Normalized defining polynomial
\( x^{38} - x^{37} - 188 x^{36} + 180 x^{35} + 15133 x^{34} - 12829 x^{33} - 695929 x^{32} + \cdots - 2836879451 \)
Invariants
Degree: | $38$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[38, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(290\!\cdots\!677\) \(\medspace = 3^{19}\cdot 191^{37}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(288.12\) | 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: | $3^{1/2}191^{37/38}\approx 288.1156178920137$ | ||
Ramified primes: | \(3\), \(191\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{573}) \) | ||
$\card{ \Gal(K/\Q) }$: | $38$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(573=3\cdot 191\) | ||
Dirichlet character group: | $\lbrace$$\chi_{573}(1,·)$, $\chi_{573}(388,·)$, $\chi_{573}(257,·)$, $\chi_{573}(136,·)$, $\chi_{573}(521,·)$, $\chi_{573}(11,·)$, $\chi_{573}(14,·)$, $\chi_{573}(275,·)$, $\chi_{573}(532,·)$, $\chi_{573}(196,·)$, $\chi_{573}(535,·)$, $\chi_{573}(25,·)$, $\chi_{573}(154,·)$, $\chi_{573}(155,·)$, $\chi_{573}(412,·)$, $\chi_{573}(413,·)$, $\chi_{573}(160,·)$, $\chi_{573}(161,·)$, $\chi_{573}(418,·)$, $\chi_{573}(419,·)$, $\chi_{573}(548,·)$, $\chi_{573}(38,·)$, $\chi_{573}(41,·)$, $\chi_{573}(298,·)$, $\chi_{573}(559,·)$, $\chi_{573}(562,·)$, $\chi_{573}(52,·)$, $\chi_{573}(437,·)$, $\chi_{573}(185,·)$, $\chi_{573}(572,·)$, $\chi_{573}(451,·)$, $\chi_{573}(452,·)$, $\chi_{573}(121,·)$, $\chi_{573}(350,·)$, $\chi_{573}(223,·)$, $\chi_{573}(316,·)$, $\chi_{573}(377,·)$, $\chi_{573}(122,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $\frac{1}{7}a^{25}-\frac{2}{7}a^{24}-\frac{2}{7}a^{23}+\frac{3}{7}a^{22}-\frac{2}{7}a^{21}+\frac{1}{7}a^{20}+\frac{3}{7}a^{19}-\frac{2}{7}a^{18}+\frac{1}{7}a^{17}+\frac{1}{7}a^{16}-\frac{1}{7}a^{15}+\frac{3}{7}a^{13}-\frac{1}{7}a^{12}-\frac{3}{7}a^{9}+\frac{1}{7}a^{8}+\frac{3}{7}a^{7}+\frac{3}{7}a^{6}-\frac{3}{7}a^{5}+\frac{1}{7}a^{4}+\frac{3}{7}a^{3}-\frac{3}{7}a^{2}+\frac{2}{7}a$, $\frac{1}{7}a^{26}+\frac{1}{7}a^{24}-\frac{1}{7}a^{23}-\frac{3}{7}a^{22}-\frac{3}{7}a^{21}-\frac{2}{7}a^{20}-\frac{3}{7}a^{19}-\frac{3}{7}a^{18}+\frac{3}{7}a^{17}+\frac{1}{7}a^{16}-\frac{2}{7}a^{15}+\frac{3}{7}a^{14}-\frac{2}{7}a^{13}-\frac{2}{7}a^{12}-\frac{3}{7}a^{10}+\frac{2}{7}a^{9}-\frac{2}{7}a^{8}+\frac{2}{7}a^{7}+\frac{3}{7}a^{6}+\frac{2}{7}a^{5}-\frac{2}{7}a^{4}+\frac{3}{7}a^{3}+\frac{3}{7}a^{2}-\frac{3}{7}a$, $\frac{1}{7}a^{27}+\frac{1}{7}a^{24}-\frac{1}{7}a^{23}+\frac{1}{7}a^{22}+\frac{3}{7}a^{20}+\frac{1}{7}a^{19}-\frac{2}{7}a^{18}-\frac{3}{7}a^{16}-\frac{3}{7}a^{15}-\frac{2}{7}a^{14}+\frac{2}{7}a^{13}+\frac{1}{7}a^{12}-\frac{3}{7}a^{11}+\frac{2}{7}a^{10}+\frac{1}{7}a^{9}+\frac{1}{7}a^{8}-\frac{1}{7}a^{6}+\frac{1}{7}a^{5}+\frac{2}{7}a^{4}-\frac{2}{7}a$, $\frac{1}{7}a^{28}+\frac{1}{7}a^{24}+\frac{3}{7}a^{23}-\frac