Normalized defining polynomial
\( x^{44} + 61 x^{40} + 1522 x^{36} + 20041 x^{32} + 150032 x^{28} + 642172 x^{24} + 1506232 x^{20} + \cdots + 1 \)
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: | \(911\!\cdots\!456\) \(\medspace = 2^{88}\cdot 23^{40}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(69.18\) | 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: | $2^{2}23^{10/11}\approx 69.1822030596691$ | ||
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: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $44$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(184=2^{3}\cdot 23\) | ||
Dirichlet character group: | $\lbrace$$\chi_{184}(1,·)$, $\chi_{184}(3,·)$, $\chi_{184}(133,·)$, $\chi_{184}(9,·)$, $\chi_{184}(139,·)$, $\chi_{184}(13,·)$, $\chi_{184}(131,·)$, $\chi_{184}(151,·)$, $\chi_{184}(25,·)$, $\chi_{184}(27,·)$, $\chi_{184}(29,·)$, $\chi_{184}(31,·)$, $\chi_{184}(35,·)$, $\chi_{184}(165,·)$, $\chi_{184}(39,·)$, $\chi_{184}(41,·)$, $\chi_{184}(173,·)$, $\chi_{184}(47,·)$, $\chi_{184}(49,·)$, $\chi_{184}(179,·)$, $\chi_{184}(55,·)$, $\chi_{184}(59,·)$, $\chi_{184}(71,·)$, $\chi_{184}(73,·)$, $\chi_{184}(75,·)$, $\chi_{184}(77,·)$, $\chi_{184}(141,·)$, $\chi_{184}(81,·)$, $\chi_{184}(163,·)$, $\chi_{184}(85,·)$, $\chi_{184}(87,·)$, $\chi_{184}(169,·)$, $\chi_{184}(93,·)$, $\chi_{184}(95,·)$, $\chi_{184}(101,·)$, $\chi_{184}(105,·)$, $\chi_{184}(167,·)$, $\chi_{184}(177,·)$, $\chi_{184}(147,·)$, $\chi_{184}(117,·)$, $\chi_{184}(119,·)$, $\chi_{184}(121,·)$, $\chi_{184}(123,·)$, $\chi_{184}(127,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{2097152}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $\frac{1}{38\!\cdots\!61}a^{40}-\frac{32\!\cdots\!57}{38\!\cdots\!61}a^{36}-\frac{90\!\cdots\!10}{38\!\cdots\!61}a^{32}-\frac{12\!\cdots\!42}{38\!\cdots\!61}a^{28}+\frac{11\!\cdots\!59}{38\!\cdots\!61}a^{24}+\frac{11\!\cdots\!44}{38\!\cdots\!61}a^{20}-\frac{29\!\cdots\!87}{38\!\cdots\!61}a^{16}+\frac{13\!\cdots\!95}{38\!\cdots\!61}a^{12}+\frac{46\!\cdots\!69}{38\!\cdots\!61}a^{8}-\frac{46\!\cdots\!78}{38\!\cdots\!61}a^{4}+\frac{14\!\cdots\!57}{38\!\cdots\!61}$, $\frac{1}{38\!\cdots\!61}a^{41}-\frac{32\!\cdots\!57}{38\!\cdots\!61}a^{37}-\frac{90\!\cdots\!10}{38\!\cdots\!61}a^{33}-\frac{12\!\cdots\!42}{38\!\cdots\!61}a^{29}+\frac{11\!\cdots\!59}{38\!\cdots\!61}a^{25}+\frac{11\!\cdots\!44}{38\!\cdots\!61}a^{21}-\frac{29\!\cdots\!87}{38\!\cdots\!61}a^{17}+\frac{13\!\cdots\!95}{38\!\cdots\!61}a^{13}+\frac{46\!\cdots\!69}{38\!\cdots\!61}a^{9}-\frac{46\!\cdots\!78}{38\!\cdots\!61}a^{5}+\frac{14\!\cdots\!57}{38\!\cdots\!61}a$, $\frac{1}{38\!\cdots\!61}a^{42}-\frac{32\!\cdots\!57}{38\!\cdots\!61}a^{38}-\frac{90\!\cdots\!10}{38\!\cdots\!61}a^{34}-\frac{12\!\cdots\!42}{38\!\cdots\!61}a^{30}+\frac{11\!\cdots\!59}{38\!\cdots\!61}a^{26}+\frac{11\!\cdots\!44}{38\!\cdots\!61}a^{22}-\frac{29\!\cdots\!87}{38\!\cdots\!61}a^{18}+\frac{13\!\cdots\!95}{38\!\cdots\!61}a^{14}+\frac{46\!\cdots\!69}{38\!\cdots\!61}a^{10}-\frac{46\!\cdots\!78}{38\!\cdots\!61}a^{6}+\frac{14\!\cdots\!57}{38\!\cdots\!61}a^{2}$, $\frac{1}{38\!\cdots\!61}a^{43}-\frac{32\!\cdots\!57}{38\!\cdots\!61}a^{39}-\frac{90\!\cdots\!10}{38\!\cdots\!61}a^{35}-\frac{12\!\cdots\!42}{38\!\cdots\!61}a^{31}+\frac{11\!\cdots\!59}{38\!\cdots\!61}a^{27}+\frac{11\!\cdots\!44}{38\!\cdots\!61}a^{23}-\frac{29\!\cdots\!87}{38\!\cdots\!61}a^{19}+\frac{13\!\cdots\!95}{38\!\cdots\!61}a^{15}+\frac{46\!\cdots\!69}{38\!\cdots\!61}a^{11}-\frac{46\!\cdots\!78}{38\!\cdots\!61}a^{7}+\frac{14\!\cdots\!57}{38\!\cdots\!61}a^{3}$
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: | \( \frac{559364217950633152513003118}{3844747107219467355553841461} a^{43} + \frac{34120009129382592888858609622}{3844747107219467355553841461} a^{39} + \frac{851278791029073106538382179100}{3844747107219467355553841461} a^{35} + \frac{11208388430103503940586671208606}{3844747107219467355553841461} a^{31} + \frac{83898538381950787425688419125632}{3844747107219467355553841461} a^{27} + \frac{359029571555533259619041645454461}{3844747107219467355553841461} a^{23} + \frac{841776480195238370324038807360508}{3844747107219467355553841461} a^{19} + \frac{982761154071312182014857524570016}{3844747107219467355553841461} a^{15} + \frac{479453552318492415604964710844738}{3844747107219467355553841461} a^{11} + \frac{70553971991710607652936942463583}{3844747107219467355553841461} a^{7} + \frac{1536643884990641141797014151784}{3844747107219467355553841461} a^{3} \) (order $8$) | 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 | $22^{2}$ | $22^{2}$ | $22^{2}$ | $22^{2}$ | $22^{2}$ | ${\href{/padicField/17.11.0.1}{11} }^{4}$ | $22^{2}$ | R | $22^{2}$ | $22^{2}$ | $22^{2}$ | ${\href{/padicField/41.11.0.1}{11} }^{4}$ | $22^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{22}$ | $22^{2}$ | $22^{2}$ |
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 $44$ | $4$ | $11$ | $88$ | |||
\(23\) | 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 | $[\ ]_{11}^{2}$ |