Normalized defining polynomial
\( x^{38} + 229 x^{36} + 22442 x^{34} + 1242325 x^{32} + 43174744 x^{30} + 989928070 x^{28} + \cdots + 149876149 \)
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: |
\(-566\!\cdots\!896\)
\(\medspace = -\,2^{38}\cdot 229^{37}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(396.98\) | 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: | $2\cdot 229^{37/38}\approx 396.9762677011813$ | ||
| 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: | \(\Q(\sqrt{-229}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $38$ | 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$$\chi_{916}(1,·)$, $\chi_{916}(643,·)$, $\chi_{916}(901,·)$, $\chi_{916}(905,·)$, $\chi_{916}(11,·)$, $\chi_{916}(15,·)$, $\chi_{916}(17,·)$, $\chi_{916}(899,·)$, $\chi_{916}(661,·)$, $\chi_{916}(795,·)$, $\chi_{916}(671,·)$, $\chi_{916}(161,·)$, $\chi_{916}(165,·)$, $\chi_{916}(431,·)$, $\chi_{916}(691,·)$, $\chi_{916}(53,·)$, $\chi_{916}(57,·)$, $\chi_{916}(415,·)$, $\chi_{916}(61,·)$, $\chi_{916}(501,·)$, $\chi_{916}(583,·)$, $\chi_{916}(333,·)$, $\chi_{916}(855,·)$, $\chi_{916}(729,·)$, $\chi_{916}(859,·)$, $\chi_{916}(915,·)$, $\chi_{916}(863,·)$, $\chi_{916}(627,·)$, $\chi_{916}(225,·)$, $\chi_{916}(187,·)$, $\chi_{916}(485,·)$, $\chi_{916}(273,·)$, $\chi_{916}(289,·)$, $\chi_{916}(751,·)$, $\chi_{916}(755,·)$, $\chi_{916}(245,·)$, $\chi_{916}(121,·)$, $\chi_{916}(255,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{262144}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $\frac{1}{107}a^{26}-\frac{6}{107}a^{24}+\frac{53}{107}a^{22}+\frac{14}{107}a^{20}-\frac{34}{107}a^{18}+\frac{41}{107}a^{16}+\frac{30}{107}a^{14}+\frac{23}{107}a^{12}+\frac{16}{107}a^{8}-\frac{46}{107}a^{6}-\frac{2}{107}a^{4}+\frac{29}{107}a^{2}-\frac{19}{107}$, $\frac{1}{107}a^{27}-\frac{6}{107}a^{25}+\frac{53}{107}a^{23}+\frac{14}{107}a^{21}-\frac{34}{107}a^{19}+\frac{41}{107}a^{17}+\frac{30}{107}a^{15}+\frac{23}{107}a^{13}+\frac{16}{107}a^{9}-\frac{46}{107}a^{7}-\frac{2}{107}a^{5}+\frac{29}{107}a^{3}-\frac{19}{107}a$, $\frac{1}{107}a^{28}+\frac{17}{107}a^{24}+\frac{11}{107}a^{22}+\frac{50}{107}a^{20}+\frac{51}{107}a^{18}-\frac{45}{107}a^{16}-\frac{11}{107}a^{14}+\frac{31}{107}a^{12}+\frac{16}{107}a^{10}+\frac{50}{107}a^{8}+\frac{43}{107}a^{6}+\frac{17}{107}a^{4}+\frac{48}{107}a^{2}-\frac{7}{107}$, $\frac{1}{107}a^{29}+\frac{17}{107}a^{25}+\frac{11}{107}a^{23}+\frac{50}{107}a^{21}+\frac{51}{107}a^{19}-\frac{45}{107}a^{17}-\frac{11}{107}a^{15}+\frac{31}{107}a^{13}+\frac{16}{107}a^{11}+\frac{50}{107}a^{9}+\frac{43}{107}a^{7}+\frac{17}{107}a^{5}+\frac{48}{107}a^{3}-\frac{7}{107}a$, $\frac{1}{107}a^{30}+\frac{6}{107}a^{24}+\frac{5}{107}a^{22}+\frac{27}{107}a^{20}-\frac{2}{107}a^{18}+\frac{41}{107}a^{16}-\frac{51}{107}a^{14}+\frac{53}{107}a^{12}+\frac{50}{107}a^{10}-\frac{15}{107}a^{8}+\frac{50}{107}a^{6}-\frac{25}{107}a^{4}+\frac{35}{107}a^{2}+\frac{2}{107}$, $\frac{1}{107}a^{31}+\frac{6}{107}a^{25}+\frac{5}{107}a^{23}+\frac{27}{107}a^{21}-\frac{2}{107}a^{19}+\frac{41}{107}a^{17}-\frac{51}{107}a^{15}+\frac{53}{107}a^{13}+\frac{50}{107}a^{11}-\frac{15}{107}a^{9}+\frac{50}{107}a^{7}-\frac{25}{107}a^{5}+\frac{35}{107}a^{3}+\frac{2}{107}a$, $\frac{1}{9523}a^{32}+\frac{3}{9523}a^{30}-\frac{6}{9523}a^{28}+\frac{8}{9523}a^{26}-\frac{3408}{9523}a^{24}-\frac{2165}{9523}a^{22}+\frac{2268}{9523}a^{20}-\frac{4084}{9523}a^{18}-\frac{860}{9523}a^{16}-\frac{3933}{9523}a^{14}-\frac{2499}{9523}a^{12}+\frac{4319}{9523}a^{10}-\frac{3045}{9523}a^{8}+\frac{3092}{9523}a^{6}+\frac{389}{9523}a^{4}+\frac{2766}{9523}a^{2}-\frac{4163}{9523}$, $\frac{1}{9523}a^{33}+\frac{3}{9523}a^{31}-\frac{6}{9523}a^{29}+\frac{8}{9523}a^{27}-\frac{3408}{9523}a^{25}-\frac{2165}{9523}a^{23}+\frac{2268}{9523}a^{21}-\frac{4084}{9523}a^{19}-\frac{860}{9523}a^{17}-\frac{3933}{9523}a^{15}-\frac{2499}{9523}a^{13}+\frac{4319}{9523}a^{11}-\frac{3045}{9523}a^{9}+\frac{3092}{9523}a^{7}+\frac{389}{9523}a^{5}+\frac{2766}{9523}a^{3}-\frac{4163}{9523}a$, $\frac{1}{127495457203}a^{34}-\frac{3762954}{127495457203}a^{32}+\frac{594272018}{127495457203}a^{30}-\frac{271586296}{127495457203}a^{28}+\frac{458836508}{127495457203}a^{26}-\frac{37659860813}{127495457203}a^{24}-\frac{21551733898}{127495457203}a^{22}-\frac{54484420657}{127495457203}a^{20}+\frac{21826424541}{127495457203}a^{18}+\frac{61232976004}{127495457203}a^{16}+\frac{9113298200}{127495457203}a^{14}-\frac{39977086622}{127495457203}a^{12}+\frac{22236079087}{127495457203}a^{10}+\frac{15642261184}{127495457203}a^{8}+\frac{34855167983}{127495457203}a^{6}-\frac{37594057250}{127495457203}a^{4}+\frac{20944557106}{127495457203}a^{2}-\frac{35681719383}{127495457203}$, $\frac{1}{127495457203}a^{35}-\frac{3762954}{127495457203}a^{33}+\frac{594272018}{127495457203}a^{31}-\frac{271586296}{127495457203}a^{29}+\frac{458836508}{127495457203}a^{27}-\frac{37659860813}{127495457203}a^{25}-\frac{21551733898}{127495457203}a^{23}-\frac{54484420