{3}{7}a^{22}-\frac{2}{7}a^{21}+\frac{2}{7}a^{19}+\frac{2}{7}a^{18}+\frac{3}{7}a^{17}+\frac{3}{7}a^{16}-\frac{1}{7}a^{15}+\frac{2}{7}a^{14}-\frac{2}{7}a^{13}-\frac{2}{7}a^{12}+\frac{2}{7}a^{11}+\frac{1}{7}a^{10}-\frac{3}{7}a^{9}-\frac{1}{7}a^{8}+\frac{3}{7}a^{7}-\frac{2}{7}a^{6}-\frac{2}{7}a^{5}-\frac{1}{7}a^{4}-\frac{3}{7}a^{3}+\frac{1}{7}a^{2}-\frac{2}{7}a$, $\frac{1}{7}a^{29}-\frac{2}{7}a^{24}-\frac{1}{7}a^{23}+\frac{2}{7}a^{22}+\frac{2}{7}a^{21}+\frac{1}{7}a^{20}-\frac{1}{7}a^{19}-\frac{2}{7}a^{18}+\frac{2}{7}a^{17}-\frac{2}{7}a^{16}+\frac{3}{7}a^{15}-\frac{2}{7}a^{14}+\frac{2}{7}a^{13}+\frac{3}{7}a^{12}+\frac{1}{7}a^{11}-\frac{3}{7}a^{10}+\frac{2}{7}a^{9}+\frac{2}{7}a^{8}+\frac{2}{7}a^{7}+\frac{2}{7}a^{6}+\frac{2}{7}a^{5}+\frac{3}{7}a^{4}-\frac{2}{7}a^{3}+\frac{1}{7}a^{2}-\frac{2}{7}a$, $\frac{1}{763}a^{30}-\frac{29}{763}a^{29}+\frac{22}{763}a^{28}+\frac{4}{763}a^{27}-\frac{9}{763}a^{26}+\frac{31}{763}a^{25}-\frac{307}{763}a^{24}+\frac{323}{763}a^{23}-\frac{223}{763}a^{22}+\frac{54}{109}a^{21}+\frac{110}{763}a^{20}+\frac{292}{763}a^{19}+\frac{57}{763}a^{18}+\frac{257}{763}a^{17}-\frac{358}{763}a^{16}+\frac{72}{763}a^{15}-\frac{253}{763}a^{14}+\frac{215}{763}a^{13}-\frac{99}{763}a^{12}-\frac{50}{109}a^{11}-\frac{22}{763}a^{10}+\frac{360}{763}a^{9}-\frac{44}{763}a^{8}+\frac{40}{109}a^{7}-\frac{151}{763}a^{6}+\frac{264}{763}a^{5}+\frac{326}{763}a^{4}-\frac{355}{763}a^{3}+\frac{376}{763}a^{2}-\frac{349}{763}a-\frac{8}{109}$, $\frac{1}{763}a^{31}+\frac{53}{763}a^{29}-\frac{12}{763}a^{28}-\frac{2}{763}a^{27}-\frac{12}{763}a^{26}+\frac{47}{763}a^{25}+\frac{20}{109}a^{24}-\frac{339}{763}a^{23}-\frac{29}{109}a^{22}+\frac{9}{109}a^{21}-\frac{6}{763}a^{20}+\frac{132}{763}a^{19}+\frac{275}{763}a^{18}+\frac{17}{109}a^{17}-\frac{282}{763}a^{16}+\frac{200}{763}a^{15}-\frac{146}{763}a^{14}+\frac{32}{763}a^{13}-\frac{60}{763}a^{12}-\frac{362}{763}a^{11}+\frac{158}{763}a^{10}+\frac{41}{763}a^{9}+\frac{312}{763}a^{8}-\frac{45}{109}a^{7}+\frac{354}{763}a^{6}+\frac{25}{763}a^{5}-\frac{275}{763}a^{4}+\frac{13}{109}a^{2}+\frac{69}{763}a-\frac{14}{109}$, $\frac{1}{763}a^{32}-\frac{1}{763}a^{29}+\frac{31}{763}a^{28}-\frac{6}{763}a^{27}-\frac{3}{109}a^{26}+\frac{23}{763}a^{25}+\frac{18}{763}a^{24}+\frac{48}{109}a^{23}+\frac{219}{763}a^{22}-\frac{202}{763}a^{21}-\frac{139}{763}a^{20}-\frac{268}{763}a^{19}-\frac{68}{763}a^{18}+\frac{267}{763}a^{17}+\frac{208}{763}a^{16}-\frac{21}{109}a^{15}+\frac{34}{763}a^{14}-\frac{17}{109}a^{13}-\frac{20}{763}a^{12}-\frac{149}{763}a^{11}-\frac{101}{763}a^{10}-\frac{347}{763}a^{9}-\frac{163}{763}a^{8}+\frac{229}{763}a^{7}-\frac{38}{763}a^{6}+\frac{12}{763}a^{5}-\frac{165}{763}a^{4}-\frac{60}{763}a^{3}+\frac{306}{763}a^{2}-\frac{349}{763}a-\frac{12}{109}$, $\frac{1}{763}a^{33}+\frac{2}{763}a^{29}+\frac{16}{763}a^{28}-\frac{17}{763}a^{27}+\frac{2}{109