Normalized defining polynomial
\( x^{38} - x^{37} + 118 x^{36} - 435 x^{35} + 6131 x^{34} - 35235 x^{33} + 222008 x^{32} + \cdots + 34\!\cdots\!89 \)
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: |
\(-239\!\cdots\!903\)
\(\medspace = -\,3^{19}\cdot 229^{37}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(343.79\) | 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: | $3^{1/2}229^{37/38}\approx 343.79153252875494$ | ||
| Ramified primes: |
\(3\), \(229\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-687}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $38$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(687=3\cdot 229\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{687}(256,·)$, $\chi_{687}(1,·)$, $\chi_{687}(644,·)$, $\chi_{687}(394,·)$, $\chi_{687}(11,·)$, $\chi_{687}(398,·)$, $\chi_{687}(271,·)$, $\chi_{687}(16,·)$, $\chi_{687}(401,·)$, $\chi_{687}(661,·)$, $\chi_{687}(26,·)$, $\chi_{687}(286,·)$, $\chi_{687}(671,·)$, $\chi_{687}(416,·)$, $\chi_{687}(289,·)$, $\chi_{687}(676,·)$, $\chi_{687}(293,·)$, $\chi_{687}(43,·)$, $\chi_{687}(686,·)$, $\chi_{687}(431,·)$, $\chi_{687}(176,·)$, $\chi_{687}(562,·)$, $\chi_{687}(566,·)$, $\chi_{687}(185,·)$, $\chi_{687}(61,·)$, $\chi_{687}(68,·)$, $\chi_{687}(454,·)$, $\chi_{687}(212,·)$, $\chi_{687}(214,·)$, $\chi_{687}(473,·)$, $\chi_{687}(475,·)$, $\chi_{687}(233,·)$, $\chi_{687}(619,·)$, $\chi_{687}(626,·)$, $\chi_{687}(502,·)$, $\chi_{687}(121,·)$, $\chi_{687}(125,·)$, $\chi_{687}(511,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{262144}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $\frac{1}{89}a^{26}-\frac{44}{89}a^{25}-\frac{10}{89}a^{24}-\frac{11}{89}a^{23}+\frac{37}{89}a^{22}-\frac{43}{89}a^{21}+\frac{18}{89}a^{20}+\frac{14}{89}a^{19}-\frac{15}{89}a^{18}-\frac{13}{89}a^{17}+\frac{6}{89}a^{16}+\frac{33}{89}a^{15}-\frac{6}{89}a^{14}+\frac{29}{89}a^{13}-\frac{30}{89}a^{12}-\frac{22}{89}a^{11}-\frac{24}{89}a^{10}+\frac{13}{89}a^{9}-\frac{20}{89}a^{8}+\frac{26}{89}a^{7}+\frac{18}{89}a^{6}-\frac{5}{89}a^{5}-\frac{8}{89}a^{4}-\frac{33}{89}a^{3}-\frac{2}{89}a^{2}+\frac{25}{89}a+\frac{4}{89}$, $\frac{1}{89}a^{27}+\frac{12}{89}a^{25}-\frac{6}{89}a^{24}-\frac{2}{89}a^{23}-\frac{17}{89}a^{22}-\frac{5}{89}a^{21}+\frac{5}{89}a^{20}-\frac{22}{89}a^{19}+\frac{39}{89}a^{18}-\frac{32}{89}a^{17}+\frac{30}{89}a^{16}+\frac{22}{89}a^{15}+\frac{32}{89}a^{14}-\frac{7}{89}a^{12}-\frac{13}{89}a^{11}+\frac{25}{89}a^{10}+\frac{18}{89}a^{9}+\frac{36}{89}a^{8}+\frac{5}{89}a^{7}-\frac{14}{89}a^{6}+\frac{39}{89}a^{5}-\frac{29}{89}a^{4}-\frac{30}{89}a^{3}+\frac{26}{89}a^{2}+\frac{36}{89}a-\frac{2}{89}$, $\frac{1}{89}a^{28}-\frac{12}{89}a^{25}+\frac{29}{89}a^{24}+\frac{26}{89}a^{23}-\frac{4}{89}a^{22}-\frac{13}{89}a^{21}+\frac{29}{89}a^{20}-\frac{40}{89}a^{19}