Normalized defining polynomial
\( x^{38} + 684 x^{36} + 197828 x^{34} + 31840200 x^{32} + 3173576992 x^{30} + 207620468672 x^{28} + \cdots + 43\!\cdots\!12 \)
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: |
\(-321\!\cdots\!848\)
\(\medspace = -\,2^{57}\cdot 19^{73}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(809.28\) | 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: | $2^{3/2}19^{73/38}\approx 809.2784274164313$ | ||
| Ramified primes: |
\(2\), \(19\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-38}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $38$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(2888=2^{3}\cdot 19^{2}\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{2888}(1,·)$, $\chi_{2888}(645,·)$, $\chi_{2888}(2433,·)$, $\chi_{2888}(1673,·)$, $\chi_{2888}(2317,·)$, $\chi_{2888}(913,·)$, $\chi_{2888}(1557,·)$, $\chi_{2888}(2585,·)$, $\chi_{2888}(153,·)$, $\chi_{2888}(797,·)$, $\chi_{2888}(1825,·)$, $\chi_{2888}(37,·)$, $\chi_{2888}(305,·)$, $\chi_{2888}(1065,·)$, $\chi_{2888}(1709,·)$, $\chi_{2888}(2737,·)$, $\chi_{2888}(949,·)$, $\chi_{2888}(1977,·)$, $\chi_{2888}(2621,·)$, $\chi_{2888}(1217,·)$, $\chi_{2888}(1861,·)$, $\chi_{2888}(457,·)$, $\chi_{2888}(1101,·)$, $\chi_{2888}(2129,·)$, $\chi_{2888}(341,·)$, $\chi_{2888}(1369,·)$, $\chi_{2888}(2013,·)$, $\chi_{2888}(2469,·)$, $\chi_{2888}(609,·)$, $\chi_{2888}(1253,·)$, $\chi_{2888}(2281,·)$, $\chi_{2888}(493,·)$, $\chi_{2888}(189,·)$, $\chi_{2888}(1521,·)$, $\chi_{2888}(2165,·)$, $\chi_{2888}(761,·)$, $\chi_{2888}(1405,·)$, $\chi_{2888}(2773,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{262144}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{4}a^{5}$, $\frac{1}{8}a^{6}$, $\frac{1}{8}a^{7}$, $\frac{1}{16}a^{8}$, $\frac{1}{16}a^{9}$, $\frac{1}{32}a^{10}$, $\frac{1}{32}a^{11}$, $\frac{1}{64}a^{12}$, $\frac{1}{64}a^{13}$, $\frac{1}{128}a^{14}$, $\frac{1}{128}a^{15}$, $\frac{1}{256}a^{16}$, $\frac{1}{256}a^{17}$, $\frac{1}{512}a^{18}$, $\frac{1}{512}a^{19}$, $\frac{1}{1024}a^{20}$, $\frac{1}{1024}a^{21}$, $\frac{1}{2048}a^{22}$, $\frac{1}{2048}a^{23}$, $\frac{1}{4096}a^{24}$, $\frac{1}{4096}a^{25}$, $\frac{1}{8192}a^{26}$, $\frac{1}{8192}a^{27}$, $\frac{1}{16384}a^{28}$, $\frac{1}{16384}a^{29}$, $\frac{1}{32768}a^{30}$, $\frac{1}{32768}a^{31}$, $\frac{1}{65536}a^{32}$, $\frac{1}{65536}a^{33}$, $\frac{1}{16646144}a^{34}+\frac{15}{8323072}a^{32}+\frac{1}{520192}a^{30}+\frac{5}{1040384}a^{28}+\frac{1}{260096}a^{26}+\frac{3}{260096}a^{24}-\frac{7}{32512}a^{22}-\frac{5}{32512}a^{20}+\frac{3}{8128}a^{18}+\frac{31}{32512}a^{16}-\frac{11}{8128}a^{14}-\frac{27}{4064}a^{12}+\frac{7}{4064}a^{10}+\frac{31}{1016}a^{8}+\frac