Normalized defining polynomial
\( x^{38} + 217 x^{36} + 20630 x^{34} + 1139565 x^{32} + 40850264 x^{30} + 1004488522 x^{28} + \cdots + 63175314409 \)
Invariants
| Degree: | $38$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 19]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-247\!\cdots\!624\)
\(\medspace = -\,2^{38}\cdot 229^{36}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(344.08\) | 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: | $2\cdot 229^{18/19}\approx 344.0833124846288$ | ||
| Ramified primes: |
\(2\), \(229\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $38$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(916=2^{2}\cdot 229\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{916}(1,·)$, $\chi_{916}(515,·)$, $\chi_{916}(901,·)$, $\chi_{916}(519,·)$, $\chi_{916}(905,·)$, $\chi_{916}(271,·)$, $\chi_{916}(273,·)$, $\chi_{916}(661,·)$, $\chi_{916}(791,·)$, $\chi_{916}(27,·)$, $\chi_{916}(289,·)$, $\chi_{916}(731,·)$, $\chi_{916}(165,·)$, $\chi_{916}(43,·)$, $\chi_{916}(619,·)$, $\chi_{916}(53,·)$, $\chi_{916}(57,·)$, $\chi_{916}(683,·)$, $\chi_{916}(443,·)$, $\chi_{916}(61,·)$, $\chi_{916}(703,·)$, $\chi_{916}(579,·)$, $\chi_{916}(161,·)$, $\chi_{916}(203,·)$, $\chi_{916}(333,·)$, $\chi_{916}(729,·)$, $\chi_{916}(475,·)$, $\chi_{916}(459,·)$, $\chi_{916}(225,·)$, $\chi_{916}(485,·)$, $\chi_{916}(17,·)$, $\chi_{916}(747,·)$, $\chi_{916}(623,·)$, $\chi_{916}(245,·)$, $\chi_{916}(501,·)$, $\chi_{916}(121,·)$, $\chi_{916}(447,·)$, $\chi_{916}(511,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{262144}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $\frac{1}{89}a^{32}+\frac{14}{89}a^{30}-\frac{20}{89}a^{28}-\frac{3}{89}a^{26}+\frac{40}{89}a^{24}-\frac{17}{89}a^{22}-\frac{33}{89}a^{20}-\frac{36}{89}a^{18}-\frac{7}{89}a^{16}+\frac{43}{89}a^{14}-\frac{40}{89}a^{12}+\frac{37}{89}a^{8}+\frac{7}{89}a^{6}-\frac{14}{89}a^{4}+\frac{33}{89}a^{2}+\frac{25}{89}$, $\frac{1}{89}a^{33}+\frac{14}{89}a^{31}-\frac{20}{89}a^{29}-\frac{3}{89}a^{27}+\frac{40}{89}a^{25}-\frac{17}{89}a^{23}-\frac{33}{89}a^{21}-\frac{36}{89}a^{19}-\frac{7}{89}a^{17}+\frac{43}{89}a^{15}-\frac{40}{89}a^{13}+\frac{37}{89}a^{9}+\frac{7}{89}a^{7}-\frac{14}{89}a^{5}+\frac{33}{89}a^{3}+\frac{25}{89}a$, $\frac{1}{9523}a^{34}-\frac{34}{9523}a^{32}-\frac{1582}{9523}a^{30}+\frac{957}{9523}a^{28}-\frac{3821}{9523}a^{26}-\frac{1225}{9523}a^{24}+\frac{2207}{9523}a^{22}-\frac{3703}{9523}a^{20}-\frac{1483}{9523}a^{18}-\frac{1846}{9523}a^{16}-\frac{2905}{9523}a^{14}-\frac{1373}{9523}a^{12}+\frac{3241}{9523}a^{10}+\frac{2147}{9523}a^{8}-\frac{3020}{9523}a^{6}-\frac{2677}{9523}a^{4}-\frac{2627}{9523}a^{2}-\frac{844}{9523}$, $\frac{1}{9523}a^{35}-\frac{34}{9523}a^{33}-\frac{1582}{9523}a^{31}+\frac{957}{9523}a^{29}-\frac{3821}{9523}a^{27}-\frac{1225}{9523}a^{25}+\frac{2207}{9523}a^{23}-\frac{3703}{9523}a^{21}-\frac{1483}{9523}a^{19}-\frac{1846}{9523}a^{17}-\frac{2905}{9523}a^{15}-\frac{1373}{9523}a^{13}+\frac{3241}{9523}a^{11}+\frac{2147}{9523}a^{9}-\frac{3020}{9523}a^{7}-\frac{2677}{9523}a^{5}-\frac{2627}{9523}a^{3}-\frac{844}{9523}a$, $\frac{1}{75\!