Normalized defining polynomial
\( x^{44} + 69 x^{40} + 1978 x^{36} + 30521 x^{32} + 274712 x^{28} + 1463260 x^{24} + 4481688 x^{20} + \cdots + 529 \)
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: | \(482\!\cdots\!224\) \(\medspace = 2^{88}\cdot 23^{42}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(79.78\) | 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: | $2^{2}23^{21/22}\approx 79.77946278015138$ | ||
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$$\chi_{184}(1,·)$, $\chi_{184}(171,·)$, $\chi_{184}(5,·)$, $\chi_{184}(9,·)$, $\chi_{184}(11,·)$, $\chi_{184}(19,·)$, $\chi_{184}(149,·)$, $\chi_{184}(151,·)$, $\chi_{184}(25,·)$, $\chi_{184}(155,·)$, $\chi_{184}(157,·)$, $\chi_{184}(31,·)$, $\chi_{184}(37,·)$, $\chi_{184}(167,·)$, $\chi_{184}(41,·)$, $\chi_{184}(43,·)$, $\chi_{184}(45,·)$, $\chi_{184}(47,·)$, $\chi_{184}(49,·)$, $\chi_{184}(51,·)$, $\chi_{184}(53,·)$, $\chi_{184}(55,·)$, $\chi_{184}(61,·)$, $\chi_{184}(181,·)$, $\chi_{184}(67,·)$, $\chi_{184}(71,·)$, $\chi_{184}(73,·)$, $\chi_{184}(119,·)$, $\chi_{184}(81,·)$, $\chi_{184}(83,·)$, $\chi_{184}(87,·)$, $\chi_{184}(91,·)$, $\chi_{184}(95,·)$, $\chi_{184}(99,·)$, $\chi_{184}(105,·)$, $\chi_{184}(107,·)$, $\chi_{184}(109,·)$, $\chi_{184}(177,·)$, $\chi_{184}(169,·)$, $\chi_{184}(121,·)$, $\chi_{184}(127,·)$, $\chi_{184}(125,·)$, $\chi_{184}(39,·)$, $\chi_{184}(21,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{2097152}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{23}a^{22}$, $\frac{1}{23}a^{23}$, $\frac{1}{23}a^{24}$, $\frac{1}{23}a^{25}$, $\frac{1}{23}a^{26}$, $\frac{1}{23}a^{27}$, $\frac{1}{23}a^{28}$, $\frac{1}{23}a^{29}$, $\frac{1}{23}a^{30}$, $\frac{1}{23}a^{31}$, $\frac{1}{23}a^{32}$, $\frac{1}{23}a^{33}$, $\frac{1}{23}a^{34}$, $\frac{1}{23}a^{35}$, $\frac{1}{23}a^{36}$, $\frac{1}{23}a^{37}$, $\frac{1}{23}a^{38}$, $\frac{1}{23}a^{39}$, $\frac{1}{20\!\cdots\!91}a^{40}-\frac{37\!\cdots\!86}{20\!\cdots\!91}a^{36}-\frac{61\!\cdots\!97}{20\!\cdots\!91}a^{32}+\frac{31\!\cdots\!38}{20\!\cdots\!91}a^{28}-\frac{27\!\cdots\!29}{20\!\cdots\!91}a^{24}-\frac{94\!\cdots\!24}{88\!\cdots\!17}a^{20}+\frac{16\!\cdots\!76}{88\!\cdots\!17}a^{16}+\frac{35\!\cdots\!34}{88\!\cdots\!17}a^{12}+\frac{37\!\cdots\!73}{88\!\cdots\!17}a^{8}-\frac{26\!\cdots\!52}{88\!\cdots\!17}a^{4}-\frac{17\!\cdots\!45}{88\!\cdots\!17}$, $\frac{1}{20\!\cdots\!91}a^{41}-\frac{37\!\cdots\!86}{20\!\cdots\!91}a^{37}-\frac{61\!\cdots\!97}{20\!\cdots\!91}a^{33}+\frac{31\!\cdots\!38}{20\!\cdots\!91}a^{29}-\frac{27\!\cdots\!29}{20\!\cdots\!91}a^{25}-\frac{94\!\cdots\!24}{88\!\cdots\!17}a^{21}+\frac{16\!