68963377421466}a^{19}+\frac{24054128117923}{31346989737030}a^{17}+\frac{802567611490871}{114938962369110}a^{15}+\frac{17\!\cdots\!68}{57469481184555}a^{13}+\frac{77\!\cdots\!13}{114938962369110}a^{11}+\frac{107132008701193}{1741499429835}a^{9}+\frac{23\!\cdots\!63}{344816887107330}a^{7}-\frac{60\!\cdots\!63}{344816887107330}a^{5}-\frac{18\!\cdots\!62}{172408443553665}a^{3}-\frac{20\!\cdots\!31}{344816887107330}a$, $\frac{25925802319}{31346989737030}a^{21}+\frac{3026160688909}{68963377421466}a^{19}+\frac{156752752797353}{172408443553665}a^{17}+\frac{10\!\cdots\!17}{114938962369110}a^{15}+\frac{59\!\cdots\!57}{114938962369110}a^{13}+\frac{17\!\cdots\!71}{114938962369110}a^{11}+\frac{94\!\cdots\!27}{38312987456370}a^{9}+\frac{54\!\cdots\!11}{344816887107330}a^{7}-\frac{78\!\cdots\!78}{172408443553665}a^{5}-\frac{15\!\cdots\!74}{172408443553665}a^{3}-\frac{755658953707357}{31346989737030}a$, $\frac{544954387}{2554199163758}a^{20}+\frac{27618095893}{2554199163758}a^{18}+\frac{531126955633}{2554199163758}a^{16}+\frac{436070988753}{232199923978}a^{14}+\frac{10140926397035}{1277099581879}a^{12}+\frac{32502492862997}{2554199163758}a^{10}-\frac{9711172956048}{1277099581879}a^{8}-\frac{106469055792333}{2554199163758}a^{6}-\frac{563425148967}{21109083998}a^{4}+\frac{14668002582352}{1277099581879}a^{2}+\frac{25520913133951}{2554199163758}$, $\frac{52609464967}{344816887107330}a^{21}+\frac{566141848883}{68963377421466}a^{19}+\frac{29811811313794}{172408443553665}a^{17}+\frac{18926250382531}{10448996579010}a^{15}+\frac{11\!\cdots\!21}{114938962369110}a^{13}+\frac{32\!\cdots\!13}{114938962369110}a^{11}+\frac{13\!\cdots\!31}{38312987456370}a^{9}+\frac{76546988497703}{31346989737030}a^{7}-\frac{51\!\cdots\!14}{172408443553665}a^{5}-\frac{19\!\cdots\!97}{172408443553665}a^{3}+\frac{20\!\cdots\!89}{344816887107330}a$, $\frac{77\!\cdots\!99}{689633774214660}a^{21}-\frac{11\!\cdots\!24}{1277099581879}a^{20}+\frac{21\!\cdots\!15}{34481688710733}a^{19}-\frac{62\!\cdots\!79}{1277099581879}a^{18}+\frac{95\!\cdots\!51}{689633774214660}a^{17}-\frac{27\!\cdots\!27}{2554199163758}a^{16}+\frac{90\!\cdots\!83}{57469481184555}a^{15}-\frac{15\!\cdots\!35}{1277099581879}a^{14}+\frac{95\!\cdots\!01}{949908779910}a^{13}-\frac{20\!\cdots\!77}{2554199163758}a^{12}+\frac{43\!\cdots\!13}{114938962369110}a^{11}-\frac{76\!\cdots\!51}{2554199163758}a^{10}+\frac{31\!\cdots\!21}{38312987456370}a^{9}-\frac{16\!\cdots\!21}{2554199163758}a^{8}+\frac{18\!\cdots\!34}{172408443553665}a^{7}-\frac{10\!\cdots\!28}{1277099581879}a^{6}+\frac{52\!\cdots\!19}{689633774214660}a^{5}-\frac{15\!\cdots\!15}{2554199163758}a^{4}+\frac{82\!\cdots\!21}{31346989737030}a^{3}-\frac{53\!\cdots\!15}{2554199163758}a^{2}+\frac{22\!\cdots\!83}{689633774214660}a-\frac{32\!\cdots\!47}{1277099581879}$ (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: | \( 712861842202 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{10}\cdot 712861842202 \cdot 8}{2\cdot\sqrt{1580153645858805893796455628954412194463744}}\cr\approx \mathstrut & 0.870110432986160 \end{aligned}\] (assuming GRH)
Galois group
$C_2^{10}.F_{11}$ (as 22T34):
A solvable group of order 112640 |
The 44 conjugacy class representatives for $C_2^{10}.F_{11}$ |
Character table for $C_2^{10}.F_{11}$ is not computed |
Intermediate fields
11.11.4910318845910094848.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 22 sibling: | data not computed |
Degree 44 siblings: | data not computed |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | $20{,}\,{\href{/padicField/3.2.0.1}{2} }$ | $20{,}\,{\href{/padicField/5.2.0.1}{2} }$ | R | R | ${\href{/padicField/13.10.0.1}{10} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.5.0.1}{5} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.11.0.1}{11} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.5.0.1}{5} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{6}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | $20{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.10.0.1}{10} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.10.0.1}{10} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | Deg $22$ | $22$ | $1$ | $36$ | |||
\(7\) | $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
7.10.5.1 | $x^{10} + 2401 x^{2} - 67228$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
7.10.5.1 | $x^{10} + 2401 x^{2} - 67228$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
\(11\) | 11.11.11.6 | $x^{11} + 11 x + 11$ | $11$ | $1$ | $11$ | $F_{11}$ | $[11/10]_{10}$ |
11.11.11.6 | $x^{11} + 11 x + 11$ | $11$ | $1$ | $11$ | $F_{11}$ | $[11/10]_{10}$ |