}a^{26}+\frac{7}{109}a^{25}+\frac{29}{763}a^{24}-\frac{221}{763}a^{23}+\frac{338}{763}a^{22}+\frac{239}{763}a^{21}-\frac{158}{763}a^{20}+\frac{32}{109}a^{19}+\frac{324}{763}a^{18}-\frac{298}{763}a^{17}+\frac{258}{763}a^{16}+\frac{106}{763}a^{15}-\frac{372}{763}a^{14}+\frac{195}{763}a^{13}-\frac{248}{763}a^{12}+\frac{312}{763}a^{11}-\frac{369}{763}a^{10}+\frac{197}{763}a^{9}+\frac{185}{763}a^{8}+\frac{242}{763}a^{7}-\frac{139}{763}a^{6}+\frac{99}{763}a^{5}+\frac{38}{109}a^{4}-\frac{7}{109}a^{3}+\frac{27}{763}a^{2}+\frac{330}{763}a-\frac{8}{109}$, $\frac{1}{763}a^{34}-\frac{5}{109}a^{29}+\frac{48}{763}a^{28}+\frac{6}{763}a^{27}-\frac{6}{109}a^{26}-\frac{33}{763}a^{25}-\frac{152}{763}a^{24}+\frac{237}{763}a^{23}-\frac{296}{763}a^{22}-\frac{260}{763}a^{21}+\frac{113}{763}a^{20}-\frac{369}{763}a^{19}+\frac{351}{763}a^{18}+\frac{289}{763}a^{17}-\frac{268}{763}a^{16}+\frac{29}{763}a^{15}+\frac{47}{763}a^{14}-\frac{19}{109}a^{13}+\frac{183}{763}a^{12}-\frac{323}{763}a^{11}+\frac{241}{763}a^{10}+\frac{228}{763}a^{9}+\frac{221}{763}a^{8}-\frac{45}{763}a^{7}-\frac{362}{763}a^{6}-\frac{153}{763}a^{5}-\frac{156}{763}a^{4}+\frac{43}{109}a^{3}+\frac{2}{109}a^{2}+\frac{206}{763}a+\frac{16}{109}$, $\frac{1}{763}a^{35}+\frac{2}{109}a^{29}+\frac{13}{763}a^{28}-\frac{11}{763}a^{27}-\frac{3}{109}a^{26}-\frac{48}{763}a^{25}-\frac{53}{109}a^{24}+\frac{3}{7}a^{23}-\frac{31}{109}a^{22}+\frac{263}{763}a^{21}+\frac{211}{763}a^{20}+\frac{216}{763}a^{19}-\frac{5}{763}a^{18}+\frac{1}{109}a^{17}-\frac{293}{763}a^{16}+\frac{60}{763}a^{15}+\frac{24}{109}a^{14}-\frac{249}{763}a^{13}+\frac{136}{763}a^{12}-\frac{19}{763}a^{11}-\frac{106}{763}a^{10}-\frac{41}{763}a^{9}+\frac{159}{763}a^{8}-\frac{45}{763}a^{7}+\frac{12}{763}a^{6}+\frac{37}{763}a^{5}-\frac{170}{763}a^{4}-\frac{312}{763}a^{3}-\frac{41}{763}a^{2}-\frac{4}{763}a+\frac{47}{109}$, $\frac{1}{1370639588843}a^{36}+\frac{833090371}{1370639588843}a^{35}-\frac{556156977}{1370639588843}a^{34}+\frac{704988761}{1370639588843}a^{33}+\frac{648807200}{1370639588843}a^{32}+\frac{750279711}{1370639588843}a^{31}-\frac{712377978}{1370639588843}a^{30}+\frac{7689553381}{1370639588843}a^{29}-\frac{83682178585}{1370639588843}a^{28}-\frac{88840402270}{1370639588843}a^{27}+\frac{504445485}{195805655549}a^{26}+\frac{85862574660}{1370639588843}a^{25}+\frac{381333541629}{1370639588843}a^{24}+\frac{802953904}{1370639588843}a^{23}+\frac{204823429009}{1370639588843}a^{22}-\frac{149150188562}{1370639588843}a^{21}+\frac{74670012202}{195805655549}a^{20}+\frac{7003252694}{195805655549}a^{19}-\frac{604737816362}{1370639588843}a^{18}-\frac{345010856341}{1370639588843}a^{17}+\frac{426709361642}{1370639588843}a^{16}-\frac{639083872715}{1370639588843}a^{15}+\frac{91079744481}{195805655549}a^{14}+\frac{479531287259}{1370639588843}a^{13}+\frac{217261920087}{1370639588843}a^{