Normalized defining polynomial
\( x^{44} + 178 x^{42} + 14240 x^{40} + 678536 x^{38} + 21520912 x^{36} + 481058528 x^{34} + \cdots + 373293056 \)
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: |
\(491\!\cdots\!016\)
\(\medspace = 2^{66}\cdot 89^{43}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(227.32\) | 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: | $2^{3/2}89^{43/44}\approx 227.31635408801262$ | ||
| Ramified primes: |
\(2\), \(89\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{89}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $44$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(712=2^{3}\cdot 89\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{712}(1,·)$, $\chi_{712}(643,·)$, $\chi_{712}(265,·)$, $\chi_{712}(435,·)$, $\chi_{712}(401,·)$, $\chi_{712}(403,·)$, $\chi_{712}(601,·)$, $\chi_{712}(25,·)$, $\chi_{712}(539,·)$, $\chi_{712}(545,·)$, $\chi_{712}(411,·)$, $\chi_{712}(555,·)$, $\chi_{712}(427,·)$, $\chi_{712}(307,·)$, $\chi_{712}(99,·)$, $\chi_{712}(177,·)$, $\chi_{712}(691,·)$, $\chi_{712}(57,·)$, $\chi_{712}(187,·)$, $\chi_{712}(449,·)$, $\chi_{712}(603,·)$, $\chi_{712}(195,·)$, $\chi_{712}(673,·)$, $\chi_{712}(73,·)$, $\chi_{712}(587,·)$, $\chi_{712}(81,·)$, $\chi_{712}(339,·)$, $\chi_{712}(441,·)$, $\chi_{712}(345,·)$, $\chi_{712}(347,·)$, $\chi_{712}(97,·)$, $\chi_{712}(227,·)$, $\chi_{712}(131,·)$, $\chi_{712}(105,·)$, $\chi_{712}(107,·)$, $\chi_{712}(289,·)$, $\chi_{712}(625,·)$, $\chi_{712}(659,·)$, $\chi_{712}(707,·)$, $\chi_{712}(489,·)$, $\chi_{712}(153,·)$, $\chi_{712}(121,·)$, $\chi_{712}(217,·)$, $\chi_{712}(123,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{2097152}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{4}a^{5}$, $\frac{1}{8}a^{6}$, $\frac{1}{8}a^{7}$, $\frac{1}{16}a^{8}$, $\frac{1}{16}a^{9}$, $\frac{1}{32}a^{10}$, $\frac{1}{32}a^{11}$, $\frac{1}{64}a^{12}$, $\frac{1}{64}a^{13}$, $\frac{1}{128}a^{14}$, $\frac{1}{128}a^{15}$, $\frac{1}{256}a^{16}$, $\frac{1}{256}a^{17}$, $\frac{1}{512}a^{18}$, $\frac{1}{512}a^{19}$, $\frac{1}{1024}a^{20}$, $\frac{1}{1024}a^{21}$, $\frac{1}{2048}a^{22}$, $\frac{1}{2048}a^{23}$, $\frac{1}{4096}a^{24}$, $\frac{1}{4096}a^{25}$, $\frac{1}{8192}a^{26}$, $\frac{1}{8192}a^{27}$, $\frac{1}{16384}a^{28}$, $\frac{1}{16384}a^{29}$, $\frac{1}{32768}a^{30}$, $\frac{1}{32768}a^{31}$, $\frac{1}{65536}a^{32}$, $\frac{1}{65536}a^{33}$, $\frac{1}{131072}a^{34}$, $\frac{1}{131072}a^{35}$, $\frac{1}{46923776}a^{36}+\frac{25}{23461888}a^{34}-\frac{29}{11730944}a^{32}+\frac{33}{5865472}a^{30}-\frac{5}{733184}a^{28}-\frac{1}{1466368}a^{26}+\frac{1}{183296}a^{24}+\frac{29}{183296}a^{22}+\frac{31}{91648}a^{20}-\frac{51}{91648}a^{18}-\frac{3}{45824}a^{16}+\frac{21}{11456}a^{14}-\frac{1}{