Normalized defining polynomial
\( x^{22} - 9 x^{21} + 125 x^{20} - 813 x^{19} + 6745 x^{18} - 34987 x^{17} + 218003 x^{16} - 939340 x^{15} + 4767272 x^{14} - 17367998 x^{13} + 74585461 x^{12} - 230625416 x^{11} + 854744494 x^{10} - 2225080031 x^{9} + 7186857343 x^{8} - 15398214635 x^{7} + 43486274233 x^{6} - 73186706990 x^{5} + 180412985566 x^{4} - 216014851131 x^{3} + 461997703697 x^{2} - 301326000030 x + 553445530681 \)
Invariants
| Degree: | $22$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 11]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-544838009528791582903069339296517113846865239=-\,3^{11}\cdot 13^{11}\cdot 23^{20}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $108.01$ | 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: | $3, 13, 23$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(897=3\cdot 13\cdot 23\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{897}(1,·)$, $\chi_{897}(196,·)$, $\chi_{897}(584,·)$, $\chi_{897}(77,·)$, $\chi_{897}(469,·)$, $\chi_{897}(662,·)$, $\chi_{897}(857,·)$, $\chi_{897}(859,·)$, $\chi_{897}(545,·)$, $\chi_{897}(547,·)$, $\chi_{897}(740,·)$, $\chi_{897}(742,·)$, $\chi_{897}(233,·)$, $\chi_{897}(430,·)$, $\chi_{897}(623,·)$, $\chi_{897}(625,·)$, $\chi_{897}(818,·)$, $\chi_{897}(116,·)$, $\chi_{897}(118,·)$, $\chi_{897}(311,·)$, $\chi_{897}(508,·)$, $\chi_{897}(703,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{21667} a^{20} - \frac{10125}{21667} a^{19} - \frac{1770}{21667} a^{18} + \frac{5638}{21667} a^{17} - \frac{3009}{21667} a^{16} - \frac{10583}{21667} a^{15} - \frac{3279}{21667} a^{14} - \frac{9190}{21667} a^{13} + \frac{5742}{21667} a^{12} - \frac{9153}{21667} a^{11} - \frac{2402}{21667} a^{10} + \frac{6835}{21667} a^{9} + \frac{347}{21667} a^{8} - \frac{10459}{21667} a^{7} + \frac{6120}{21667} a^{6} - \frac{7347}{21667} a^{5} + \frac{2623}{21667} a^{4} - \frac{8009}{21667} a^{3} + \frac{3639}{21667} a^{2} - \frac{7193}{21667} a - \frac{59}{461}$, $\frac{1}{31174953691808677408476279205170597050875468724194942811656041895458337528876838740081} a^{21} - \frac{706348140538723528892763509980930761053467237223945488630215203769108359318904351}{31174953691808677408476279205170597050875468724194942811656041895458337528876838740081} a^{20} + \frac{11106405825122703977914293926518361461756293517503448541325411955918136257817440005556}{31174953691808677408476279205170597050875468724194942811656041895458337528876838740081} a^{19} - \frac{15053245083545142614912280686788367626270002968604315876188477890744844228766774283204}{31174953691808677408476279205170597050875468724194942811656041895458337528876838740081} a^{18} - \frac{5545146616098815034438862251287890227524355320773815741843006576347472568424173362553}{31174953691808677408476279205170597050875468724194942811656041895458337528876838740081} a^{17} - \frac{14665652700066501460797064089916113775472906671044717981075076019452111398442710435687}{31174953691808677408476279205170597050875468724194942811656041895458337528876838740081} a^{16} + \frac{3845223012348592835748340119158745026119483082446902844544359227199400275177257834943}{31174953691808677408476279205170597050875468724194942811656041895458337528876838740081} a^{15} - \frac{4369080438577548486650687787553019877984344583389723096196179534854939449203069312109}{31174953691808677408476279205170597050875468724194942811656041895458337528876838740081} a^{14} - \frac{9612402792618016404951353053214920141717004181885977218058298790457100041406152080230}{31174953691808677408476279205170597050875468724194942811656041895458337528876838740081} a^{13} - \frac{6345342155995240958940734040634534609914733402071647238728909634194349304208264311576}{31174953691808677408476279205170597050875468724194942811656041895458337528876838740081} a^{12} - \frac{5802888773290536226436975862763884730872670007166239676807630344023808220685074779547}{31174953691808677408476279205170597050875468724194942811656041895458337528876838740081} a^{11} + \frac{542005852585167315938321433313405714399153467655001498647458139400329983408305467212}{31174953691808677408476279205170597050875468724194942811656041895458337528876838740081} a^{10} - \frac{13118683567146685886956798060415525892643229273745103327918430037939647287650384474853}{31174953691808677408476279205170597050875468724194942811656041895458337528876838740081} a^{9} - \frac{8524136841883804494742380133091051827372570143777783810463754417878129292489041143575}{31174953691808677408476279205170597050875468724194942811656041895458337528876838740081} a^{8} - \frac{406052102882895663467658393992682683706885605078453454507058303229738047564644231839}{31174953691808677408476279205170597050875468724194942811656041895458337528876838740081} a^{7} + \frac{328719204614337295664393626742736415490760563958350733772449860183895663466895355219}{31174953691808677408476279205170597050875468724194942811656041895458337528876838740081} a^{6} + \frac{14540672099402997495682493497522999575396486090757253234087374335631485564654719933428}{31174953691808677408476279205170597050875468724194942811656041895458337528876838740081} a^{5} + \frac{15333002686236959630546881283582917021359533586262547464801868619012354138911888543526}{31174953691808677408476279205170597050875468724194942811656041895458337528876838740081} a^{4} + \frac{1103176402956591724800009611998770239847072043625954743093726133687687136123620707796}{31174953691808677408476279205170597050875468724194942811656041895458337528876838740081} a^{3} + \frac{5880200463654986250715967928102801876123510692846731825589052433191170031067746141314}{31174953691808677408476279205170597050875468724194942811656041895458337528876838740081} a^{2} - \frac{7724896300398493707274056827800168968826594293259126778487163569807275984468334388646}{31174953691808677408476279205170597050875468724194942811656041895458337528876838740081} a + \frac{250834474647489702653523997661782042208173616714176170128761762115984586391930784540}{663296887059759093797367642663204192571818483493509421524596636073581649550571037023}$
Class group and class number
Not computed
Unit group
| Rank: | $10$ | 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: | Not computed | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | Not computed | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 22 |
| The 22 conjugacy class representatives for $C_{22}$ |
| Character table for $C_{22}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-39}) \), \(\Q(\zeta_{23})^+\) |
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 | ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ | R | ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ | $22$ | ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ | R | $22$ | $22$ | R | $22$ | $22$ | $22$ | ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/47.1.0.1}{1} }^{22}$ | $22$ | ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| 3 | Data not computed | ||||||
| 13 | Data not computed | ||||||
| 23 | Data not computed | ||||||