Normalized defining polynomial
\( x^{29} - x^{28} - 448 x^{27} + 107 x^{26} + 76138 x^{25} - 43560 x^{24} - 6896718 x^{23} + \cdots - 27\!\cdots\!09 \)
Invariants
| Degree: | $29$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[29, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(127\!\cdots\!161\)
\(\medspace = 929^{28}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(733.96\) | 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: | $929^{28/29}\approx 733.956539804396$ | ||
| Ramified primes: |
\(929\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $29$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(929\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{929}(1,·)$, $\chi_{929}(261,·)$, $\chi_{929}(454,·)$, $\chi_{929}(72,·)$, $\chi_{929}(521,·)$, $\chi_{929}(524,·)$, $\chi_{929}(719,·)$, $\chi_{929}(400,·)$, $\chi_{929}(148,·)$, $\chi_{929}(537,·)$, $\chi_{929}(539,·)$, $\chi_{929}(20,·)$, $\chi_{929}(352,·)$, $\chi_{929}(673,·)$, $\chi_{929}(347,·)$, $\chi_{929}(511,·)$, $\chi_{929}(807,·)$, $\chi_{929}(173,·)$, $\chi_{929}(304,·)$, $\chi_{929}(561,·)$, $\chi_{929}(437,·)$, $\chi_{929}(201,·)$, $\chi_{929}(568,·)$, $\chi_{929}(212,·)$, $\chi_{929}(506,·)$, $\chi_{929}(379,·)$, $\chi_{929}(445,·)$, $\chi_{929}(830,·)$, $\chi_{929}(575,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{101}a^{24}+\frac{13}{101}a^{23}-\frac{1}{101}a^{22}-\frac{4}{101}a^{21}+\frac{13}{101}a^{20}+\frac{4}{101}a^{19}-\frac{32}{101}a^{18}-\frac{49}{101}a^{17}+\frac{30}{101}a^{16}-\frac{9}{101}a^{15}-\frac{32}{101}a^{14}-\frac{20}{101}a^{13}-\frac{22}{101}a^{12}-\frac{26}{101}a^{11}+\frac{17}{101}a^{10}-\frac{11}{101}a^{9}-\frac{13}{101}a^{8}-\frac{34}{101}a^{7}-\frac{23}{101}a^{6}-\frac{40}{101}a^{5}-\frac{10}{101}a^{4}-\frac{30}{101}a^{3}-\frac{28}{101}a^{2}-\frac{41}{101}a-\frac{39}{101}$, $\frac{1}{101}a^{25}+\frac{32}{101}a^{23}+\frac{9}{101}a^{22}-\frac{36}{101}a^{21}+\frac{37}{101}a^{20}+\frac{17}{101}a^{19}-\frac{37}{101}a^{18}-\frac{40}{101}a^{17}+\frac{5}{101}a^{16}-\frac{16}{101}a^{15}-\frac{8}{101}a^{14}+\frac{36}{101}a^{13}-\frac{43}{101}a^{12}-\frac{49}{101}a^{11}-\frac{30}{101}a^{10}+\frac{29}{101}a^{9}+\frac{34}{101}a^{8}+\frac{15}{101}a^{7}-\frac{44}{101}a^{6}+\frac{5}{101}a^{5}-\frac{1}{101}a^{4}-\frac{42}{101}a^{3}+\frac{20}{101}a^{2}-\frac{11}{101}a+\frac{2}{101}$, $\frac{1}{101}a^{26}-\frac{3}{101}a^{23}-\frac{4}{101}a^{22}-\frac{37}{101}a^{21}+\frac{5}{101}a^{20}+\frac{37}{101}a^{19}-\frac{26}{101}a^{18}-\frac{43}{101}a^{17}+\frac{34}{101}a^{16}-\frac{23}{101}a^{15}+\frac{50}{101}a^{14}-\frac{9}{101}a^{13}+\frac{49}{101}a^{12}-\frac{6}{101}a^{11}-\frac{10}{101}a^{10}-\