5728}a^{12}-\frac{11}{5728}a^{10}-\frac{3}{716}a^{8}-\frac{3}{179}a^{6}+\frac{9}{358}a^{4}+\frac{77}{358}a^{2}+\frac{85}{179}$, $\frac{1}{46923776}a^{37}+\frac{25}{23461888}a^{35}-\frac{29}{11730944}a^{33}+\frac{33}{5865472}a^{31}-\frac{5}{733184}a^{29}-\frac{1}{1466368}a^{27}+\frac{1}{183296}a^{25}+\frac{29}{183296}a^{23}+\frac{31}{91648}a^{21}-\frac{51}{91648}a^{19}-\frac{3}{45824}a^{17}+\frac{21}{11456}a^{15}-\frac{1}{5728}a^{13}-\frac{11}{5728}a^{11}-\frac{3}{716}a^{9}-\frac{3}{179}a^{7}+\frac{9}{358}a^{5}+\frac{77}{358}a^{3}+\frac{85}{179}a$, $\frac{1}{93847552}a^{38}+\frac{31}{11730944}a^{34}+\frac{21}{5865472}a^{32}+\frac{25}{2932736}a^{30}-\frac{19}{1466368}a^{28}+\frac{29}{1466368}a^{26}-\frac{21}{366592}a^{24}+\frac{11}{91648}a^{22}+\frac{5}{91648}a^{20}+\frac{19}{91648}a^{18}-\frac{31}{22912}a^{16}+\frac{11}{11456}a^{14}+\frac{39}{11456}a^{12}+\frac{21}{1432}a^{10}-\frac{41}{1432}a^{8}+\frac{81}{1432}a^{6}-\frac{15}{716}a^{4}-\frac{25}{179}a^{2}+\frac{23}{179}$, $\frac{1}{93847552}a^{39}+\frac{31}{11730944}a^{35}+\frac{21}{5865472}a^{33}+\frac{25}{2932736}a^{31}-\frac{19}{1466368}a^{29}+\frac{29}{1466368}a^{27}-\frac{21}{366592}a^{25}+\frac{11}{91648}a^{23}+\frac{5}{91648}a^{21}+\frac{19}{91648}a^{19}-\frac{31}{22912}a^{17}+\frac{11}{11456}a^{15}+\frac{39}{11456}a^{13}+\frac{21}{1432}a^{11}-\frac{41}{1432}a^{9}+\frac{81}{1432}a^{7}-\frac{15}{716}a^{5}-\frac{25}{179}a^{3}+\frac{23}{179}a$, $\frac{1}{89905954816}a^{40}-\frac{25}{22476488704}a^{38}+\frac{233}{22476488704}a^{36}-\frac{22411}{11238244352}a^{34}+\frac{21273}{5619122176}a^{32}+\frac{5833}{702390272}a^{30}+\frac{14917}{702390272}a^{28}-\frac{10229}{351195136}a^{26}-\frac{2063}{43899392}a^{24}-\frac{2185}{43899392}a^{22}+\frac{15849}{87798784}a^{20}-\frac{14387}{43899392}a^{18}+\frac{21941}{21949696}a^{16}+\frac{2255}{5487424}a^{14}-\frac{19183}{2743712}a^{12}-\frac{20713}{1371856}a^{10}+\frac{12153}{1371856}a^{8}+\frac{21545}{685928}a^{6}-\frac{19671}{342964}a^{4}-\frac{17215}{85741}a^{2}-\frac{30649}{85741}$, $\frac{1}{89905954816}a^{41}-\frac{25}{22476488704}a^{39}+\frac{233}{22476488704}a^{37}-\frac{22411}{11238244352}a^{35}+\frac{21273}{5619122176}a^{33}+\frac{5833}{702390272}a^{31}+\frac{14917}{702390272}a^{29}-\frac{10229}{351195136}a^{27}-\frac{2063}{43899392}a^{25}-\frac{2185}{43899392}a^{23}+\frac{15849}{87798784}a^{21}-\frac{14387}{43899392}a^{19}+\frac{21941}{21949696}a^{17}+\frac{2255}{5487424}a^{15}-\frac{19183}{2743712}a^{13}-\frac{20713}{1371856}a^{11}+\frac{12153}{1371856}a^{9}+\frac{21545}{685928}a^{7}-\frac{19671}{342964}a^{5}-\frac{17215}{85741}a^{3}-\frac{30649}{85741}a$, $\frac{1}{37\!\cdots\!04}a^{42}-\frac{38\!\cdots\!87}{92\!\cdots\!76}a^{40}+\frac{27\!\cdots\!63}{23\!\cdots\!44}a^{38}+\frac{42\!