Normalized defining polynomial
\( x^{38} - x^{37} - 111 x^{36} + 252 x^{35} + 5215 x^{34} - 18518 x^{33} - 127217 x^{32} + 667591 x^{31} + \cdots - 6131569 \)
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: |
\(206\!\cdots\!109\)
\(\medspace = 229^{37}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(198.49\) | 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: | $229^{37/38}\approx 198.48813385059066$ | ||
| Ramified primes: |
\(229\)
| 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: | \(229\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{229}(1,·)$, $\chi_{229}(4,·)$, $\chi_{229}(44,·)$, $\chi_{229}(11,·)$, $\chi_{229}(15,·)$, $\chi_{229}(16,·)$, $\chi_{229}(17,·)$, $\chi_{229}(26,·)$, $\chi_{229}(27,·)$, $\chi_{229}(161,·)$, $\chi_{229}(165,·)$, $\chi_{229}(168,·)$, $\chi_{229}(169,·)$, $\chi_{229}(42,·)$, $\chi_{229}(43,·)$, $\chi_{229}(172,·)$, $\chi_{229}(176,·)$, $\chi_{229}(53,·)$, $\chi_{229}(57,·)$, $\chi_{229}(186,·)$, $\chi_{229}(187,·)$, $\chi_{229}(60,·)$, $\chi_{229}(61,·)$, $\chi_{229}(64,·)$, $\chi_{229}(68,·)$, $\chi_{229}(202,·)$, $\chi_{229}(203,·)$, $\chi_{229}(212,·)$, $\chi_{229}(213,·)$, $\chi_{229}(214,·)$, $\chi_{229}(185,·)$, $\chi_{229}(218,·)$, $\chi_{229}(225,·)$, $\chi_{229}(228,·)$, $\chi_{229}(104,·)$, $\chi_{229}(108,·)$, $\chi_{229}(121,·)$, $\chi_{229}(125,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $\frac{1}{7787737}a^{36}-\frac{2637475}{7787737}a^{35}+\frac{615065}{7787737}a^{34}-\frac{2506933}{7787737}a^{33}+\frac{1207515}{7787737}a^{32}+\frac{119873}{7787737}a^{31}+\frac{1574909}{7787737}a^{30}+\frac{3751364}{7787737}a^{29}-\frac{755682}{7787737}a^{28}+\frac{3174239}{7787737}a^{27}-\frac{96975}{7787737}a^{26}+\frac{913786}{7787737}a^{25}-\frac{3040176}{7787737}a^{24}-\frac{1682192}{7787737}a^{23}-\frac{1864227}{7787737}a^{22}-\frac{3799130}{7787737}a^{21}+\frac{1276549}{7787737}a^{20}+\frac{62153}{7787737}a^{19}-\frac{2536689}{7787737}a^{18}+\frac{2163242}{7787737}a^{17}-\frac{945908}{7787737}a^{16}+\frac{2582888}{7787737}a^{15}-\frac{2012910}{7787737}a^{14}-\frac{1100709}{7787737}a^{13}+\frac{1058155}{7787737}a^{12}+\frac{1861100}{7787737}a^{11}-\frac{37384}{7787737}a^{10}-\frac{3344653}{7787737}a^{9}+\frac{205591}{7787737}a^{8}-\frac{727001}{7787737}a^{7}-\frac{3708005}{7787737}a^{6}+\frac{2492503}{7787737}a^{5}+\frac{1487749}{7787737}a^{4}-\frac{1386641}{7787737}a^{3}-\frac{567980}{7787737}a^{2}-\frac{2451025}{7787737}a-\frac{3398}{17041}$, $\frac{1}{47\!\cdots\!31}a^{37}-\frac{63\!\cdots\!55}{47\!\cdots\!31}a^{36}+\frac{12\!\cdots\!53}{47\!\cdots\!31}a^{35}+\frac{16\!\cdots\!33}{47\!\cdots\!31}a^{34}+\frac{83\!\cdots\!92}{47\!\cdots\!31}a^{33}+\frac{15\!\cdots\!28}{47\!\cdots\!31}a^{32}+\frac{21\!\cdots\!91}{47\!\cdots\!31}a^{31}+\frac{12\!\cdots\!93}{47\!\cdots\!31}a^{30}-\frac{14\!\cdots\!17}{47\!\cdots\!31}a^{29}-\frac{11\!\cdots\!28}{47\!\cdots\!31}a^{28}+\frac{18\!\cdots\!84}{47\!\cdots\!31}a^{27}+\frac{12\!\cdots\!86}{47\!\cdots\!31}a^{26}+\frac{20\!\cdots\!89}{47\!\cdots\!31}a^{25}+\frac{31\!\cdots\!58}{47\!\cdots\!31}a^{24}-\frac{55\!\cdots\!39}{47\!\cdots\!31}a^{23}+\frac{20\!\cdots\!60}{47\!\cdots\!31}a^{22}-\frac{17\!\cdots\!90}{47\!\cdots\!31}a^{21}-\frac{15\!\cdots\!77}{47\!\cdots\!31}a^{20}-\frac{22\!\cdots\!85}{47\!\cdots\!31}a^{19}-\frac{13\!\cdots\!35}{47\!\cdots\!31}a^{18}+\frac{49\!\cdots\!11}{47\!\cdots\!31}a^{17}+\frac{12\!\cdots\!61}{47\!\cdots\!31}a^{16}-\frac{39\!\cdots\!76}{47\!\cdots\!31}a^{15}+\frac{45\!\cdots\!41}{47\!\cdots\!31}a^{14}+\frac{13\!\cdots\!35}{47\!\cdots\!31}a^{13}+\frac{23\!\cdots\!35}{47\!\cdots\!31}a^{12}-\frac{16\!\cdots\!90}{47\!\cdots\!31}a^{11}+\frac{14\!\cdots\!36}{47\!\cdots\!31}a^{10}+\frac{49\!\cdots\!76}{47\!\cdots\!31}a^{9}+\frac{18\!\cdots\!06}{47\!\cdots\!31}a^{8}-\frac{13\!\cdots\!96}{47\!\cdots\!31}a^{7}-\frac{47\!\cdots\!00}{47\!\cdots\!31}a^{6}+\frac{23\!\cdots\!02}{47\!\cdots\!31}a^{5}-\frac{18\!\cdots\!16}{47\!\cdots\!31}a^{4}+\frac{88\!\cdots\!05}{47\!\cdots\!31}a^{3}-\frac{12\!\cdots\!21}{47\!\cdots\!31}a^{2}+\frac{77\!\cdots\!83}{47\!\cdots\!31}a+\frac{18\!\cdots\!44}{10\!\cdots\!83}$
| 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
| \(\Q(\sqrt{229}) \), 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 | $38$ | $19^{2}$ | $19^{2}$ | $38$ | $19^{2}$ | $38$ | $19^{2}$ | $19^{2}$ | $38$ | $38$ | $38$ | $19^{2}$ | $38$ | $19^{2}$ | $38$ | $19^{2}$ | $38$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(229\)
| Deg $38$ | $38$ | $1$ | $37$ |