Global number field 22.22.2053696021140625408463434849100950458853228544.1

• Label: 22.22.2053696021...8544.1
• Degree: $22$
• Signature: $[22, 0]$
• Discriminant: $2^{22}\cdot 11^{11}\cdot 23^{20}$
• Root discriminant: $114.73$
• Ramified primes: $2, 11, 23$
• Class number: $1$ (GRH)
• Class group: Trivial (GRH)
• Galois group: $C_{22}$ (as 22T1)

Normalized defining polynomial

\( x^{22} - 2 x^{21} - 140 x^{20} + 258 x^{19} + 8459 x^{18} - 14168 x^{17} - 289511 x^{16} + 433206 x^{15} + 6197354 x^{14} - 8106226 x^{13} - 86544524 x^{12} + 96213194 x^{11} + 799094606 x^{10} - 727527486 x^{9} - 4847415983 x^{8} + 3434361596 x^{7} + 18832650088 x^{6} - 9628969964 x^{5} - 44480164414 x^{4} + 14436561400 x^{3} + 57439955172 x^{2} - 8797078896 x - 30789776159 \)

Invariants

Degree:		$22$
Signature:		$[22, 0]$
Discriminant:		\(2053696021140625408463434849100950458853228544=2^{22}\cdot 11^{11}\cdot 23^{20}\)
Root discriminant:		$114.73$
Ramified primes:		$2, 11, 23$
This field is Galois and abelian over $\Q$.
Conductor:		\(1012=2^{2}\cdot 11\cdot 23\)
Dirichlet character group:		$\lbrace$$\chi_{1012}(1,·)$, $\chi_{1012}(131,·)$, $\chi_{1012}(133,·)$, $\chi_{1012}(967,·)$, $\chi_{1012}(265,·)$, $\chi_{1012}(395,·)$, $\chi_{1012}(397,·)$, $\chi_{1012}(791,·)$, $\chi_{1012}(923,·)$, $\chi_{1012}(837,·)$, $\chi_{1012}(353,·)$, $\chi_{1012}(219,·)$, $\chi_{1012}(485,·)$, $\chi_{1012}(749,·)$, $\chi_{1012}(177,·)$, $\chi_{1012}(969,·)$, $\chi_{1012}(307,·)$, $\chi_{1012}(439,·)$, $\chi_{1012}(441,·)$, $\chi_{1012}(87,·)$, $\chi_{1012}(351,·)$, $\chi_{1012}(703,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{137} a^{20} - \frac{63}{137} a^{19} - \frac{59}{137} a^{18} + \frac{17}{137} a^{17} + \frac{42}{137} a^{16} + \frac{9}{137} a^{15} + \frac{63}{137} a^{14} - \frac{14}{137} a^{13} + \frac{58}{137} a^{12} + \frac{11}{137} a^{11} - \frac{58}{137} a^{10} - \frac{20}{137} a^{9} - \frac{11}{137} a^{8} + \frac{67}{137} a^{7} - \frac{26}{137} a^{6} - \frac{57}{137} a^{5} - \frac{66}{137} a^{4} + \frac{42}{137} a^{3} - \frac{41}{137} a^{2} - \frac{65}{137} a - \frac{58}{137}$, $\frac{1}{51972370556198186516736858107832233152744047107232063071536697099411832973160117336507} a^{21} - \frac{112031046588349797427435357545952163866309840881966623986193375885122691369577989907}{51972370556198186516736858107832233152744047107232063071536697099411832973160117336507} a^{20} + \frac{8573428031849360497050092653674491841004333188827153127802070987628208094615952470214}{51972370556198186516736858107832233152744047107232063071536697099411832973160117336507} a^{19} - \frac{42269160842788553239605335499794098535436232143323929549662159278719619459059680950}{226953583214839242431165319248175690623336450249921672801470293010532021716856407583} a^{18} + \frac{18007241305710816119484597769181690963328875727980514044275400832206638568024332511439}{51972370556198186516736858107832233152744047107232063071536697099411832973160117336507} a^{17} - \frac{17468552473094906508116131958132261851883876959199936513871894959930727374465973639907}{51972370556198186516736858107832233152744047107232063071536697099411832973160117336507} a^{16} + \frac{6144157552324326723319995236970015443530253911953296278455011139115825897416293059283}{51972370556198186516736858107832233152744047107232063071536697099411832973160117336507} a^{15} - \frac{517473156599460404078401642411901071727339373716507777228514889446967301951489754265}{51972370556198186516736858107832233152744047107232063071536697099411832973160117336507} a^{14} - \frac{18645004803438949122724367643077192638852189663862238270471