\cdots\!83}{46\!\cdots\!88}a^{36}-\frac{36\!\cdots\!89}{29\!\cdots\!68}a^{34}+\frac{84\!\cdots\!41}{11\!\cdots\!72}a^{32}-\frac{35\!\cdots\!81}{29\!\cdots\!68}a^{30}+\frac{61\!\cdots\!19}{29\!\cdots\!68}a^{28}+\frac{58\!\cdots\!73}{72\!\cdots\!92}a^{26}-\frac{14\!\cdots\!49}{18\!\cdots\!48}a^{24}-\frac{42\!\cdots\!57}{18\!\cdots\!48}a^{22}-\frac{11\!\cdots\!45}{45\!\cdots\!12}a^{20}-\frac{20\!\cdots\!91}{90\!\cdots\!24}a^{18}-\frac{20\!\cdots\!79}{22\!\cdots\!56}a^{16}-\frac{87\!\cdots\!97}{22\!\cdots\!56}a^{14}+\frac{60\!\cdots\!15}{11\!\cdots\!28}a^{12}-\frac{12\!\cdots\!91}{14\!\cdots\!16}a^{10}-\frac{10\!\cdots\!65}{17\!\cdots\!27}a^{8}+\frac{10\!\cdots\!07}{17\!\cdots\!27}a^{6}+\frac{39\!\cdots\!91}{70\!\cdots\!08}a^{4}+\frac{75\!\cdots\!13}{35\!\cdots\!54}a^{2}-\frac{82\!\cdots\!57}{17\!\cdots\!27}$, $\frac{1}{37\!\cdots\!04}a^{43}-\frac{38\!\cdots\!87}{92\!\cdots\!76}a^{41}+\frac{27\!\cdots\!63}{23\!\cdots\!44}a^{39}+\frac{42\!\cdots\!83}{46\!\cdots\!88}a^{37}-\frac{36\!\cdots\!89}{29\!\cdots\!68}a^{35}+\frac{84\!\cdots\!41}{11\!\cdots\!72}a^{33}-\frac{35\!\cdots\!81}{29\!\cdots\!68}a^{31}+\frac{61\!\cdots\!19}{29\!\cdots\!68}a^{29}+\frac{58\!\cdots\!73}{72\!\cdots\!92}a^{27}-\frac{14\!\cdots\!49}{18\!\cdots\!48}a^{25}-\frac{42\!\cdots\!57}{18\!\cdots\!48}a^{23}-\frac{11\!\cdots\!45}{45\!\cdots\!12}a^{21}-\frac{20\!\cdots\!91}{90\!\cdots\!24}a^{19}-\frac{20\!\cdots\!79}{22\!\cdots\!56}a^{17}-\frac{87\!\cdots\!97}{22\!\cdots\!56}a^{15}+\frac{60\!\cdots\!15}{11\!\cdots\!28}a^{13}-\frac{12\!\cdots\!91}{14\!\cdots\!16}a^{11}-\frac{10\!\cdots\!65}{17\!\cdots\!27}a^{9}+\frac{10\!\cdots\!07}{17\!\cdots\!27}a^{7}+\frac{39\!\cdots\!91}{70\!\cdots\!08}a^{5}+\frac{75\!\cdots\!13}{35\!\cdots\!54}a^{3}-\frac{82\!\cdots\!57}{17\!\cdots\!27}a$
| 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: |
\( -1 \)
(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: | 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
| A cyclic group of order 44 |
| The 44 conjugacy class representatives for $C_{44}$ |
| Character table for $C_{44}$ |
Intermediate fields
| \(\Q(\sqrt{89}) \), 4.0.45118016.1, 11.11.31181719929966183601.1, 22.22.86534669543385676516186776267386878120889.1 |
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 | $44$ | ${\href{/padicField/5.11.0.1}{11} }^{4}$ | $44$ | ${\href{/padicField/11.11.0.1}{11} }^{4}$ | $44$ | $22^{2}$ | $44$ | $44$ | $44$ | $44$ | ${\href{/padicField/37.4.0.1}{4} }^{11}$ | $44$ | $44$ | ${\href{/padicField/47.11.0.1}{11} }^{4}$ | ${\href{/padicField/53.11.0.1}{11} }^{4}$ | $44$ |
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 $22$ | $2$ | $11$ | $33$ | |||
| Deg $22$ | $2$ | $11$ | $33$ | ||||
|
\(89\)
| Deg $44$ | $44$ | $1$ | $43$ |