Normalized defining polynomial
\( x^{45} - 9 x^{44} - 117 x^{43} + 1356 x^{42} + 4455 x^{41} - 86382 x^{40} + 4596 x^{39} + \cdots - 155181097 \)
Invariants
Degree: | $45$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[45, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(148\!\cdots\!969\) \(\medspace = 3^{110}\cdot 31^{36}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(228.76\) | 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: | $3^{22/9}31^{4/5}\approx 228.76298537732032$ | ||
Ramified primes: | \(3\), \(31\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $45$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(837=3^{3}\cdot 31\) | ||
Dirichlet character group: | $\lbrace$$\chi_{837}(256,·)$, $\chi_{837}(1,·)$, $\chi_{837}(4,·)$, $\chi_{837}(652,·)$, $\chi_{837}(655,·)$, $\chi_{837}(16,·)$, $\chi_{837}(529,·)$, $\chi_{837}(535,·)$, $\chi_{837}(280,·)$, $\chi_{837}(388,·)$, $\chi_{837}(283,·)$, $\chi_{837}(157,·)$, $\chi_{837}(376,·)$, $\chi_{837}(667,·)$, $\chi_{837}(295,·)$, $\chi_{837}(808,·)$, $\chi_{837}(814,·)$, $\chi_{837}(559,·)$, $\chi_{837}(562,·)$, $\chi_{837}(436,·)$, $\chi_{837}(442,·)$, $\chi_{837}(187,·)$, $\chi_{837}(574,·)$, $\chi_{837}(373,·)$, $\chi_{837}(64,·)$, $\chi_{837}(70,·)$, $\chi_{837}(97,·)$, $\chi_{837}(202,·)$, $\chi_{837}(715,·)$, $\chi_{837}(721,·)$, $\chi_{837}(466,·)$, $\chi_{837}(163,·)$, $\chi_{837}(469,·)$, $\chi_{837}(343,·)$, $\chi_{837}(349,·)$, $\chi_{837}(94,·)$, $\chi_{837}(481,·)$, $\chi_{837}(745,·)$, $\chi_{837}(748,·)$, $\chi_{837}(109,·)$, $\chi_{837}(622,·)$, $\chi_{837}(628,·)$, $\chi_{837}(190,·)$, $\chi_{837}(760,·)$, $\chi_{837}(250,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $\frac{1}{5}a^{36}-\frac{2}{5}a^{35}-\frac{1}{5}a^{34}-\frac{2}{5}a^{33}+\frac{1}{5}a^{32}-\frac{2}{5}a^{31}-\frac{1}{5}a^{30}+\frac{2}{5}a^{29}-\frac{1}{5}a^{28}+\frac{2}{5}a^{27}+\frac{1}{5}a^{26}-\frac{1}{5}a^{25}-\frac{1}{5}a^{24}-\frac{2}{5}a^{23}+\frac{1}{5}a^{22}+\frac{2}{5}a^{21}+\frac{2}{5}a^{20}+\frac{2}{5}a^{19}+\frac{1}{5}a^{18}+\frac{1}{5}a^{17}-\frac{1}{5}a^{16}+\frac{2}{5}a^{15}-\frac{1}{5}a^{14}-\frac{2}{5}a^{13}-\frac{2}{5}a^{12}+\frac{2}{5}a^{11}+\frac{2}{5}a^{10}-\frac{1}{5}a^{9}+\frac{1}{5}a^{8}+\frac{2}{5}a^{7}+\frac{1}{5}a^{5}-\frac{2}{5}a^{4}+\frac{2}{5}a^{2}+\frac{1}{5}a+\frac{1}{5}$, $\frac{1}{5}a^{37}+\frac{1}{5}a^{34}+\frac{2}{5}a^{33}-\frac{2}{5}a^{29}+\frac{1}{5}a^{26}+\frac{2}{5}a^{25}+\frac{1}{5}a^{24}+\frac{2}{5}a^{23}-\frac{1}{5}a^{22}+\frac{1}{5}a^{21}+\frac{1}{5}a^{20}-\frac{2}{5}a^{18}+\frac{1}{5}a^{17}-\frac{2}{5}a^{15}+\frac{1}{5}a^{14}-\frac{1}{5}a^{13}-\frac{2}{5}a^{12}+\frac{1}{5}a^{11}-\frac{2}{5}a^{10}-\frac{1}{5}a^{9}-\frac{1}{5}a^{8}-\frac{1}{5}a^{7}+\frac{1}{5}a^{6}+\frac{1}{5}a^{4}+\frac{2}{5}a^{3}-\frac{2}{5}a+\frac{2}{5}$, $\frac{1}{5}a^{38}+\frac{1}{5}a^{35}+\frac{2}{5}a^{34}-\frac{2}{5}a^{30}+\frac{1}{5}a^{27}+\frac{2}{5}a^{26}+\frac{1}{5}a^{25}+\frac{2}{5}a^{24}-\frac{1}{5}a^{23}+\frac{1}{5}a^{22}+\frac{1}{5}a^{21}-\frac{2}{5}a^{19}+\frac{1}{5}a^{18}-\frac{2}{5}a^{16}+\frac{1}{5}a^{15}-\frac{1}{5}a^{14}-\frac{2}{5}a^{13}+\frac{1}{5}a^{12}-\frac{2}{5}a^{11}-\frac{1}{5}a^{10}-\frac{1}{5}a^{9}-\frac{1}{5}a^{8}+\frac{1}{5}a^{7}+\frac{1}{5}a^{5}+\frac{2}{5}a^{4}-\frac{2}{5}a^{2}+\frac{2}{5}a$, $\frac{1}{5}a^{39}-\frac{1}{5}a^{35}+\frac{1}{5}a^{34}+\frac{2}{5}a^{33}-\frac{1}{5}a^{32}+\frac{1}{5}a^{30}-\frac{2}{5}a^{29}+\frac{2}{5}a^{28}-\frac{2}{5}a^{25}-\frac{2}{5}a^{23}-\frac{2}{5}a^{21}+\frac{1}{5}a^{20}-\frac{1}{5}a^{19}-\frac{1}{5}a^{18}+\frac{2}{5}a^{17}+\frac{2}{5}a^{16}+\frac{2}{5}a^{15}-\frac{1}{5}a^{14}-\frac{2}{5}a^{13}+\frac{2}{5}a^{11}+\frac{2}{5}a^{10}-\frac{2}{5}a^{7}+\frac{1}{5}a^{6}+\frac{1}{5}a^{5}+\frac{2}{5}a^{4}-\frac{2}{5}a^{3}-\frac{1}{5}a-\frac{1}{5}$, $\frac{1}{5}a^{40}-\frac{1}{5}a^{35}+\frac{1}{5}a^{34}+\frac{2}{5}a^{33}+\frac{1}{5}a^{32}-\frac{1}{5}a^{31}+\frac{2}{5}a^{30}-\frac{1}{5}a^{29}-\frac{1}{5}a^{28}+\frac{2}{5}a^{27}-\frac{1}{5}a^{26}-\frac{1}{5}a^{25}+\frac{2}{5}a^{24}-\frac{2}{5}a^{23}-\frac{1}{5}a^{22}-\frac{2}{5}a^{21}+\frac{1}{5}a^{20}+\frac{1}{5}a^{19}-\frac{2}{5}a^{18}-\frac{2}{5}a^{17}+\frac{1}{5}a^{16}+\frac{1}{5}a^{15}+\frac{2}{5}a^{14}-\frac{2}{5}a^{13}-\frac{1}{5}a^{11}+\frac{2}{5}a^{10}-\frac{1}{5}a^{9}-\frac{1}{5}a^{8}-\frac{2}{5}a^{7}+\frac{1}{5}a^{6}-\frac{2}{5}a^{5}+\frac{1}{5}a^{4}+\frac{1}{5}a^{2}+\frac{1}{5}$, $\frac{1}{5}a^{41}-\frac{1}{5}a^{35}+\frac{1}{5}a^{34}-\frac{1}{5}a^{33}-\frac{2}{5}a^{30}+\frac{1}{5}a^{29}+\frac{1}{5}a^{28}+\frac{1}{5}a^{27}+\frac{1}{5}a^{25}+\frac{2}{5}a^{24}+\frac{2}{5}a^{23}-\frac{1}{5}a^{22}-\frac{2}{5}a^{21}-\frac{2}{5}a^{20}-\frac{1}{5}a^{18}+\frac{2}{5}a^{17}-\frac{1}{5}a^{15}+\frac{2}{5}a^{14}-\frac{2}{5}a^{13}+\frac{2}{5}a^{12}-\frac{1}{5}a^{11}+\frac{1}{5}a^{10}-\frac{2}{5}a^{9}-\frac{1}{5}a^{8}-\frac{2}{5}a^{7}-\frac{2}{5}a^{6}+\frac{2}{5}a^{5}-\frac{2}{5}a^{4}+\frac{1}{5}a^{3}+\frac{2}{5}a^{2}+\frac{2}{5}a+\frac{1}{5}$, $\frac{1}{2536915}a^{42}-\frac{225354}{2536915}a^{41}-\frac{250386}{2536915}a^{40}+\frac{178183}{2536915}a^{39}-\frac{116036}{2536915}a^{38}-\frac{2993}{507383}a^{37}-\frac{79711}{2536915}a^{36}+\frac{467062}{2536915}a^{35}+\frac{125147}{507383}a^{34}-\frac{211527}{2536915}a^{33}+\frac{502301}{2536915}a^{32}-\frac{920006}{2536915}a^{31}-\frac{918368}{2536915}a^{30