Normalized defining polynomial
\( x^{22} - 2 x^{21} - 129 x^{20} + 238 x^{19} + 7154 x^{18} - 12008 x^{17} - 223655 x^{16} + 335630 x^{15} + 4347432 x^{14} - 5704484 x^{13} - 54729020 x^{12} + 61006720 x^{11} + 451520904 x^{10} - 411532688 x^{9} - 2421562269 x^{8} + 1712218138 x^{7} + 8216704351 x^{6} - 4172120122 x^{5} - 16723460175 x^{4} + 5354124650 x^{3} + 18354424206 x^{2} - 2750460712 x - 8251382159 \)
Invariants
| Degree: | $22$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[22, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(719807926798998083647106533713510400000000000=2^{33}\cdot 5^{11}\cdot 23^{20}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $109.39$ | 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: | $2, 5, 23$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(920=2^{3}\cdot 5\cdot 23\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{920}(1,·)$, $\chi_{920}(841,·)$, $\chi_{920}(761,·)$, $\chi_{920}(909,·)$, $\chi_{920}(269,·)$, $\chi_{920}(81,·)$, $\chi_{920}(469,·)$, $\chi_{920}(121,·)$, $\chi_{920}(601,·)$, $\chi_{920}(29,·)$, $\chi_{920}(869,·)$, $\chi_{920}(721,·)$, $\chi_{920}(41,·)$, $\chi_{920}(749,·)$, $\chi_{920}(669,·)$, $\chi_{920}(561,·)$, $\chi_{920}(629,·)$, $\chi_{920}(349,·)$, $\chi_{920}(361,·)$, $\chi_{920}(441,·)$, $\chi_{920}(509,·)$, $\chi_{920}(829,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{19300221896409597797582452444000554654012075326466861072138124648665489889693046751431} a^{21} - \frac{833188369923228737641957738305272299633548352777180041220137136096767548733471051187}{19300221896409597797582452444000554654012075326466861072138124648665489889693046751431} a^{20} - \frac{578301048064703998788250181533002554938055911583018640341931147527804548544275143284}{19300221896409597797582452444000554654012075326466861072138124648665489889693046751431} a^{19} - \frac{5640429922256404538270143525204296473769637197515416876699671553597105142831078739593}{19300221896409597797582452444000554654012075326466861072138124648665489889693046751431} a^{18} + \frac{77711923999206618999229355777030522620257890054696179859890935069174525792010047099}{19300221896409597797582452444000554654012075326466861072138124648665489889693046751431} a^{17} - \frac{5165885672836109819427529623617482018364122384297158456686813106767019922614639348384}{19300221896409597797582452444000554654012075326466861072138124648665489889693046751431} a^{16} + \frac{9353730568445525426479300855695251126074842434501236345067741144705713668559136944130}{19300221896409597797582452444000554654012075326466861072138124648665489889693046751431} a^{15} + \frac{5354376799239761322755286438773198490063560521374681901735611807632255390936927509448}{19300221896409597797582452444000554654012075326466861072138124648665489889693046751431} a^{14} + \frac{238802914259018137394393362872007192700705678255630659506117360533543675904698725760}{19300221896409597797582452444000554654012075326466861072138124648665489889693046751431} a^{13} + \frac{4200582833704331404480420580507826413516545273908668213663057273702636829734727230948}{19300221896409597797582452444000554654012075326466861072138124648665489889693046751431} a^{12} - \frac{5940765303480554059714057536794924539875482374993653645992609256039965716095379452470}{19300221896409597797582452444000554654012075326466861072138124648665489889693046751431} a^{11} - \frac{4660199603611431626441678011906319766794071820546808241985442843831500323117172868407}{19300221896409597797582452444000554654012075326466861072138124648665489889693046751431} a^{10} + \frac{7420032117863351035643219673221140960894399165187548423800153046770553486933477502131}{19300221896409597797582452444000554654012075326466861072138124648665489889693046751431} a^{9} - \frac{3301044070061201620182854590949652933622754381286155852481331147621862464415260640852}{19300221896409597797582452444000554654012075326466861072138124648665489889693046751431} a^{8} - \frac{1966798763146694778397891075592716182562163820903133534162866747698089372108845415544}{19300221896409597797582452444000554654012075326466861072138124648665489889693046751431} a^{7} - \frac{6110254405325885700473922147634790363270765303427708254124023627349305885936505435861}{19300221896409597797582452444000554654012075326466861072138124648665489889693046751431} a^{6} + \frac{3661834918345453971689200681564641048734036788173430394072342994399688809879867865025}{19300221896409597797582452444000554654012075326466861072138124648665489889693046751431} a^{5} - \frac{9471413053522446600037093824533187852028889761659246841776770422986059916670667940110}{19300221896409597797582452444000554654012075326466861072138124648665489889693046751431} a^{4} + \frac{1550224209100394494591056037506872074749390860386236175679368363618864873708998276577}{19300221896409597797582452444000554654012075326466861072138124648665489889693046751431} a^{3} - \frac{4498219265434771919747047785986760097250990824547341608237430691331179287243085856085}{19300221896409597797582452444000554654012075326466861072138124648665489889693046751431} a^{2} - \frac{2094846191268953114575721557729942785848920246447117413888685177374704913241337844024}{19300221896409597797582452444000554654012075326466861072138124648665489889693046751431} a + \frac{7101713892333188832430060368332383092179429321774714319497164923790657085403760788752}{19300221896409597797582452444000554654012075326466861072138124648665489889693046751431}$
Class group and class number
$C_{2}$, which has order $2$ (assuming GRH)
Unit group
| Rank: | $21$ | 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: | 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) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 642628850080310.2 \) (assuming GRH) | 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
| \(\Q(\sqrt{10}) \), \(\Q(\zeta_{23})^+\) |
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 | ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ | R | $22$ | $22$ | ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ | $22$ | $22$ | R | $22$ | ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{11}$ | ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ | $22$ |
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 | ||||||
| 5 | Data not computed | ||||||
| 23 | Data not computed | ||||||