frac{18}{101}a^{9}+\frac{27}{101}a^{8}+\frac{34}{101}a^{7}+\frac{34}{101}a^{6}-\frac{34}{101}a^{5}-\frac{25}{101}a^{4}-\frac{30}{101}a^{3}-\frac{24}{101}a^{2}+\frac{1}{101}a+\frac{36}{101}$, $\frac{1}{19897}a^{27}+\frac{63}{19897}a^{26}-\frac{77}{19897}a^{25}-\frac{49}{19897}a^{24}+\frac{5027}{19897}a^{23}+\frac{6033}{19897}a^{22}-\frac{7349}{19897}a^{21}+\frac{4985}{19897}a^{20}-\frac{8682}{19897}a^{19}-\frac{5743}{19897}a^{18}-\frac{8249}{19897}a^{17}-\frac{1262}{19897}a^{16}-\frac{460}{19897}a^{15}+\frac{3411}{19897}a^{14}-\frac{9743}{19897}a^{13}-\frac{8958}{19897}a^{12}+\frac{7207}{19897}a^{11}-\frac{4776}{19897}a^{10}-\frac{3642}{19897}a^{9}-\frac{2507}{19897}a^{8}+\frac{9150}{19897}a^{7}+\frac{9079}{19897}a^{6}+\frac{8681}{19897}a^{5}-\frac{9855}{19897}a^{4}-\frac{6289}{19897}a^{3}-\frac{46}{19897}a^{2}-\frac{299}{19897}a+\frac{6130}{19897}$, $\frac{1}{10\!\cdots\!27}a^{28}-\frac{82\!\cdots\!22}{10\!\cdots\!27}a^{27}-\frac{37\!\cdots\!99}{10\!\cdots\!27}a^{26}+\frac{39\!\cdots\!22}{10\!\cdots\!27}a^{25}-\frac{50\!\cdots\!45}{10\!\cdots\!27}a^{24}+\frac{32\!\cdots\!26}{10\!\cdots\!27}a^{23}-\frac{47\!\cdots\!69}{10\!\cdots\!27}a^{22}+\frac{40\!\cdots\!13}{10\!\cdots\!27}a^{21}+\frac{26\!\cdots\!88}{10\!\cdots\!27}a^{20}+\frac{96\!\cdots\!10}{10\!\cdots\!27}a^{19}-\frac{16\!\cdots\!31}{10\!\cdots\!27}a^{18}+\frac{40\!\cdots\!00}{10\!\cdots\!27}a^{17}+\frac{14\!\cdots\!22}{10\!\cdots\!27}a^{16}+\frac{47\!\cdots\!59}{10\!\cdots\!27}a^{15}+\frac{22\!\cdots\!91}{10\!\cdots\!27}a^{14}-\frac{21\!\cdots\!85}{10\!\cdots\!27}a^{13}+\frac{93\!\cdots\!97}{10\!\cdots\!27}a^{12}+\frac{37\!\cdots\!61}{10\!\cdots\!27}a^{11}+\frac{13\!\cdots\!18}{10\!\cdots\!27}a^{10}+\frac{12\!\cdots\!48}{10\!\cdots\!27}a^{9}+\frac{26\!\cdots\!62}{10\!\cdots\!27}a^{8}-\frac{43\!\cdots\!97}{10\!\cdots\!27}a^{7}-\frac{32\!\cdots\!00}{10\!\cdots\!27}a^{6}-\frac{16\!\cdots\!66}{10\!\cdots\!27}a^{5}+\frac{39\!\cdots\!08}{10\!\cdots\!27}a^{4}-\frac{22\!\cdots\!33}{10\!\cdots\!27}a^{3}+\frac{39\!\cdots\!48}{10\!\cdots\!27}a^{2}-\frac{24\!\cdots\!93}{10\!\cdots\!27}a-\frac{52\!\cdots\!73}{10\!\cdots\!27}$
| Monogenic: | No | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
not computed
Unit group
| Rank: | $28$ | 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 29 |
| The 29 conjugacy class representatives for $C_{29}$ |
| Character table for $C_{29}$ is not computed |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $29$ | $29$ | $29$ | $29$ | $29$ | $29$ | $29$ | $29$ | $29$ | $29$ | $29$ | $29$ | $29$ | $29$ | $29$ | $29$ | $29$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(929\)
| Deg $29$ | $29$ | $1$ | $28$ |