\cdots\!76}{88\!\cdots\!17}a^{17}+\frac{35\!\cdots\!34}{88\!\cdots\!17}a^{13}+\frac{37\!\cdots\!73}{88\!\cdots\!17}a^{9}-\frac{26\!\cdots\!52}{88\!\cdots\!17}a^{5}-\frac{17\!\cdots\!45}{88\!\cdots\!17}a$, $\frac{1}{20\!\cdots\!91}a^{42}-\frac{37\!\cdots\!86}{20\!\cdots\!91}a^{38}-\frac{61\!\cdots\!97}{20\!\cdots\!91}a^{34}+\frac{31\!\cdots\!38}{20\!\cdots\!91}a^{30}-\frac{27\!\cdots\!29}{20\!\cdots\!91}a^{26}-\frac{39\!\cdots\!18}{20\!\cdots\!91}a^{22}+\frac{16\!\cdots\!76}{88\!\cdots\!17}a^{18}+\frac{35\!\cdots\!34}{88\!\cdots\!17}a^{14}+\frac{37\!\cdots\!73}{88\!\cdots\!17}a^{10}-\frac{26\!\cdots\!52}{88\!\cdots\!17}a^{6}-\frac{17\!\cdots\!45}{88\!\cdots\!17}a^{2}$, $\frac{1}{20\!\cdots\!91}a^{43}-\frac{37\!\cdots\!86}{20\!\cdots\!91}a^{39}-\frac{61\!\cdots\!97}{20\!\cdots\!91}a^{35}+\frac{31\!\cdots\!38}{20\!\cdots\!91}a^{31}-\frac{27\!\cdots\!29}{20\!\cdots\!91}a^{27}-\frac{39\!\cdots\!18}{20\!\cdots\!91}a^{23}+\frac{16\!\cdots\!76}{88\!\cdots\!17}a^{19}+\frac{35\!\cdots\!34}{88\!\cdots\!17}a^{15}+\frac{37\!\cdots\!73}{88\!\cdots\!17}a^{11}-\frac{26\!\cdots\!52}{88\!\cdots\!17}a^{7}-\frac{17\!\cdots\!45}{88\!\cdots\!17}a^{3}$
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: | \( \frac{2159754831367634313897}{20449419753638432679768691} a^{42} + \frac{149047263848576640299926}{20449419753638432679768691} a^{38} + \frac{4273665809431876600479477}{20449419753638432679768691} a^{34} + \frac{2868081116953286494019238}{889105206679931855642117} a^{30} + \frac{594053187211700567170909452}{20449419753638432679768691} a^{26} + \frac{3166997370442965220709483272}{20449419753638432679768691} a^{22} + \frac{422408163695724435246179921}{889105206679931855642117} a^{18} + \frac{694100615457853045962198762}{889105206679931855642117} a^{14} + \frac{536315973527527086028235889}{889105206679931855642117} a^{10} + \frac{156054186041719117834635830}{889105206679931855642117} a^{6} + \frac{12830402239602083026931771}{889105206679931855642117} a^{2} \) (order $4$) | 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 $C_2\times C_{22}$ is not computed |
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 | $22^{2}$ | ${\href{/padicField/5.11.0.1}{11} }^{4}$ | $22^{2}$ | $22^{2}$ | $22^{2}$ | $22^{2}$ | $22^{2}$ | R | $22^{2}$ | $22^{2}$ | ${\href{/padicField/37.11.0.1}{11} }^{4}$ | ${\href{/padicField/41.11.0.1}{11} }^{4}$ | $22^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{22}$ | ${\href{/padicField/53.11.0.1}{11} }^{4}$ | $22^{2}$ |
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 $44$ | $4$ | $11$ | $88$ | |||
\(23\) | Deg $44$ | $22$ | $2$ | $42$ |