Normalized defining polynomial
\( x^{44} - x^{43} + 51 x^{42} - 72 x^{41} + 566 x^{40} - 1823 x^{39} - 5918 x^{38} - 45125 x^{37} + \cdots + 9380892213701 \)
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: |
\(871\!\cdots\!125\)
\(\medspace = 5^{33}\cdot 89^{42}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(242.67\) | 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: | $5^{3/4}89^{21/22}\approx 242.6659674557602$ | ||
| Ramified primes: |
\(5\), \(89\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $44$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(445=5\cdot 89\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{445}(256,·)$, $\chi_{445}(1,·)$, $\chi_{445}(4,·)$, $\chi_{445}(133,·)$, $\chi_{445}(134,·)$, $\chi_{445}(263,·)$, $\chi_{445}(269,·)$, $\chi_{445}(271,·)$, $\chi_{445}(16,·)$, $\chi_{445}(401,·)$, $\chi_{445}(22,·)$, $\chi_{445}(156,·)$, $\chi_{445}(413,·)$, $\chi_{445}(278,·)$, $\chi_{445}(292,·)$, $\chi_{445}(39,·)$, $\chi_{445}(299,·)$, $\chi_{445}(434,·)$, $\chi_{445}(177,·)$, $\chi_{445}(306,·)$, $\chi_{445}(179,·)$, $\chi_{445}(437,·)$, $\chi_{445}(57,·)$, $\chi_{445}(186,·)$, $\chi_{445}(443,·)$, $\chi_{445}(317,·)$, $\chi_{445}(64,·)$, $\chi_{445}(194,·)$, $\chi_{445}(203,·)$, $\chi_{445}(73,·)$, $\chi_{445}(331,·)$, $\chi_{445}(162,·)$, $\chi_{445}(334,·)$, $\chi_{445}(87,·)$, $\chi_{445}(88,·)$, $\chi_{445}(91,·)$, $\chi_{445}(348,·)$, $\chi_{445}(222,·)$, $\chi_{445}(352,·)$, $\chi_{445}(228,·)$, $\chi_{445}(364,·)$, $\chi_{445}(367,·)$, $\chi_{445}(121,·)$, $\chi_{445}(378,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{2097152}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$, $\frac{1}{210263330603}a^{42}+\frac{86374319650}{210263330603}a^{41}+\frac{58402938669}{210263330603}a^{40}-\frac{103027668576}{210263330603}a^{39}-\frac{62382945342}{210263330603}a^{38}+\frac{4767334415}{210263330603}a^{37}-\frac{75650898133}{210263330603}a^{36}+\frac{82562465189}{210263330603}a^{35}-\frac{53232215545}{210263330603}a^{34}-\frac{102482714472}{210263330603}a^{33}+\frac{28957602617}{210263330603}a^{32}+\frac{104820047174}{210263330603}a^{31}+\frac{5738319175}{210263330603}a^{30}+\frac{23238614055}{210263330603}a^{29}+\frac{52980079450}{210263330603}a^{28}+\frac{31200190187}{210263330603}a^{27}+\frac{71235850674}{210263330603}a^{26}-\frac{95753142401}{210263330603}a^{25}+\frac{75609993357}{210263330603}a^{24}-\frac{46193821326}{210263330603}a^{23}-\frac{101215507307}{210263330603}a^{22}-\frac{58768359090}{210263330603}a^{21}-\frac{15629409434}{210263330603}a^{20}+\frac{6713434155}{210263330603}a^{19}-\frac{102356883470}{210263330603}a^{18}-\frac{80021844649}{210263330603}a^{17}-\frac{71500365357}{210263330603}a^{16}-\frac{46136863854}{210263330603}a^{15}-\frac{98685967407}{210263330603}a^{14}-\frac{11514720379}{210263330603}a^{13}+\frac{95607792270}{210263330603}a^{12}+\frac{39065077242}{210263330603}a^{11}-\frac{31123888088}{210263330603}a^{10}-\frac{96148531684}{210263330603}a^{9}-\frac{22271477098}{210263330603}a^{8}+\frac{76154887550}{210263330603}a^{7}+\frac{103282392914}{210263330603}a^{6}+\frac{87595562806}{210263330603}a^{5}-\frac{2405980949}{5682792719}a^{4}-\frac{20120505834}{210263330603}a^{3}-\frac{69565839262}{210263330603}a^{2}+\frac{77421794523}{210263330603}a-\frac{49988316487}{210263330603}$, $\frac{1}{23\!