}-\frac{384288}{2536915}a^{29}+\frac{15494}{2536915}a^{28}+\frac{216943}{2536915}a^{27}+\frac{78881}{507383}a^{26}-\frac{866433}{2536915}a^{25}-\frac{56657}{507383}a^{24}-\frac{1218792}{2536915}a^{23}-\frac{870953}{2536915}a^{22}-\frac{1205484}{2536915}a^{21}-\frac{234942}{507383}a^{20}+\frac{702932}{2536915}a^{19}+\frac{690114}{2536915}a^{18}+\frac{77003}{507383}a^{17}+\frac{614031}{2536915}a^{16}-\frac{31239}{507383}a^{15}-\frac{1119864}{2536915}a^{14}+\frac{808968}{2536915}a^{13}+\frac{245050}{507383}a^{12}-\frac{197956}{2536915}a^{11}-\frac{518826}{2536915}a^{10}-\frac{1002586}{2536915}a^{9}+\frac{322329}{2536915}a^{8}+\frac{651611}{2536915}a^{7}+\frac{416507}{2536915}a^{6}+\frac{906634}{2536915}a^{5}-\frac{1153298}{2536915}a^{4}+\frac{475007}{2536915}a^{3}-\frac{222221}{507383}a^{2}+\frac{29253}{2536915}a+\frac{428487}{2536915}$, $\frac{1}{947991860285}a^{43}-\frac{143333}{947991860285}a^{42}+\frac{50939682921}{947991860285}a^{41}+\frac{45754886559}{947991860285}a^{40}-\frac{4761287697}{189598372057}a^{39}-\frac{66891765978}{947991860285}a^{38}-\frac{57478540271}{947991860285}a^{37}-\frac{9603924160}{189598372057}a^{36}-\frac{347515677589}{947991860285}a^{35}-\frac{268974497034}{947991860285}a^{34}-\frac{10976359285}{189598372057}a^{33}+\frac{122395319188}{947991860285}a^{32}+\frac{261951501421}{947991860285}a^{31}-\frac{247909636551}{947991860285}a^{30}-\frac{356413749093}{947991860285}a^{29}+\frac{359930960123}{947991860285}a^{28}-\frac{196378684071}{947991860285}a^{27}-\frac{91691532787}{189598372057}a^{26}+\frac{326833404499}{947991860285}a^{25}-\frac{102047831784}{947991860285}a^{24}+\frac{40883125981}{947991860285}a^{23}+\frac{236702650382}{947991860285}a^{22}-\frac{3545778507}{189598372057}a^{21}+\frac{366817937253}{947991860285}a^{20}+\frac{351675191292}{947991860285}a^{19}+\frac{330739708628}{947991860285}a^{18}+\frac{191702900784}{947991860285}a^{17}+\frac{334406938899}{947991860285}a^{16}-\frac{433120349031}{947991860285}a^{15}-\frac{76805408392}{189598372057}a^{14}+\frac{212291147102}{947991860285}a^{13}-\frac{168409854344}{947991860285}a^{12}+\frac{185528855878}{947991860285}a^{11}+\frac{279301253969}{947991860285}a^{10}-\frac{115812405683}{947991860285}a^{9}-\frac{16574487922}{947991860285}a^{8}-\frac{261231212113}{947991860285}a^{7}+\frac{450286291369}{947991860285}a^{6}+\frac{335981740774}{947991860285}a^{5}-\frac{441713663722}{947991860285}a^{4}+\frac{273966559067}{947991860285}a^{3}+\frac{82019160309}{947991860285}a^{2}+\frac{4258332729}{947991860285}a-\frac{382130398654}{947991860285}$, $\frac{1}{21\!\cdots\!45}a^{44}-\frac{61\!\cdots\!09}{43\!\cdots\!69}a^{43}-\frac{41\!