Normalized defining polynomial
\( x^{44} + 92 x^{42} + 3818 x^{40} + 94760 x^{38} + 1573384 x^{36} + 18537632 x^{34} + 160562632 x^{32} + \cdots + 1083392 \)
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: |
\(414\!\cdots\!608\)
\(\medspace = 2^{121}\cdot 23^{42}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(134.17\) | 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: | $2^{11/4}23^{21/22}\approx 134.17252852539286$ | ||
| Ramified primes: |
\(2\), \(23\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{2}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $44$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(368=2^{4}\cdot 23\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{368}(1,·)$, $\chi_{368}(5,·)$, $\chi_{368}(257,·)$, $\chi_{368}(9,·)$, $\chi_{368}(109,·)$, $\chi_{368}(237,·)$, $\chi_{368}(21,·)$, $\chi_{368}(25,·)$, $\chi_{368}(157,·)$, $\chi_{368}(289,·)$, $\chi_{368}(37,·)$, $\chi_{368}(305,·)$, $\chi_{368}(41,·)$, $\chi_{368}(45,·)$, $\chi_{368}(49,·)$, $\chi_{368}(309,·)$, $\chi_{368}(265,·)$, $\chi_{368}(185,·)$, $\chi_{368}(61,·)$, $\chi_{368}(53,·)$, $\chi_{368}(193,·)$, $\chi_{368}(181,·)$, $\chi_{368}(353,·)$, $\chi_{368}(73,·)$, $\chi_{368}(205,·)$, $\chi_{368}(333,·)$, $\chi_{368}(81,·)$, $\chi_{368}(341,·)$, $\chi_{368}(105,·)$, $\chi_{368}(361,·)$, $\chi_{368}(221,·)$, $\chi_{368}(293,·)$, $\chi_{368}(225,·)$, $\chi_{368}(229,·)$, $\chi_{368}(209,·)$, $\chi_{368}(233,·)$, $\chi_{368}(365,·)$, $\chi_{368}(177,·)$, $\chi_{368}(189,·)$, $\chi_{368}(245,·)$, $\chi_{368}(169,·)$, $\chi_{368}(121,·)$, $\chi_{368}(125,·)$, $\chi_{368}(149,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{2097152}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{4}a^{8}$, $\frac{1}{4}a^{9}$, $\frac{1}{4}a^{10}$, $\frac{1}{4}a^{11}$, $\frac{1}{8}a^{12}$, $\frac{1}{8}a^{13}$, $\frac{1}{8}a^{14}$, $\frac{1}{8}a^{15}$, $\frac{1}{16}a^{16}$, $\frac{1}{16}a^{17}$, $\frac{1}{16}a^{18}$, $\frac{1}{16}a^{19}$, $\frac{1}{32}a^{20}$, $\frac{1}{32}a^{21}$, $\frac{1}{736}a^{22}$, $\frac{1}{736}a^{23}$, $\frac{1}{1472}a^{24}$, $\frac{1}{1472}a^{25}$, $\frac{1}{1472}a^{26}$, $\frac{1}{1472}a^{27}$, $\frac{1}{2944}a^{28}$, $\frac{1}{2944}a^{29}$, $\frac{1}{2944}a^{30}$, $\frac{1}{2944}a^{31}$, $\frac{1}{5888}a^{32}$, $\frac{1}{5888}a^{33}$, $\frac{1}{276736}a^{34}-\frac{19}{276736}a^{32}-\frac{1}{8648}a^{30}-\frac{3}{69184}a^{28}-\frac{11}{69184}a^{26}-\frac{11}{34592}a^{24}-\frac{1}{4324}a^{22}-\frac{1}{188}a^{20}-\frac{1}{47}a^{18}-\frac{11}{376}a^{16}+\frac{3}{188}a^{14}-\frac{1}{47}a^{12}+\frac{3}{47}a^{10}+\frac{1}{47}