\cdots\!61}a^{36}-\frac{38\!\cdots\!11}{75\!\cdots\!61}a^{34}+\frac{26\!\cdots\!42}{75\!\cdots\!61}a^{32}-\frac{57\!\cdots\!05}{75\!\cdots\!61}a^{30}-\frac{12\!\cdots\!19}{75\!\cdots\!61}a^{28}+\frac{32\!\cdots\!03}{75\!\cdots\!61}a^{26}-\frac{12\!\cdots\!94}{75\!\cdots\!61}a^{24}+\frac{21\!\cdots\!86}{75\!\cdots\!61}a^{22}+\frac{17\!\cdots\!55}{75\!\cdots\!61}a^{20}-\frac{15\!\cdots\!37}{75\!\cdots\!61}a^{18}-\frac{29\!\cdots\!45}{75\!\cdots\!61}a^{16}-\frac{35\!\cdots\!60}{75\!\cdots\!61}a^{14}-\frac{31\!\cdots\!13}{75\!\cdots\!61}a^{12}+\frac{20\!\cdots\!36}{75\!\cdots\!61}a^{10}-\frac{10\!\cdots\!35}{35\!\cdots\!51}a^{8}-\frac{12\!\cdots\!18}{75\!\cdots\!61}a^{6}+\frac{23\!\cdots\!60}{75\!\cdots\!61}a^{4}-\frac{13\!\cdots\!78}{75\!\cdots\!61}a^{2}+\frac{18\!\cdots\!95}{75\!\cdots\!61}$, $\frac{1}{19\!\cdots\!67}a^{37}-\frac{81\!\cdots\!50}{19\!\cdots\!67}a^{35}-\frac{90\!\cdots\!76}{19\!\cdots\!67}a^{33}+\frac{74\!\cdots\!10}{19\!\cdots\!67}a^{31}+\frac{14\!\cdots\!19}{19\!\cdots\!67}a^{29}+\frac{29\!\cdots\!69}{19\!\cdots\!67}a^{27}-\frac{31\!\cdots\!76}{19\!\cdots\!67}a^{25}-\frac{27\!\cdots\!07}{19\!\cdots\!67}a^{23}+\frac{29\!\cdots\!32}{19\!\cdots\!67}a^{21}+\frac{89\!\cdots\!45}{19\!\cdots\!67}a^{19}-\frac{66\!\cdots\!60}{19\!\cdots\!67}a^{17}+\frac{13\!\cdots\!67}{19\!\cdots\!67}a^{15}+\frac{46\!\cdots\!89}{19\!\cdots\!67}a^{13}-\frac{26\!\cdots\!12}{19\!\cdots\!67}a^{11}+\frac{36\!\cdots\!11}{90\!\cdots\!97}a^{9}-\frac{13\!\cdots\!63}{19\!\cdots\!67}a^{7}-\frac{50\!\cdots\!57}{19\!\cdots\!67}a^{5}+\frac{89\!\cdots\!26}{19\!\cdots\!67}a^{3}+\frac{28\!\cdots\!61}{19\!\cdots\!67}a$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
not computed
Unit group
| Rank: | $18$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( \frac{46853782422954686141310318886114072746648594869935230304454647360499}{19052220748576276413799550376002203682511875846412990539296689881526630523168121} a^{37} + \frac{10102903940206627179928491667851699239309302493691921290270561643050699}{19052220748576276413799550376002203682511875846412990539296689881526630523168121} a^{35} + \frac{952684298567841396756319184941891007496649998674236069267902472948848044}{19052220748576276413799550376002203682511875846412990539296689881526630523168121} a^{33} + \frac{52077626694473833444426839598140789924246953076424035076396477578113397700}{19052220748576276413799550376002203682511875846412990539296689881526630523168121} a^{31} + \frac{1841825638751966374225628181606262713569014609390387843644122893158878626841}{19052220748576276413799550376002203682511875846412990539296689881526630523168121} a^{29} + \frac{44499536288702500972615850996938924208131307616351973347101771302986618649278}{19052220748576276413799550376002203682511875846412990539296689881526630523168121} a^{27} + \frac{755