-\frac{30}{89}a^{18}+\frac{8}{89}a^{17}+\frac{39}{89}a^{16}-\frac{8}{89}a^{15}-\frac{17}{89}a^{14}+\frac{1}{89}a^{13}-\frac{9}{89}a^{12}+\frac{22}{89}a^{11}+\frac{39}{89}a^{10}-\frac{31}{89}a^{9}-\frac{22}{89}a^{8}+\frac{30}{89}a^{7}+\frac{1}{89}a^{6}+\frac{31}{89}a^{5}-\frac{23}{89}a^{4}-\frac{23}{89}a^{3}-\frac{29}{89}a^{2}-\frac{35}{89}a+\frac{41}{89}$, $\frac{1}{89}a^{29}+\frac{35}{89}a^{25}-\frac{5}{89}a^{24}+\frac{42}{89}a^{23}-\frac{14}{89}a^{22}-\frac{42}{89}a^{21}-\frac{2}{89}a^{20}-\frac{40}{89}a^{19}+\frac{6}{89}a^{18}-\frac{28}{89}a^{17}-\frac{25}{89}a^{16}+\frac{23}{89}a^{15}+\frac{18}{89}a^{14}-\frac{17}{89}a^{13}+\frac{18}{89}a^{12}+\frac{42}{89}a^{11}+\frac{37}{89}a^{10}-\frac{44}{89}a^{9}-\frac{32}{89}a^{8}-\frac{43}{89}a^{7}-\frac{20}{89}a^{6}+\frac{6}{89}a^{5}-\frac{30}{89}a^{4}+\frac{20}{89}a^{3}+\frac{30}{89}a^{2}-\frac{15}{89}a-\frac{41}{89}$, $\frac{1}{89}a^{30}+\frac{22}{89}a^{25}+\frac{36}{89}a^{24}+\frac{15}{89}a^{23}-\frac{2}{89}a^{22}-\frac{10}{89}a^{21}+\frac{42}{89}a^{20}-\frac{39}{89}a^{19}-\frac{37}{89}a^{18}-\frac{15}{89}a^{17}-\frac{9}{89}a^{16}+\frac{20}{89}a^{15}+\frac{15}{89}a^{14}-\frac{18}{89}a^{13}+\frac{24}{89}a^{12}+\frac{6}{89}a^{11}-\frac{5}{89}a^{10}-\frac{42}{89}a^{9}+\frac{34}{89}a^{8}-\frac{40}{89}a^{7}-\frac{1}{89}a^{6}-\frac{33}{89}a^{5}+\frac{33}{89}a^{4}+\frac{28}{89}a^{3}-\frac{34}{89}a^{2}-\frac{26}{89}a+\frac{38}{89}$, $\frac{1}{9523}a^{31}+\frac{28}{9523}a^{30}-\frac{2}{9523}a^{29}+\frac{20}{9523}a^{28}-\frac{18}{9523}a^{27}+\frac{53}{9523}a^{26}+\frac{364}{9523}a^{25}-\frac{2416}{9523}a^{24}+\frac{2418}{9523}a^{23}+\frac{31}{107}a^{22}-\frac{144}{9523}a^{21}-\frac{748}{9523}a^{20}-\frac{2087}{9523}a^{19}+\frac{18}{9523}a^{18}-\frac{1731}{9523}a^{17}-\frac{4028}{9523}a^{16}-\frac{3187}{9523}a^{15}-\frac{4474}{9523}a^{14}+\frac{1185}{9523}a^{13}+\frac{4553}{9523}a^{12}-\frac{997}{9523}a^{11}-\frac{3785}{9523}a^{10}+\frac{2143}{9523}a^{9}+\frac{4163}{9523}a^{8}+\frac{1883}{9523}a^{7}+\frac{364}{9523}a^{6}+\frac{2420}{9523}a^{5}+\frac{2606}{9523}a^{4}-\frac{1835}{9523}a^{3}+\frac{255}{9523}a^{2}+\frac{2416}{9523}a+\frac{1503}{9523}$, $\frac{1}{9523}a^{32}-\frac{37}{9523}a^{30}-\frac{31}{9523}a^{29}-\frac{43}{9523}a^{28}+\frac{22}{9523}a^{27}-\frac{50}{9523}a^{26}-\frac{2657}{9523}a^{25}+\frac{837}{9523}a^{24}+\frac{2144}{9523}a^{23}-\frac{2282}{9523}a^{22}-\frac{2387}{9523}a^{21}-\frac{3078}{9523}a^{20}+\frac{781}{9523}a^{19}+\frac{2152}{9523}a^{18}-\frac{3924}{9523}a^{17}+\frac{2490}{9523}a^{16}+\frac{2265}{9523}a^{15}-\frac{1622}{9523}a^{14}+\frac{798}{9523}a^{13}-\frac{2756}{9523}a^{12}+\frac{270}{9523}a^{11}-\frac{3478}{9523}a^{10}+\frac{334}{9523}a^{9}+\frac{4624}{9523}a^{8}+\frac{1568}{9523}a^{7}+\frac{1858}{9523}a^{6}+\frac{4610}{9523}a^{5}-\frac{4611}{9523}a^{4}+\frac{810}{9523}a^{3}+\frac{1696}{9523}a^{2}-\frac{19}{9523}a-\frac{996}{9523}$, $\frac{1}{9523}a^{33}+\frac{42}{9523}a^{30}-\frac{10}{9523}a^{29}+\frac{13}{9523}a^{28}+\frac{33}{9523}a^{27}+\frac{53}{9523}a^{26}+\frac{930}{9523}a^{25}-\frac{3788}{9523}a^{24}+\frac{407}{9523}a^{23}+\frac{3824}{9523}a^{22}-\frac{916}{9523}a^{21}+\frac{4135}{9523}a^{20}+\frac{1224}{9523}a^{19}-\frac{2723}{9523}a^{18}-\frac{4098}{9523}a^{17}-\frac{181}{9523}a^{16}-\frac{3446}{9523}a^{15}-\frac{2207}{9523}a^{14}+\frac{1392}{9523}a^{13}+\frac{2774}{9523}a^{12}+\frac{1363}{9523}a^{11}+\frac{28}{107}a^{10}+\frac{4200}{9523}a^{9}-\frac{4473}{9523}a^{8}+\frac{1444}{9523}a^{7}+\frac{102}{9523}a^{6}-\frac{4202}{9523}a^{5}+\frac{4142}{9523}a^{4}+\frac{2816}{9523}a^{3}-\frac{6}{89}a^{2}+\frac{2796}{9523}a+\frac{4465}{9523}$, $\frac{1}{9523}a^{34}-\frac{9}{9523}a^{30}-\frac{10}{9523}a^{29}+\frac{49}{9523}a^{28}-\frac{47}{9523}a^{27}-\frac{12}{9523}a^{26}+\frac{2217}{9523}a^{25}+\frac{15}{9523}a^{24}+\frac{1457}{9523}a^{23}-\frac{1876}{9523}a^{22}-\frac{2015}{9523}a^{21}+\frac{2145}{9523}a^{20}-\frac{1739}{9523}a^{19}-\frac{3570}{9523}a^{18}-\frac{774}{9523}a^{17}+\frac{1806}{9523}a^{16}-\frac{1996}{9523}a^{15}+\frac{3013}{9523}a^{14}+\frac{298}{9523}a^{13}-\frac{3790}{9523}a^{12}+\frac{1031}{9523}a^{11}+\frac{1172}{9523}a^{10}-\frac{2566}{9523}a^{9}+\frac{4218}{9523}a^{8}-\frac{18}{9523}a^{7}-\frac{1621}{9523}a^{6}+\frac{2119}{9523}a^{5}-\frac{3060}{9523}a^{4}+\frac{4203}{9523}a^{3}+\frac{3963}{9523}a^{2}-\frac{2312}{9523}a-\frac{638}{9523}$, $\frac{1}{9523}a^{35}+\frac{28}{9523}a^{30}+\frac{31}{9523}a^{29}+\frac{26}{9523}a^{28}+\frac{40}{9523}a^{27}+\frac{19}{9523}a^{26}-\frac{3664}{9523}a^{25}+\frac{3895}{9523}a^{24}-\frac{4724}{9523}a^{23}-\frac{2757}{9523}a^{22}+\frac{4059}{9523}a^{21}-\frac{981}{9523}a^{20}-\frac{4270}{9523}a^{19}+\frac{1849}{9523}a^{18}-\frac{2538}{9523}a^{17}-\frac{2510}{9523}a^{16}+\frac{1615}{9523}a^{15}+\frac{585}{9523}a^{14}-\frac{294}{9523}a^{13}+\frac{2311}{9523}a^{12}-\frac{2986}{9523}a^{11}+\frac{1247}{9523}a^{10}-\frac{4636}{9523}a^{9}-\frac{1499}{9523}a^{8}-\frac{189}{9523}a^{7}+\frac{1971}{9523}a^{6}-\frac{3429}{9523}a^{5}+\frac{158}{9523}a^{4}-\frac{889}{9523}a^{3}+\frac{2230}{9523}a^{2}-\frac{187}{9523}a-\frac{597}{9523}$, $\frac{1}{387328979}a^{36}-\frac{3037}{387328979}a^{35}-\frac{9097}{387328979}a^{34}+\frac{11995}{387328979}a^{33}-\frac{11849}{387328979}a^{32}+\frac{15927}{387328979}a^{31}+\frac{828699}{387328979}a^{30}+\frac{1518890}{387328979}a^{29}+\frac{1754945}{387328979}a^{28}+\frac{2126013}{387328979}a^{27}-\frac{1922654}{387328979}a^{26}+\frac{2114166}{387328979}a^{25}-\frac{69225020}{387328979}a^{24}-\frac{11519224}{387328979}a^{23}-\frac{29964832}{387328979}a^{22}-\frac{102790022}{387328979}a^{21}+\frac{1262720