{57}{1016}a^{6}+\frac{23}{508}a^{4}-\frac{37}{254}a^{2}-\frac{5}{127}$, $\frac{1}{16646144}a^{35}+\frac{15}{8323072}a^{33}+\frac{1}{520192}a^{31}+\frac{5}{1040384}a^{29}+\frac{1}{260096}a^{27}+\frac{3}{260096}a^{25}-\frac{7}{32512}a^{23}-\frac{5}{32512}a^{21}+\frac{3}{8128}a^{19}+\frac{31}{32512}a^{17}-\frac{11}{8128}a^{15}-\frac{27}{4064}a^{13}+\frac{7}{4064}a^{11}+\frac{31}{1016}a^{9}+\frac{57}{1016}a^{7}+\frac{23}{508}a^{5}-\frac{37}{254}a^{3}-\frac{5}{127}a$, $\frac{1}{14\!\cdots\!32}a^{36}-\frac{11\!\cdots\!27}{70\!\cdots\!16}a^{34}+\frac{21\!\cdots\!41}{35\!\cdots\!08}a^{32}-\frac{10\!\cdots\!65}{17\!\cdots\!04}a^{30}-\frac{57\!\cdots\!07}{44\!\cdots\!76}a^{28}+\frac{14\!\cdots\!03}{44\!\cdots\!76}a^{26}+\frac{79\!\cdots\!37}{22\!\cdots\!88}a^{24}+\frac{65\!\cdots\!77}{55\!\cdots\!72}a^{22}-\frac{10\!\cdots\!71}{27\!\cdots\!36}a^{20}-\frac{25\!\cdots\!97}{27\!\cdots\!36}a^{18}-\frac{44\!\cdots\!45}{13\!\cdots\!68}a^{16}+\frac{21\!\cdots\!85}{34\!\cdots\!92}a^{14}-\frac{16\!\cdots\!95}{34\!\cdots\!92}a^{12}+\frac{19\!\cdots\!39}{13\!\cdots\!48}a^{10}-\frac{11\!\cdots\!13}{43\!\cdots\!24}a^{8}-\frac{60\!\cdots\!83}{10\!\cdots\!06}a^{6}+\frac{22\!\cdots\!83}{10\!\cdots\!06}a^{4}+\frac{86\!\cdots\!57}{10\!\cdots\!06}a^{2}+\frac{15\!\cdots\!65}{53\!\cdots\!03}$, $\frac{1}{29\!\cdots\!48}a^{37}-\frac{38\!\cdots\!81}{14\!\cdots\!24}a^{35}+\frac{97\!\cdots\!41}{74\!\cdots\!12}a^{33}+\frac{18\!\cdots\!99}{23\!\cdots\!16}a^{31}+\frac{17\!\cdots\!39}{92\!\cdots\!64}a^{29}-\frac{37\!\cdots\!91}{92\!\cdots\!64}a^{27}-\frac{84\!\cdots\!89}{46\!\cdots\!32}a^{25}+\frac{51\!\cdots\!83}{23\!\cdots\!16}a^{23}-\frac{13\!\cdots\!65}{36\!\cdots\!44}a^{21}+\frac{21\!\cdots\!01}{36\!\cdots\!44}a^{19}+\frac{17\!\cdots\!37}{28\!\cdots\!52}a^{17}-\frac{24\!\cdots\!67}{14\!\cdots\!76}a^{15}+\frac{53\!\cdots\!87}{45\!\cdots\!68}a^{13}+\frac{13\!\cdots\!33}{28\!\cdots\!72}a^{11}+\frac{95\!\cdots\!11}{90\!\cdots\!36}a^{9}-\frac{48\!\cdots\!21}{11\!\cdots\!17}a^{7}-\frac{13\!\cdots\!54}{11\!\cdots\!17}a^{5}+\frac{40\!\cdots\!11}{22\!\cdots\!34}a^{3}-\frac{55\!\cdots\!16}{11\!\cdots\!17}a$
| 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: |
\( -1 \)
(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 $C_{38}$ |
Intermediate fields
| \(\Q(\sqrt{-38}) \), 19.19.10842505080063916320800450434338728415281531281.1 |
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 | $19^{2}$ | $38$ | $19^{2}$ | $38$ | $19^{2}$ | $19^{2}$ | R | $19^{2}$ | $19^{2}$ | $38$ | $19^{2}$ | $38$ | $38$ | $19^{2}$ | $19^{2}$ | $19^{2}$ |
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 $38$ | $2$ | $19$ | $57$ | |||
|
\(19\)
| Deg $38$ | $38$ | $1$ | $73$ |