12}+\frac{5196343346}{195805655549}a^{11}+\frac{613471234725}{1370639588843}a^{10}+\frac{610227836935}{1370639588843}a^{9}-\frac{111995596197}{1370639588843}a^{8}-\frac{540066546438}{1370639588843}a^{7}-\frac{394756406594}{1370639588843}a^{6}-\frac{92916546647}{1370639588843}a^{5}+\frac{518409606985}{1370639588843}a^{4}-\frac{469238922909}{1370639588843}a^{3}+\frac{239234853605}{1370639588843}a^{2}-\frac{79324356280}{195805655549}a-\frac{5044819}{66442367}$, $\frac{1}{67\!\cdots\!89}a^{37}+\frac{35\!\cdots\!50}{96\!\cdots\!27}a^{36}-\frac{43\!\cdots\!66}{67\!\cdots\!89}a^{35}-\frac{27\!\cdots\!45}{67\!\cdots\!89}a^{34}-\frac{19\!\cdots\!61}{67\!\cdots\!89}a^{33}+\frac{45\!\cdots\!26}{96\!\cdots\!27}a^{32}-\frac{32\!\cdots\!54}{67\!\cdots\!89}a^{31}-\frac{25\!\cdots\!16}{67\!\cdots\!89}a^{30}-\frac{24\!\cdots\!01}{67\!\cdots\!89}a^{29}-\frac{40\!\cdots\!39}{67\!\cdots\!89}a^{28}+\frac{28\!\cdots\!79}{67\!\cdots\!89}a^{27}+\frac{21\!\cdots\!72}{67\!\cdots\!89}a^{26}-\frac{16\!\cdots\!04}{67\!\cdots\!89}a^{25}-\frac{76\!\cdots\!77}{67\!\cdots\!89}a^{24}-\frac{58\!\cdots\!70}{67\!\cdots\!89}a^{23}-\frac{12\!\cdots\!98}{96\!\cdots\!27}a^{22}-\frac{17\!\cdots\!40}{67\!\cdots\!89}a^{21}-\frac{16\!\cdots\!74}{13\!\cdots\!61}a^{20}-\frac{21\!\cdots\!40}{67\!\cdots\!89}a^{19}+\frac{30\!\cdots\!80}{67\!\cdots\!89}a^{18}+\frac{91\!\cdots\!80}{67\!\cdots\!89}a^{17}+\frac{68\!\cdots\!77}{67\!\cdots\!89}a^{16}-\frac{26\!\cdots\!58}{67\!\cdots\!89}a^{15}+\frac{23\!\cdots\!00}{67\!\cdots\!89}a^{14}+\frac{16\!\cdots\!34}{67\!\cdots\!89}a^{13}+\frac{14\!\cdots\!42}{67\!\cdots\!89}a^{12}-\frac{60\!\cdots\!03}{67\!\cdots\!89}a^{11}+\frac{28\!\cdots\!71}{67\!\cdots\!89}a^{10}-\frac{24\!\cdots\!92}{67\!\cdots\!89}a^{9}+\frac{18\!\cdots\!32}{67\!\cdots\!89}a^{8}+\frac{13\!\cdots\!30}{67\!\cdots\!89}a^{7}-\frac{37\!\cdots\!72}{67\!\cdots\!89}a^{6}-\frac{28\!\cdots\!87}{67\!\cdots\!89}a^{5}-\frac{26\!\cdots\!18}{67\!\cdots\!89}a^{4}-\frac{43\!\cdots\!73}{96\!\cdots\!27}a^{3}+\frac{13\!\cdots\!88}{67\!\cdots\!89}a^{2}+\frac{36\!\cdots\!67}{13\!\cdots\!61}a-\frac{46\!\cdots\!99}{32\!\cdots\!41}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
not computed
Unit group
Rank: | $37$ | 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 38 |
The 38 conjugacy class representatives for $C_{38}$ |
Character table for $C_{38}$ is not computed |
Intermediate fields
\(\Q(\sqrt{573}) \), 19.19.114445997944945591651333831028437092270721.1 |
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 | $38$ | R | $38$ | ${\href{/padicField/7.2.0.1}{2} }^{19}$ | $19^{2}$ | $19^{2}$ | $38$ | $38$ | $38$ | $19^{2}$ | $38$ | $38$ | $19^{2}$ | $19^{2}$ | $19^{2}$ | $19^{2}$ | $38$ |
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 $38$ | $2$ | $19$ | $19$ | |||
\(191\) | Deg $38$ | $38$ | $1$ | $37$ |