Normalized defining polynomial
\( x^{44} - 21 x^{42} + 251 x^{40} - 2052 x^{38} + 12691 x^{36} - 61778 x^{34} + 243629 x^{32} - 788303 x^{30} + \cdots + 1 \)
Invariants
| Degree: | $44$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 22]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(1625926291579854267093042018571578660715394142508043230799997286804887850450944\)
\(\medspace = 2^{44}\cdot 3^{22}\cdot 23^{40}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(59.91\) | 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: | $2\cdot 3^{1/2}23^{10/11}\approx 59.91354533944697$ | ||
| Ramified primes: |
\(2\), \(3\), \(23\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $44$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(276=2^{2}\cdot 3\cdot 23\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{276}(1,·)$, $\chi_{276}(131,·)$, $\chi_{276}(133,·)$, $\chi_{276}(257,·)$, $\chi_{276}(265,·)$, $\chi_{276}(139,·)$, $\chi_{276}(13,·)$, $\chi_{276}(271,·)$, $\chi_{276}(259,·)$, $\chi_{276}(151,·)$, $\chi_{276}(25,·)$, $\chi_{276}(29,·)$, $\chi_{276}(31,·)$, $\chi_{276}(35,·)$, $\chi_{276}(167,·)$, $\chi_{276}(41,·)$, $\chi_{276}(95,·)$, $\chi_{276}(173,·)$, $\chi_{276}(47,·)$, $\chi_{276}(49,·)$, $\chi_{276}(179,·)$, $\chi_{276}(55,·)$, $\chi_{276}(185,·)$, $\chi_{276}(59,·)$, $\chi_{276}(193,·)$, $\chi_{276}(197,·)$, $\chi_{276}(71,·)$, $\chi_{276}(73,·)$, $\chi_{276}(119,·)$, $\chi_{276}(77,·)$, $\chi_{276}(269,·)$, $\chi_{276}(209,·)$, $\chi_{276}(163,·)$, $\chi_{276}(85,·)$, $\chi_{276}(215,·)$, $\chi_{276}(223,·)$, $\chi_{276}(187,·)$, $\chi_{276}(101,·)$, $\chi_{276}(233,·)$, $\chi_{276}(239,·)$, $\chi_{276}(211,·)$, $\chi_{276}(169,·)$, $\chi_{276}(121,·)$, $\chi_{276}(127,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{2097152}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $\frac{1}{38503}a^{40}-\frac{14974}{38503}a^{38}-\frac{7545}{38503}a^{36}-\frac{11609}{38503}a^{34}+\frac{12115}{38503}a^{32}+\frac{3952}{38503}a^{30}+\frac{7568}{38503}a^{28}-\frac{9852}{38503}a^{26}+\frac{3271}{38503}a^{24}-\frac{2590}{38503}a^{22}-\frac{4438}{38503}a^{20}+\frac{12705}{38503}a^{18}-\frac{13257}{38503}a^{16}+\frac{10861}{38503}a^{14}-\frac{18128}{38503}a^{12}-\frac{5037}{38503}a^{10}+\frac{16826}{38503}a^{8}-\frac{11491}{38503}a^{6}-\frac{15191}{38503}a^{4}-\frac{10591}{38503}a^{2}+\frac{10893}{38503}$, $\frac{1}{38503}a^{41}-\frac{14974}{38503}a^{39}-\frac{7545}{38503}a^{37}-\frac{11609}{38503}a^{35}+\frac{12115}{38503}a^{33}+\frac{3952}{38503}a^{31}+\frac{7568}{38503}a^{29}-\frac{9852}{38503}a^{27}+\frac{3271}{38503}a^{25}-\frac{2590}{38503}a^{23}-\frac{4438}{38503}a^{21}+\frac{12705}{38503}a^{19}-\frac{13257}{38503}a^{17}+\frac{10861}{38503}a^{15}-\frac{18128}{38503}a^{13}-\frac{5037}{38503}a^{11}+\frac{16826}{38503}a^{9}-\frac{11491}{38503}a^{7}-\frac{15191}{38503}a^{5}-\frac{10591}{38503}a^{3}+\frac{10893}{38503}a$, $\frac{1}{39\!\cdots\!53}a^{42}-\frac{19\!\cdots\!75}{39\!\cdots\!53}a^{40}-\frac{15\!\cdots\!50}{39\!\cdots\!53}a^{38}-\frac{60\!\cdots\!86}{39\!\cdots\!53}a^{36}+\frac{96\!\cdots\!98}{39\!\cdots\!53}a^{34}-\frac{42\!\cdots\!51}{39\!\cdots\!53}a^{32}+\frac{65\!