a^{8}-\frac{21}{94}a^{6}-\frac{3}{47}a^{4}-\frac{10}{47}a^{2}+\frac{2}{47}$, $\frac{1}{276736}a^{35}-\frac{19}{276736}a^{33}-\frac{1}{8648}a^{31}-\frac{3}{69184}a^{29}-\frac{11}{69184}a^{27}-\frac{11}{34592}a^{25}-\frac{1}{4324}a^{23}-\frac{1}{188}a^{21}-\frac{1}{47}a^{19}-\frac{11}{376}a^{17}+\frac{3}{188}a^{15}-\frac{1}{47}a^{13}+\frac{3}{47}a^{11}+\frac{1}{47}a^{9}-\frac{21}{94}a^{7}-\frac{3}{47}a^{5}-\frac{10}{47}a^{3}+\frac{2}{47}a$, $\frac{1}{553472}a^{36}+\frac{15}{276736}a^{32}-\frac{7}{69184}a^{30}-\frac{21}{138368}a^{28}+\frac{1}{34592}a^{26}+\frac{9}{34592}a^{24}+\frac{5}{8648}a^{22}+\frac{1}{752}a^{20}-\frac{11}{376}a^{18}-\frac{15}{752}a^{16}+\frac{3}{188}a^{14}-\frac{17}{376}a^{12}+\frac{11}{94}a^{10}+\frac{17}{188}a^{8}+\frac{9}{94}a^{6}-\frac{10}{47}a^{4}+\frac{19}{47}$, $\frac{1}{553472}a^{37}+\frac{15}{276736}a^{33}-\frac{7}{69184}a^{31}-\frac{21}{138368}a^{29}+\frac{1}{34592}a^{27}+\frac{9}{34592}a^{25}+\frac{5}{8648}a^{23}+\frac{1}{752}a^{21}-\frac{11}{376}a^{19}-\frac{15}{752}a^{17}+\frac{3}{188}a^{15}-\frac{17}{376}a^{13}+\frac{11}{94}a^{11}+\frac{17}{188}a^{9}+\frac{9}{94}a^{7}-\frac{10}{47}a^{5}+\frac{19}{47}a$, $\frac{1}{553472}a^{38}+\frac{11}{138368}a^{32}-\frac{1}{8648}a^{30}-\frac{5}{69184}a^{26}-\frac{3}{34592}a^{24}-\frac{11}{17296}a^{22}-\frac{9}{752}a^{20}-\frac{5}{376}a^{18}+\frac{13}{752}a^{16}-\frac{13}{376}a^{14}+\frac{23}{376}a^{12}-\frac{11}{94}a^{10}+\frac{5}{188}a^{8}+\frac{13}{94}a^{6}-\frac{2}{47}a^{4}-\frac{19}{47}a^{2}+\frac{17}{47}$, $\frac{1}{553472}a^{39}+\frac{11}{138368}a^{33}-\frac{1}{8648}a^{31}-\frac{5}{69184}a^{27}-\frac{3}{34592}a^{25}-\frac{11}{17296}a^{23}-\frac{9}{752}a^{21}-\frac{5}{376}a^{19}+\frac{13}{752}a^{17}-\frac{13}{376}a^{15}+\frac{23}{376}a^{13}-\frac{11}{94}a^{11}+\frac{5}{188}a^{9}+\frac{13}{94}a^{7}-\frac{2}{47}a^{5}-\frac{19}{47}a^{3}+\frac{17}{47}a$, $\frac{1}{1106944}a^{40}+\frac{5}{276736}a^{32}-\frac{3}{34592}a^{30}+\frac{7}{69184}a^{28}-\frac{1}{3008}a^{26}-\frac{15}{69184}a^{24}+\frac{11}{17296}a^{22}-\frac{1}{94}a^{20}+\frac{9}{376}a^{18}-\frac{3}{376}a^{16}+\frac{2}{47}a^{14}+\frac{19}{376}a^{12}-\frac{3}{47}a^{10}+\frac{4}{47}a^{8}-\frac{3}{47}a^{6}+\frac{1}{47}a^{2}-\frac{22}{47}$, $\frac{1}{1106944}a^{41}+\frac{5}{276736}a^{33}-\frac{3}{34592}a^{31}+\frac{7}{69184}a^{29}-\frac{1}{3008}a^{27}-\frac{15}{69184}a^{25}+\frac{11}{17296}a^{23}-\frac{1}{94}a^{21}+\frac{9}{376}a^{19}-\frac{3}{376}a^{17}+\frac{2}{47}a^{15}+\frac{19}{376}a^{13}-\frac{3}{47}a^{11}+\frac{4}{47}a^{9}-\frac{3}{47}a^{7}+\frac{1}{47}a^{3}-\frac{22}{47}a$, $\frac{1}{14\!\cdots\!08}a^{42}+\frac{22\!