Normalized defining polynomial
\( x^{44} + 115 x^{42} + 6095 x^{40} + 197800 x^{38} + 4405075 x^{36} + 71512750 x^{34} + \cdots + 25830078125 \)
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: |
\(319\!\cdots\!000\)
\(\medspace = 2^{44}\cdot 5^{33}\cdot 23^{42}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(133.38\) | 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: | $2\cdot 5^{3/4}23^{21/22}\approx 133.37935567613386$ | ||
| Ramified primes: |
\(2\), \(5\), \(23\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $44$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(460=2^{2}\cdot 5\cdot 23\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{460}(1,·)$, $\chi_{460}(387,·)$, $\chi_{460}(261,·)$, $\chi_{460}(7,·)$, $\chi_{460}(9,·)$, $\chi_{460}(267,·)$, $\chi_{460}(269,·)$, $\chi_{460}(143,·)$, $\chi_{460}(107,·)$, $\chi_{460}(409,·)$, $\chi_{460}(283,·)$, $\chi_{460}(29,·)$, $\chi_{460}(287,·)$, $\chi_{460}(289,·)$, $\chi_{460}(41,·)$, $\chi_{460}(43,·)$, $\chi_{460}(301,·)$, $\chi_{460}(49,·)$, $\chi_{460}(183,·)$, $\chi_{460}(441,·)$, $\chi_{460}(63,·)$, $\chi_{460}(449,·)$, $\chi_{460}(67,·)$, $\chi_{460}(327,·)$, $\chi_{460}(203,·)$, $\chi_{460}(141,·)$, $\chi_{460}(209,·)$, $\chi_{460}(83,·)$, $\chi_{460}(343,·)$, $\chi_{460}(349,·)$, $\chi_{460}(263,·)$, $\chi_{460}(227,·)$, $\chi_{460}(101,·)$, $\chi_{460}(81,·)$, $\chi_{460}(361,·)$, $\chi_{460}(103,·)$, $\chi_{460}(367,·)$, $\chi_{460}(369,·)$, $\chi_{460}(169,·)$, $\chi_{460}(121,·)$, $\chi_{460}(447,·)$, $\chi_{460}(247,·)$, $\chi_{460}(381,·)$, $\chi_{460}(383,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{2097152}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5}a^{4}$, $\frac{1}{5}a^{5}$, $\frac{1}{5}a^{6}$, $\frac{1}{5}a^{7}$, $\frac{1}{25}a^{8}$, $\frac{1}{25}a^{9}$, $\frac{1}{25}a^{10}$, $\frac{1}{25}a^{11}$, $\frac{1}{125}a^{12}$, $\frac{1}{125}a^{13}$, $\frac{1}{125}a^{14}$, $\frac{1}{125}a^{15}$, $\frac{1}{625}a^{16}$, $\frac{1}{625}a^{17}$, $\frac{1}{625}a^{18}$, $\frac{1}{625}a^{19}$, $\frac{1}{3125}a^{20}$, $\frac{1}{3125}a^{21}$, $\frac{1}{71875}a^{22}$, $\frac{1}{71875}a^{23}$, $\frac{1}{359375}a^{24}$, $\frac{1}{359375}a^{25}$, $\frac{1}{359375}a^{26}$, $\frac{1}{359375}a^{27}$, $\frac{1}{1796875}a^{28}$, $\frac{1}{1796875}a^{29}$, $\frac{1}{1796875}a^{30}$, $\frac{1}{1796875}a^{31}$, $\frac{1}{8984375}a^{32}$, $\frac{1}{8984375}a^{33}$, $\frac{1}{8984375}a^{34}$, $\frac{1}{8984375}a^{35}$, $\frac{1}{44921875}a^{36}$, $\frac{1}{44921875}a^{37}$, $\frac{1}{44921875}a^{38}$, $\frac{1}{44921875}a^{39}$, $\frac{1}{31220703125}a^{40}+\frac{38}{6244140625}a^{38}+\frac{57}{6244140625}a^{36}+\frac{37}{1248828125}a^{34}+\frac{56}{1248828125}a^{32}-\frac{8}{249765625}a^{30}-\frac{66}{249765625}a^{28}-\frac{18}{49953125}a^{26}-\frac{64}{49953125}a^{24}-\frac{12}{9990625}a^{22}-\frac{13}{86875}a^{20}-\frac{61}{86875}a^{18}-\frac{4}{86875}a^{16}-\frac{61}{17375}a^{14}+\frac{11}{17375}a^{12}-\frac{7}{3475}