523885314445208764443092656}{51972370556198186516736858107832233152744047107232063071536697099411832973160117336507} a^{13} - \frac{22082013237756509032956719005915781081339433591450458579895034771263575959062066363524}{51972370556198186516736858107832233152744047107232063071536697099411832973160117336507} a^{12} + \frac{16342054762230006213810968435642154729866850341500670418421191837936148937821055440296}{51972370556198186516736858107832233152744047107232063071536697099411832973160117336507} a^{11} + \frac{18635830436162404608822508832004258282399510490996400953307836424853516936304503609361}{51972370556198186516736858107832233152744047107232063071536697099411832973160117336507} a^{10} + \frac{3466439572105000362995942671581264595475320320523091859413348049493309795172602643401}{51972370556198186516736858107832233152744047107232063071536697099411832973160117336507} a^{9} - \frac{9247023602045294276775775643353474891298681741738948187174507421498628253303147337753}{51972370556198186516736858107832233152744047107232063071536697099411832973160117336507} a^{8} + \frac{6669610282422865223806271562565423911883294079979788956734522133852515813347517035466}{51972370556198186516736858107832233152744047107232063071536697099411832973160117336507} a^{7} + \frac{25265594985438019999048478440181619502390106755980242018468283498912591135225579561209}{51972370556198186516736858107832233152744047107232063071536697099411832973160117336507} a^{6} - \frac{17916460296260689135473483779695190219711360339972622620406472462083262257859054415090}{51972370556198186516736858107832233152744047107232063071536697099411832973160117336507} a^{5} + \frac{10320246792202800440104345644977509375642972316172178011418338344223217531876069056892}{51972370556198186516736858107832233152744047107232063071536697099411832973160117336507} a^{4} + \frac{20155017744942752192673803695821045518169467219919416820087548155143995441981066846942}{51972370556198186516736858107832233152744047107232063071536697099411832973160117336507} a^{3} + \frac{20566933669191247202211768134713984066563877067786172442736683209640325427936143861443}{51972370556198186516736858107832233152744047107232063071536697099411832973160117336507} a^{2} + \frac{20556180661829971232542231925056096908403808588009209979150707930793677554890490165557}{51972370556198186516736858107832233152744047107232063071536697099411832973160117336507} a + \frac{2407443133867651729022428855801080617932018362509548425569868935257746453719899}{11917953222435500836589052105664847483239849301576520128208280891169039632245781}$

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

Unit group

Rank:		$21$
Torsion generator:		\( -1 \) (order $2$)
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)
Regulator:		\( 3035479805294090.5 \) (assuming GRH)

Galois group

$C_{22}$ (as 22T1):

A cyclic group of order 22

The 22 conjugacy class representatives for $C_{22}$

Character table for $C_{22}$ is not computed

Intermediate fields

\(\Q(\sqrt{11}) \), \(\Q(\zeta_{23})^+\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	R	$22$	${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$	${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$	R	$22$	$22$	${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$	R	$22$	$22$	${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$	$22$	${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$	${\href{/LocalNumberField/47.2.0.1}{2} }^{11}$	${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$	$22$

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	Data not computed
23	Data not computed