\cdots\!98}{21\!\cdots\!45}a^{42}-\frac{74\!\cdots\!88}{21\!\cdots\!45}a^{41}-\frac{53\!\cdots\!64}{21\!\cdots\!45}a^{40}-\frac{33\!\cdots\!61}{43\!\cdots\!69}a^{39}+\frac{77\!\cdots\!63}{21\!\cdots\!45}a^{38}+\frac{11\!\cdots\!62}{21\!\cdots\!45}a^{37}+\frac{81\!\cdots\!01}{21\!\cdots\!45}a^{36}-\frac{54\!\cdots\!00}{43\!\cdots\!69}a^{35}-\frac{98\!\cdots\!24}{21\!\cdots\!45}a^{34}-\frac{94\!\cdots\!48}{21\!\cdots\!45}a^{33}+\frac{10\!\cdots\!76}{21\!\cdots\!45}a^{32}+\frac{69\!\cdots\!28}{21\!\cdots\!45}a^{31}+\frac{51\!\cdots\!99}{21\!\cdots\!45}a^{30}+\frac{35\!\cdots\!98}{50\!\cdots\!65}a^{29}-\frac{12\!\cdots\!36}{43\!\cdots\!69}a^{28}+\frac{21\!\cdots\!68}{21\!\cdots\!45}a^{27}-\frac{47\!\cdots\!44}{21\!\cdots\!45}a^{26}+\frac{61\!\cdots\!91}{21\!\cdots\!45}a^{25}-\frac{87\!\cdots\!52}{21\!\cdots\!45}a^{24}-\frac{44\!\cdots\!27}{21\!\cdots\!45}a^{23}-\frac{21\!\cdots\!42}{43\!\cdots\!69}a^{22}-\frac{48\!\cdots\!78}{21\!\cdots\!45}a^{21}-\frac{49\!\cdots\!12}{21\!\cdots\!45}a^{20}-\frac{77\!\cdots\!46}{21\!\cdots\!45}a^{19}+\frac{73\!\cdots\!40}{43\!\cdots\!69}a^{18}+\frac{35\!\cdots\!74}{21\!\cdots\!45}a^{17}+\frac{12\!\cdots\!59}{43\!\cdots\!69}a^{16}-\frac{38\!\cdots\!68}{43\!\cdots\!69}a^{15}-\frac{10\!\cdots\!16}{21\!\cdots\!45}a^{14}-\frac{55\!\cdots\!78}{21\!\cdots\!45}a^{13}-\frac{39\!\cdots\!21}{21\!\cdots\!45}a^{12}+\frac{61\!\cdots\!39}{21\!\cdots\!45}a^{11}+\frac{21\!\cdots\!26}{43\!\cdots\!69}a^{10}+\frac{56\!\cdots\!02}{21\!\cdots\!45}a^{9}-\frac{67\!\cdots\!31}{21\!\cdots\!45}a^{8}+\frac{80\!\cdots\!29}{21\!\cdots\!45}a^{7}+\frac{48\!\cdots\!93}{21\!\cdots\!45}a^{6}+\frac{72\!\cdots\!03}{21\!\cdots\!45}a^{5}-\frac{36\!\cdots\!58}{21\!\cdots\!45}a^{4}-\frac{48\!\cdots\!56}{21\!\cdots\!45}a^{3}-\frac{56\!\cdots\!41}{21\!\cdots\!45}a^{2}-\frac{63\!\cdots\!94}{21\!\cdots\!45}a+\frac{66\!\cdots\!14}{21\!\cdots\!45}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
not computed
Unit group
Rank: | $44$ | 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 45 |
The 45 conjugacy class representatives for $C_{45}$ |
Character table for $C_{45}$ is not computed |
Intermediate fields
\(\Q(\zeta_{9})^+\), 5.5.923521.1, \(\Q(\zeta_{27})^+\), 15.15.2746410307762150989067078161.1 |
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 | $45$ | R | ${\href{/padicField/5.9.0.1}{9} }^{5}$ | $45$ | $45$ | $45$ | $15^{3}$ | $15^{3}$ | $45$ | $45$ | R | ${\href{/padicField/37.3.0.1}{3} }^{15}$ | $45$ | $45$ | $45$ | ${\href{/padicField/53.5.0.1}{5} }^{9}$ | $45$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | Deg $45$ | $9$ | $5$ | $110$ | |||
\(31\) | Deg $45$ | $5$ | $9$ | $36$ |