657}{127495457203}a^{21}+\frac{21826424541}{127495457203}a^{19}+\frac{61232976004}{127495457203}a^{17}+\frac{9113298200}{127495457203}a^{15}-\frac{39977086622}{127495457203}a^{13}+\frac{22236079087}{127495457203}a^{11}+\frac{15642261184}{127495457203}a^{9}+\frac{34855167983}{127495457203}a^{7}-\frac{37594057250}{127495457203}a^{5}+\frac{20944557106}{127495457203}a^{3}-\frac{35681719383}{127495457203}a$, $\frac{1}{39\!\cdots\!53}a^{36}+\frac{46\!\cdots\!85}{39\!\cdots\!53}a^{34}-\frac{17\!\cdots\!96}{39\!\cdots\!53}a^{32}-\frac{96\!\cdots\!47}{39\!\cdots\!53}a^{30}-\frac{13\!\cdots\!17}{39\!\cdots\!53}a^{28}+\frac{88\!\cdots\!70}{39\!\cdots\!53}a^{26}-\frac{76\!\cdots\!74}{39\!\cdots\!53}a^{24}-\frac{14\!\cdots\!18}{39\!\cdots\!53}a^{22}+\frac{19\!\cdots\!01}{39\!\cdots\!53}a^{20}+\frac{64\!\cdots\!47}{39\!\cdots\!53}a^{18}+\frac{10\!\cdots\!22}{39\!\cdots\!53}a^{16}+\frac{15\!\cdots\!72}{39\!\cdots\!53}a^{14}+\frac{12\!\cdots\!96}{39\!\cdots\!53}a^{12}-\frac{90\!\cdots\!13}{39\!\cdots\!53}a^{10}+\frac{18\!\cdots\!74}{39\!\cdots\!53}a^{8}+\frac{13\!\cdots\!62}{39\!\cdots\!53}a^{6}-\frac{76\!\cdots\!97}{44\!\cdots\!77}a^{4}-\frac{11\!\cdots\!54}{39\!\cdots\!53}a^{2}-\frac{74\!\cdots\!27}{39\!\cdots\!53}$, $\frac{1}{32\!\cdots\!77}a^{37}+\frac{11\!\cdots\!67}{32\!\cdots\!77}a^{35}+\frac{86\!\cdots\!84}{32\!\cdots\!77}a^{33}+\frac{14\!\cdots\!95}{32\!\cdots\!77}a^{31}-\frac{14\!\cdots\!57}{32\!\cdots\!77}a^{29}+\frac{88\!\cdots\!25}{32\!\cdots\!77}a^{27}+\frac{15\!\cdots\!91}{32\!\cdots\!77}a^{25}-\frac{70\!\cdots\!17}{32\!\cdots\!77}a^{23}+\frac{61\!\cdots\!53}{32\!\cdots\!77}a^{21}-\frac{80\!\cdots\!11}{32\!\cdots\!77}a^{19}+\frac{14\!\cdots\!54}{32\!\cdots\!77}a^{17}+\frac{14\!\cdots\!29}{32\!\cdots\!77}a^{15}+\frac{77\!\cdots\!91}{32\!\cdots\!77}a^{13}-\frac{67\!\cdots\!79}{32\!\cdots\!77}a^{11}-\frac{46\!\cdots\!00}{32\!\cdots\!77}a^{9}+\frac{52\!\cdots\!78}{32\!\cdots\!77}a^{7}+\frac{63\!\cdots\!34}{32\!\cdots\!77}a^{5}-\frac{22\!\cdots\!83}{32\!\cdots\!77}a^{3}-\frac{10\!\cdots\!81}{32\!\cdots\!77}a$
| 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
| \(\Q(\sqrt{-229}) \), 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 | $38$ | $19^{2}$ | $19^{2}$ | $38$ | $38$ | $19^{2}$ | $38$ | $19^{2}$ | $38$ | $19^{2}$ | $19^{2}$ | $38$ | $38$ | $19^{2}$ | $19^{2}$ | $19^{2}$ |
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 $38$ | $2$ | $19$ | $38$ | |||
|
\(229\)
| Deg $38$ | $38$ | $1$ | $37$ |