\cdots\!33}a^{43}-\frac{37\!\cdots\!50}{23\!\cdots\!33}a^{42}-\frac{93\!\cdots\!46}{23\!\cdots\!33}a^{41}-\frac{68\!\cdots\!06}{23\!\cdots\!33}a^{40}-\frac{74\!\cdots\!55}{23\!\cdots\!33}a^{39}-\frac{65\!\cdots\!40}{23\!\cdots\!33}a^{38}-\frac{21\!\cdots\!28}{23\!\cdots\!33}a^{37}+\frac{20\!\cdots\!09}{63\!\cdots\!09}a^{36}-\frac{80\!\cdots\!87}{23\!\cdots\!33}a^{35}-\frac{90\!\cdots\!06}{23\!\cdots\!33}a^{34}+\frac{49\!\cdots\!24}{23\!\cdots\!33}a^{33}-\frac{10\!\cdots\!68}{23\!\cdots\!33}a^{32}-\frac{68\!\cdots\!03}{23\!\cdots\!33}a^{31}-\frac{10\!\cdots\!01}{23\!\cdots\!33}a^{30}+\frac{68\!\cdots\!62}{23\!\cdots\!33}a^{29}-\frac{58\!\cdots\!37}{23\!\cdots\!33}a^{28}+\frac{49\!\cdots\!54}{23\!\cdots\!33}a^{27}-\frac{68\!\cdots\!69}{23\!\cdots\!33}a^{26}+\frac{79\!\cdots\!41}{23\!\cdots\!33}a^{25}-\frac{33\!\cdots\!68}{23\!\cdots\!33}a^{24}+\frac{10\!\cdots\!10}{23\!\cdots\!33}a^{23}+\frac{69\!\cdots\!08}{23\!\cdots\!33}a^{22}+\frac{94\!\cdots\!62}{23\!\cdots\!33}a^{21}-\frac{10\!\cdots\!12}{23\!\cdots\!33}a^{20}-\frac{38\!\cdots\!81}{23\!\cdots\!33}a^{19}+\frac{57\!\cdots\!91}{23\!\cdots\!33}a^{18}+\frac{14\!\cdots\!21}{23\!\cdots\!33}a^{17}+\frac{92\!\cdots\!82}{23\!\cdots\!33}a^{16}+\frac{35\!\cdots\!26}{23\!\cdots\!33}a^{15}-\frac{47\!\cdots\!32}{23\!\cdots\!33}a^{14}+\frac{40\!\cdots\!89}{23\!\cdots\!33}a^{13}+\frac{54\!\cdots\!53}{23\!\cdots\!33}a^{12}-\frac{14\!\cdots\!57}{23\!\cdots\!33}a^{11}-\frac{55\!\cdots\!59}{23\!\cdots\!33}a^{10}+\frac{87\!\cdots\!74}{23\!\cdots\!33}a^{9}-\frac{15\!\cdots\!07}{23\!\cdots\!33}a^{8}-\frac{10\!\cdots\!17}{23\!\cdots\!33}a^{7}+\frac{12\!\cdots\!33}{63\!\cdots\!09}a^{6}-\frac{40\!\cdots\!48}{23\!\cdots\!33}a^{5}-\frac{55\!\cdots\!58}{23\!\cdots\!33}a^{4}+\frac{38\!\cdots\!35}{23\!\cdots\!33}a^{3}+\frac{17\!\cdots\!65}{23\!\cdots\!33}a^{2}-\frac{50\!\cdots\!05}{23\!\cdots\!33}a+\frac{70\!\cdots\!50}{23\!\cdots\!33}$
| 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
| \(\Q(\sqrt{5}) \), 4.0.990125.2, 11.11.31181719929966183601.1, 22.22.47475569228068862203841937471134830429736328125.1 |
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 | $44$ | $44$ | R | $44$ | ${\href{/padicField/11.11.0.1}{11} }^{4}$ | $44$ | $44$ | ${\href{/padicField/19.11.0.1}{11} }^{4}$ | $44$ | ${\href{/padicField/29.11.0.1}{11} }^{4}$ | $22^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{11}$ | $22^{2}$ | $44$ | $44$ | $44$ | ${\href{/padicField/59.11.0.1}{11} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| Deg $44$ | $4$ | $11$ | $33$ | |||
|
\(89\)
| 89.22.21.17 | $x^{22} + 1068$ | $22$ | $1$ | $21$ | 22T1 | $[\ ]_{22}$ |
| 89.22.21.17 | $x^{22} + 1068$ | $22$ | $1$ | $21$ | 22T1 | $[\ ]_{22}$ |