448454933914896194666511672901134237132831665439804818357365517908583428128}{19052220748576276413799550376002203682511875846412990539296689881526630523168121} a^{25} + \frac{9125009419698612123561015336255739906827775913342632717884246581865869739081106}{19052220748576276413799550376002203682511875846412990539296689881526630523168121} a^{23} + \frac{78513685453925739833129376680572333352840594271757121981676531450371062426645173}{19052220748576276413799550376002203682511875846412990539296689881526630523168121} a^{21} + \frac{476980479607085392817731814354244134590642962517432869708243733732774133947973346}{19052220748576276413799550376002203682511875846412990539296689881526630523168121} a^{19} + \frac{2006030891513535782641029262726037259733931789463555186670389292555658031215174091}{19052220748576276413799550376002203682511875846412990539296689881526630523168121} a^{17} + \frac{5641143728876116641539119594736003119427623316532550911184793708540814780537165006}{19052220748576276413799550376002203682511875846412990539296689881526630523168121} a^{15} + \frac{9963706879387261492643178768554303946771254463496315310544123119913074069316207298}{19052220748576276413799550376002203682511875846412990539296689881526630523168121} a^{13} + \frac{9630868874321405441195591362779458488074860530562077842363133084406978808372540800}{19052220748576276413799550376002203682511875846412990539296689881526630523168121} a^{11} + \frac{2842565466296993744604253636963240295718673754662751284968626529020590984597092349}{19052220748576276413799550376002203682511875846412990539296689881526630523168121} a^{9} - \frac{2628562655427141506201746484443852928566687731499703754027868849551884958762778760}{19052220748576276413799550376002203682511875846412990539296689881526630523168121} a^{7} - \frac{2289640601488793385546579889388712714148240811025535459279385922916734928604242213}{19052220748576276413799550376002203682511875846412990539296689881526630523168121} a^{5} - \frac{653433706385450123329553053749072913703539620819701593952288274114946846345309872}{19052220748576276413799550376002203682511875846412990539296689881526630523168121} a^{3} - \frac{105363604973823752251054079132002238121554153292581720077418413643914048783263492}{19052220748576276413799550376002203682511875846412990539296689881526630523168121} a \)
(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
| A cyclic group of order 38 |
| The 38 conjugacy class representatives for $C_{38}$ |
| Character table for $C_{38}$ |
Intermediate fields
| \(\Q(\sqrt{-1}) \), 19.19.2999429662895796650415561622892044448017561.1 |
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 | $38$ | $19^{2}$ | $38$ | $38$ | $19^{2}$ | $19^{2}$ | $38$ | $38$ | $19^{2}$ | $38$ | $19^{2}$ | $19^{2}$ | $38$ | $38$ | $19^{2}$ | $38$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $38$ | $2$ | $19$ | $38$ | |||
|
\(229\)
| Deg $19$ | $19$ | $1$ | $18$ | |||
| Deg $19$ | $19$ | $1$ | $18$ |