\cdots\!69}{60\!\cdots\!96}a^{40}+\frac{25\!\cdots\!71}{30\!\cdots\!48}a^{38}+\frac{11\!\cdots\!67}{30\!\cdots\!48}a^{36}-\frac{14\!\cdots\!27}{15\!\cdots\!24}a^{34}+\frac{25\!\cdots\!97}{76\!\cdots\!12}a^{32}+\frac{47\!\cdots\!11}{76\!\cdots\!12}a^{30}-\frac{76\!\cdots\!01}{76\!\cdots\!12}a^{28}+\frac{30\!\cdots\!81}{19\!\cdots\!28}a^{26}+\frac{44\!\cdots\!27}{16\!\cdots\!72}a^{24}+\frac{49\!\cdots\!11}{95\!\cdots\!64}a^{22}+\frac{13\!\cdots\!89}{11\!\cdots\!58}a^{20}+\frac{52\!\cdots\!87}{51\!\cdots\!46}a^{18}+\frac{83\!\cdots\!69}{41\!\cdots\!68}a^{16}-\frac{36\!\cdots\!51}{20\!\cdots\!84}a^{14}+\frac{21\!\cdots\!53}{20\!\cdots\!84}a^{12}-\frac{10\!\cdots\!49}{10\!\cdots\!92}a^{10}+\frac{98\!\cdots\!25}{10\!\cdots\!92}a^{8}-\frac{29\!\cdots\!72}{25\!\cdots\!23}a^{6}-\frac{46\!\cdots\!75}{25\!\cdots\!23}a^{4}+\frac{57\!\cdots\!12}{25\!\cdots\!23}a^{2}-\frac{72\!\cdots\!10}{25\!\cdots\!23}$, $\frac{1}{14\!\cdots\!08}a^{43}+\frac{22\!\cdots\!69}{60\!\cdots\!96}a^{41}+\frac{25\!\cdots\!71}{30\!\cdots\!48}a^{39}+\frac{11\!\cdots\!67}{30\!\cdots\!48}a^{37}-\frac{14\!\cdots\!27}{15\!\cdots\!24}a^{35}+\frac{25\!\cdots\!97}{76\!\cdots\!12}a^{33}+\frac{47\!\cdots\!11}{76\!\cdots\!12}a^{31}-\frac{76\!\cdots\!01}{76\!\cdots\!12}a^{29}+\frac{30\!\cdots\!81}{19\!\cdots\!28}a^{27}+\frac{44\!\cdots\!27}{16\!\cdots\!72}a^{25}+\frac{49\!\cdots\!11}{95\!\cdots\!64}a^{23}+\frac{13\!\cdots\!89}{11\!\cdots\!58}a^{21}+\frac{52\!\cdots\!87}{51\!\cdots\!46}a^{19}+\frac{83\!\cdots\!69}{41\!\cdots\!68}a^{17}-\frac{36\!\cdots\!51}{20\!\cdots\!84}a^{15}+\frac{21\!\cdots\!53}{20\!\cdots\!84}a^{13}-\frac{10\!\cdots\!49}{10\!\cdots\!92}a^{11}+\frac{98\!\cdots\!25}{10\!\cdots\!92}a^{9}-\frac{29\!\cdots\!72}{25\!\cdots\!23}a^{7}-\frac{46\!\cdots\!75}{25\!\cdots\!23}a^{5}+\frac{57\!\cdots\!12}{25\!\cdots\!23}a^{3}-\frac{72\!\cdots\!10}{25\!\cdots\!23}a$
| 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 $C_{44}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{2}) \), 4.0.1083392.5, \(\Q(\zeta_{23})^+\), 22.22.14741666340843480753092741810452692992.1 |
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 | $44$ | $44$ | ${\href{/padicField/7.11.0.1}{11} }^{4}$ | $44$ | $44$ | $22^{2}$ | $44$ | R | $44$ | ${\href{/padicField/31.11.0.1}{11} }^{4}$ | $44$ | $22^{2}$ | $44$ | ${\href{/padicField/47.1.0.1}{1} }^{44}$ | $44$ | $44$ |
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 $44$ | $4$ | $11$ | $121$ | |||
|
\(23\)
| 23.22.21.1 | $x^{22} + 506$ | $22$ | $1$ | $21$ | 22T1 | $[\ ]_{22}$ |
| 23.22.21.1 | $x^{22} + 506$ | $22$ | $1$ | $21$ | 22T1 | $[\ ]_{22}$ |