}{387328979}a^{20}+\frac{41019024}{387328979}a^{19}-\frac{56733827}{387328979}a^{18}+\frac{134062140}{387328979}a^{17}-\frac{118449271}{387328979}a^{16}+\frac{175774992}{387328979}a^{15}+\frac{124204446}{387328979}a^{14}-\frac{100550113}{387328979}a^{13}+\frac{65989375}{387328979}a^{12}+\frac{182783746}{387328979}a^{11}-\frac{153923732}{387328979}a^{10}+\frac{14892614}{387328979}a^{9}-\frac{152247725}{387328979}a^{8}-\frac{41614587}{387328979}a^{7}+\frac{177416809}{387328979}a^{6}+\frac{114001440}{387328979}a^{5}-\frac{7976645}{387328979}a^{4}+\frac{178035798}{387328979}a^{3}+\frac{8110883}{387328979}a^{2}-\frac{40095012}{387328979}a-\frac{10617930}{387328979}$, $\frac{1}{47\!\cdots\!53}a^{37}+\frac{65\!\cdots\!66}{53\!\cdots\!77}a^{36}-\frac{19\!\cdots\!71}{47\!\cdots\!53}a^{35}+\frac{18\!\cdots\!61}{47\!\cdots\!53}a^{34}+\frac{97\!\cdots\!11}{47\!\cdots\!53}a^{33}-\frac{13\!\cdots\!95}{47\!\cdots\!53}a^{32}-\frac{19\!\cdots\!20}{47\!\cdots\!53}a^{31}+\frac{38\!\cdots\!26}{47\!\cdots\!53}a^{30}+\frac{24\!\cdots\!36}{47\!\cdots\!53}a^{29}+\frac{12\!\cdots\!96}{47\!\cdots\!53}a^{28}+\frac{30\!\cdots\!54}{47\!\cdots\!53}a^{27}+\frac{31\!\cdots\!36}{47\!\cdots\!53}a^{26}-\frac{68\!\cdots\!20}{47\!\cdots\!53}a^{25}+\frac{51\!\cdots\!91}{47\!\cdots\!53}a^{24}+\frac{21\!\cdots\!56}{47\!\cdots\!53}a^{23}-\frac{18\!\cdots\!94}{47\!\cdots\!53}a^{22}-\frac{26\!\cdots\!08}{44\!\cdots\!79}a^{21}+\frac{40\!\cdots\!73}{47\!\cdots\!53}a^{20}+\frac{44\!\cdots\!37}{47\!\cdots\!53}a^{19}+\frac{13\!\cdots\!32}{47\!\cdots\!53}a^{18}-\frac{11\!\cdots\!05}{47\!\cdots\!53}a^{17}+\frac{25\!\cdots\!76}{47\!\cdots\!53}a^{16}+\frac{75\!\cdots\!82}{47\!\cdots\!53}a^{15}+\frac{23\!\cdots\!65}{47\!\cdots\!53}a^{14}+\frac{11\!\cdots\!81}{47\!\cdots\!53}a^{13}-\frac{16\!\cdots\!80}{47\!\cdots\!53}a^{12}+\frac{13\!\cdots\!83}{47\!\cdots\!53}a^{11}-\frac{21\!\cdots\!34}{47\!\cdots\!53}a^{10}-\frac{12\!\cdots\!79}{47\!\cdots\!53}a^{9}-\frac{22\!\cdots\!56}{47\!\cdots\!53}a^{8}-\frac{36\!\cdots\!64}{47\!\cdots\!53}a^{7}+\frac{19\!\cdots\!37}{47\!\cdots\!53}a^{6}-\frac{14\!\cdots\!64}{47\!\cdots\!53}a^{5}+\frac{54\!\cdots\!22}{47\!\cdots\!53}a^{4}-\frac{16\!\cdots\!59}{47\!\cdots\!53}a^{3}-\frac{21\!\cdots\!35}{47\!\cdots\!53}a^{2}-\frac{19\!\cdots\!90}{47\!\cdots\!53}a-\frac{18\!\cdots\!02}{47\!\cdots\!53}$
| 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 $C_{38}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-687}) \), 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 | $19^{2}$ | R | $38$ | $38$ | $38$ | $38$ | $38$ | $19^{2}$ | $19^{2}$ | $19^{2}$ | $38$ | $19^{2}$ | $19^{2}$ | $19^{2}$ | $19^{2}$ | $38$ | $19^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| Deg $38$ | $2$ | $19$ | $19$ | |||
|
\(229\)
| Deg $38$ | $38$ | $1$ | $37$ |