a^{10}-\frac{14}{3475}a^{8}+\frac{54}{695}a^{6}+\frac{12}{695}a^{4}+\frac{29}{139}a^{2}+\frac{50}{139}$, $\frac{1}{31220703125}a^{41}+\frac{38}{6244140625}a^{39}+\frac{57}{6244140625}a^{37}+\frac{37}{1248828125}a^{35}+\frac{56}{1248828125}a^{33}-\frac{8}{249765625}a^{31}-\frac{66}{249765625}a^{29}-\frac{18}{49953125}a^{27}-\frac{64}{49953125}a^{25}-\frac{12}{9990625}a^{23}-\frac{13}{86875}a^{21}-\frac{61}{86875}a^{19}-\frac{4}{86875}a^{17}-\frac{61}{17375}a^{15}+\frac{11}{17375}a^{13}-\frac{7}{3475}a^{11}-\frac{14}{3475}a^{9}+\frac{54}{695}a^{7}+\frac{12}{695}a^{5}+\frac{29}{139}a^{3}+\frac{50}{139}a$, $\frac{1}{13\!\cdots\!25}a^{42}+\frac{16407932}{13\!\cdots\!25}a^{40}+\frac{18964067483}{26\!\cdots\!25}a^{38}-\frac{4118885674}{26\!\cdots\!25}a^{36}-\frac{926153448}{21\!\cdots\!25}a^{34}+\frac{601820938}{53\!\cdots\!25}a^{32}-\frac{11481904146}{10\!\cdots\!25}a^{30}-\frac{165538382}{46\!\cdots\!75}a^{28}+\frac{1237569152}{42\!\cdots\!25}a^{26}-\frac{17097125271}{21\!\cdots\!25}a^{24}-\frac{13064830797}{42\!\cdots\!25}a^{22}-\frac{17927745913}{185673107684375}a^{20}-\frac{12866578354}{37134621536875}a^{18}+\frac{1605500536}{37134621536875}a^{16}+\frac{23650877134}{7426924307375}a^{14}-\frac{4879993064}{1485384861475}a^{12}+\frac{8842120783}{1485384861475}a^{10}+\frac{4770921064}{297076972295}a^{8}+\frac{15693037004}{297076972295}a^{6}+\frac{22412302837}{297076972295}a^{4}+\frac{7428755939}{59415394459}a^{2}-\frac{17364519196}{59415394459}$, $\frac{1}{13\!\cdots\!25}a^{43}+\frac{16407932}{13\!\cdots\!25}a^{41}+\frac{18964067483}{26\!\cdots\!25}a^{39}-\frac{4118885674}{26\!\cdots\!25}a^{37}-\frac{926153448}{21\!\cdots\!25}a^{35}+\frac{601820938}{53\!\cdots\!25}a^{33}-\frac{11481904146}{10\!\cdots\!25}a^{31}-\frac{165538382}{46\!\cdots\!75}a^{29}+\frac{1237569152}{42\!\cdots\!25}a^{27}-\frac{17097125271}{21\!\cdots\!25}a^{25}-\frac{13064830797}{42\!\cdots\!25}a^{23}-\frac{17927745913}{185673107684375}a^{21}-\frac{12866578354}{37134621536875}a^{19}+\frac{1605500536}{37134621536875}a^{17}+\frac{23650877134}{7426924307375}a^{15}-\frac{4879993064}{1485384861475}a^{13}+\frac{8842120783}{1485384861475}a^{11}+\frac{4770921064}{297076972295}a^{9}+\frac{15693037004}{297076972295}a^{7}+\frac{22412302837}{297076972295}a^{5}+\frac{7428755939}{59415394459}a^{3}-\frac{17364519196}{59415394459}a$
| 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
| \(\Q(\sqrt{5}) \), 4.0.1058000.1, \(\Q(\zeta_{23})^+\), 22.22.83796671451884098775580820361328125.1 |
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 | $44$ | R | $44$ | ${\href{/padicField/11.11.0.1}{11} }^{4}$ | $44$ | $44$ | $22^{2}$ | R | $22^{2}$ | $22^{2}$ | $44$ | ${\href{/padicField/41.11.0.1}{11} }^{4}$ | $44$ | ${\href{/padicField/47.4.0.1}{4} }^{11}$ | $44$ | ${\href{/padicField/59.11.0.1}{11} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $44$ | $2$ | $22$ | $44$ | |||
|
\(5\)
| Deg $44$ | $4$ | $11$ | $33$ | |||
|
\(23\)
| Deg $44$ | $22$ | $2$ | $42$ |