\cdots\!33}{39\!\cdots\!53}a^{30}-\frac{14\!\cdots\!11}{39\!\cdots\!53}a^{28}+\frac{12\!\cdots\!06}{39\!\cdots\!53}a^{26}+\frac{45\!\cdots\!14}{39\!\cdots\!53}a^{24}+\frac{10\!\cdots\!29}{39\!\cdots\!53}a^{22}-\frac{12\!\cdots\!55}{39\!\cdots\!53}a^{20}+\frac{12\!\cdots\!36}{39\!\cdots\!53}a^{18}-\frac{13\!\cdots\!33}{39\!\cdots\!53}a^{16}-\frac{56\!\cdots\!52}{39\!\cdots\!53}a^{14}-\frac{69\!\cdots\!19}{39\!\cdots\!53}a^{12}-\frac{13\!\cdots\!84}{39\!\cdots\!53}a^{10}+\frac{16\!\cdots\!65}{39\!\cdots\!53}a^{8}-\frac{17\!\cdots\!34}{39\!\cdots\!53}a^{6}-\frac{14\!\cdots\!05}{39\!\cdots\!53}a^{4}-\frac{17\!\cdots\!38}{39\!\cdots\!53}a^{2}+\frac{18\!\cdots\!21}{39\!\cdots\!53}$, $\frac{1}{39\!\cdots\!53}a^{43}-\frac{19\!\cdots\!75}{39\!\cdots\!53}a^{41}-\frac{15\!\cdots\!50}{39\!\cdots\!53}a^{39}-\frac{60\!\cdots\!86}{39\!\cdots\!53}a^{37}+\frac{96\!\cdots\!98}{39\!\cdots\!53}a^{35}-\frac{42\!\cdots\!51}{39\!\cdots\!53}a^{33}+\frac{65\!\cdots\!33}{39\!\cdots\!53}a^{31}-\frac{14\!\cdots\!11}{39\!\cdots\!53}a^{29}+\frac{12\!\cdots\!06}{39\!\cdots\!53}a^{27}+\frac{45\!\cdots\!14}{39\!\cdots\!53}a^{25}+\frac{10\!\cdots\!29}{39\!\cdots\!53}a^{23}-\frac{12\!\cdots\!55}{39\!\cdots\!53}a^{21}+\frac{12\!\cdots\!36}{39\!\cdots\!53}a^{19}-\frac{13\!\cdots\!33}{39\!\cdots\!53}a^{17}-\frac{56\!\cdots\!52}{39\!\cdots\!53}a^{15}-\frac{69\!\cdots\!19}{39\!\cdots\!53}a^{13}-\frac{13\!\cdots\!84}{39\!\cdots\!53}a^{11}+\frac{16\!\cdots\!65}{39\!\cdots\!53}a^{9}-\frac{17\!\cdots\!34}{39\!\cdots\!53}a^{7}-\frac{14\!\cdots\!05}{39\!\cdots\!53}a^{5}-\frac{17\!\cdots\!38}{39\!\cdots\!53}a^{3}+\frac{18\!\cdots\!21}{39\!\cdots\!53}a$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
not computed
Unit group
| Rank: | $21$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( \frac{1793299759063376835378664135048865178373688585}{39550136075673318268146636906670404265971857953} a^{43} - \frac{37524918191114127225451393946175224908107429507}{39550136075673318268146636906670404265971857953} a^{41} + \frac{447284901272128218229591301808252944778196436644}{39550136075673318268146636906670404265971857953} a^{39} - \frac{3645882836287000816414101768580673206196470843929}{39550136075673318268146636906670404265971857953} a^{37} + \frac{22480164947188205680925170226234287579609269605288}{39550136075673318268146636906670404265971857953} a^{35} - \frac{109057747165731681781192779380042573020302787841455}{39550136075673318268146636906670404265971857953} a^{33} + \frac{428454218369402889523266744705522378312522532365379}{39550136075673318268146636906670404265971857953} a^{31} - \frac{1380226261933181872822564645576170371973592506670921}{39550136075673318268146636906670404265971857953} a^{29} + \frac{3680880153832659028736431604610474723067831565699494}{39550136075673318268146636906670404265971857953} a^{27} - \frac{8135814576352391316054710021131930412486905029471824}{39550136075673318268146636906670404265971857953} a^{25} + \frac{14906096680940235386082672071811211460263861946573645}{39550136075673318268146636906670404265971857953} a^{23} - \frac{22479367063610811030950372300497610893271117186709459}{39550136075673318268146636906670404265971857953} a^{21} + \frac{27705573322205199049276348121608471713881569145400448}{39550136075673318268146636906670404265971857953} a^{19} - \frac{27444059324777363037209369190182981475233408409741898}{39550136075673318268146636906670404265971857953} a^{17} + \frac{21519004793326216917235629591870133556476659111095426}{39550136075673318268146636906670404265971857953} a^{15} - \frac{12900146249961478700999452315093622441872757730860338}{39550136075673318268146636906670404265971857953} a^{13} + \frac{5759968267918956170937644195812435949662336470191215}{39550136075673318268146636906670404265971857953} a^{11} - \frac{1750040917198848548965297061359053244112911785532794}{39550136075673318268146636906670404265971857953} a^{9} + \frac{348473274344915359648878440386430236588516863579197}{39550136075673318268146636906670404265971857953} a^{7} - \frac{26920199464397917471205835997146294435151744317003}{39550136075673318268146636906670404265971857953} a^{5} + \frac{490852689044286343341728016721638864406908608802}{39550136075673318268146636906670404265971857953} a^{3} + \frac{282675491634706018423048373549513976076133825027}{39550136075673318268146636906670404265971857953} a \)
(order $12$)
| 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: | not computed | 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: | not computed | 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 $
Galois group
$C_2\times C_{22}$ (as 44T2):
| An abelian group of order 44 |
| The 44 conjugacy class representatives for $C_2\times C_{22}$ |
| Character table for $C_2\times C_{22}$ |
Intermediate fields
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 | R | $22^{2}$ | $22^{2}$ | $22^{2}$ | ${\href{/padicField/13.11.0.1}{11} }^{4}$ | $22^{2}$ | $22^{2}$ | R | $22^{2}$ | $22^{2}$ | ${\href{/padicField/37.11.0.1}{11} }^{4}$ | $22^{2}$ | $22^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{22}$ | $22^{2}$ | $22^{2}$ |
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 $44$ | $2$ | $22$ | $44$ | |||
|
\(3\)
| Deg $44$ | $2$ | $22$ | $22$ | |||
|
\(23\)
| 23.22.20.1 | $x^{22} + 231 x^{21} + 24310 x^{20} + 1539615 x^{19} + 65271580 x^{18} + 1948237137 x^{17} + 41892053577 x^{16} + 652037968890 x^{15} + 7265319209115 x^{14} + 56290884204555 x^{13} + 287278948122936 x^{12} + 881069580352567 x^{11} + 1436394740619993 x^{10} + 1407272105659090 x^{9} + 908164935089445 x^{8} + 407525140102740 x^{7} + 130953632793951 x^{6} + 31291611122541 x^{5} + 17709125163290 x^{4} + 131485551467820 x^{3} + 905728303498040 x^{2} + 3760233666129578 x + 7096289029978197$ | $11$ | $2$ | $20$ | 22T1 | $[\ ]_{11}^{2}$ |
| 23.22.20.1 | $x^{22} + 231 x^{21} + 24310 x^{20} + 1539615 x^{19} + 65271580 x^{18} + 1948237137 x^{17} + 41892053577 x^{16} + 652037968890 x^{15} + 7265319209115 x^{14} + 56290884204555 x^{13} + 287278948122936 x^{12} + 881069580352567 x^{11} + 1436394740619993 x^{10} + 1407272105659090 x^{9} + 908164935089445 x^{8} + 407525140102740 x^{7} + 130953632793951 x^{6} + 31291611122541 x^{5} + 17709125163290 x^{4} + 131485551467820 x^{3} + 905728303498040 x^{2} + 3760233666129578 x + 7096289029978197$ | $11$ | $2$ | $20$ | 22T1 | $[\ ]_{11}^{2}$ |