Normalized defining polynomial
\( x^{29} + 4x - 4 \)
Invariants
Degree: | $29$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[1, 14]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(8948737504120671477993534661511237990596936204288\) \(\medspace = 2^{28}\cdot 27325532669\cdot 12\!\cdots\!17\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(48.75\) | 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: | $2^{28/29}27325532669^{1/2}1219981524656697632736241282817^{1/2}\approx 3.56541917804482e+20$ | ||
Ramified primes: | \(2\), \(27325532669\), \(12199\!\cdots\!82817\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{33336\!\cdots\!48573}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2}a^{15}$, $\frac{1}{2}a^{16}$, $\frac{1}{2}a^{17}$, $\frac{1}{2}a^{18}$, $\frac{1}{2}a^{19}$, $\frac{1}{2}a^{20}$, $\frac{1}{2}a^{21}$, $\frac{1}{2}a^{22}$, $\frac{1}{2}a^{23}$, $\frac{1}{2}a^{24}$, $\frac{1}{2}a^{25}$, $\frac{1}{2}a^{26}$, $\frac{1}{2}a^{27}$, $\frac{1}{18}a^{28}+\frac{1}{9}a^{27}+\frac{2}{9}a^{26}-\frac{1}{18}a^{25}-\frac{1}{9}a^{24}-\frac{2}{9}a^{23}+\frac{1}{18}a^{22}+\frac{1}{9}a^{21}+\frac{2}{9}a^{20}-\frac{1}{18}a^{19}-\frac{1}{9}a^{18}-\frac{2}{9}a^{17}+\frac{1}{18}a^{16}+\frac{1}{9}a^{15}+\frac{2}{9}a^{14}+\frac{4}{9}a^{13}-\frac{1}{9}a^{12}-\frac{2}{9}a^{11}-\frac{4}{9}a^{10}+\frac{1}{9}a^{9}+\frac{2}{9}a^{8}+\frac{4}{9}a^{7}-\frac{1}{9}a^{6}-\frac{2}{9}a^{5}-\frac{4}{9}a^{4}+\frac{1}{9}a^{3}+\frac{2}{9}a^{2}+\frac{4}{9}a+\frac{1}{9}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $14$ | 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: | $\frac{1}{2}a^{15}+1$, $a-1$, $\frac{1}{2}a^{21}+\frac{1}{2}a^{20}+\frac{1}{2}a^{19}+\frac{1}{2}a^{18}+\frac{1}{2}a^{17}+\frac{1}{2}a^{16}+\frac{1}{2}a^{15}+1$, $\frac{1}{6}a^{28}+\frac{1}{3}a^{27}+\frac{1}{6}a^{26}-\frac{1}{6}a^{25}-\frac{1}{3}a^{24}-\frac{1}{6}a^{23}+\frac{1}{6}a^{22}+\frac{1}{3}a^{21}+\frac{1}{6}a^{20}-\frac{1}{6}a^{19}-\frac{1}{3}a^{18}-\frac{1}{6}a^{17}+\frac{1}{6}a^{16}-\frac{1}{6}a^{15}-\frac{1}{3}a^{14}+\frac{1}{3}a^{13}+\frac{2}{3}a^{12}+\frac{1}{3}a^{11}-\frac{1}{3}a^{10}-\frac{2}{3}a^{9}-\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{2}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{2}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{2}{3}a^{28}+\frac{1}{3}a^{27}+\frac{2}{3}a^{26}+\frac{1}{3}a^{25}+\frac{2}{3}a^{24}+\frac{1}{3}a^{23}+\frac{2}{3}a^{22}+\frac{1}{3}a^{21}+\frac{2}{3}a^{20}+\frac{1}{3}a^{19}-\frac{1}{3}a^{18}+\frac{1}{3}a^{17}-\frac{1}{3}a^{16}+\frac{1}{3}a^{15}-\frac{1}{3}a^{14}+\frac{1}{3}a^{13}-\frac{1}{3}a^{12}+\frac{1}{3}a^{11}-\frac{1}{3}a^{10}-\frac{2}{3}a^{9}+\frac{2}{3}a^{8}-\frac{2}{3}a^{7}+\frac{2}{3}a^{6}-\frac{2}{3}a^{5}+\frac{2}{3}a^{4}-\frac{2}{3}a^{3}+\frac{2}{3}a^{2}-\frac{2}{3}a+\frac{7}{3}$, $2a^{28}+2a^{27}+2a^{26}+2a^{25}+\frac{3}{2}a^{24}+\frac{3}{2}a^{23}+\frac{3}{2}a^{22}+\frac{3}{2}a^{21}+\frac{3}{2}a^{20}+\frac{3}{2}a^{19}+a^{18}+a^{17}+\frac{3}{2}a^{16}+a^{15}+a^{14}+a^{13}+a^{12}+a^{11}+a^{10}+a^{9}+a^{7}+a^{6}+a^{5}+a^{3}+a+9$, $\frac{1}{9}a^{28}-\frac{5}{18}a^{27}-\frac{1}{18}a^{26}-\frac{1}{9}a^{25}-\frac{13}{18}a^{24}+\frac{1}{18}a^{23}-\frac{7}{18}a^{22}-\frac{7}{9}a^{21}-\frac{1}{18}a^{20}-\frac{11}{18}a^{19}-\frac{2}{9}a^{18}-\frac{4}{9}a^{17}-\frac{8}{9}a^{16}+\frac{13}{18}a^{15}-\frac{5}{9}a^{14}-\frac{10}{9}a^{13}+\frac{7}{9}a^{12}-\frac{4}{9}a^{11}+\frac{1}{9}a^{10}+\frac{2}{9}a^{9}-\frac{5}{9}a^{8}+\frac{17}{9}a^{7}-\frac{2}{9}a^{6}-\frac{4}{9}a^{5}+\frac{19}{9}a^{4}-\frac{7}{9}a^{3}+\frac{4}{9}a^{2}+\frac{8}{9}a-\frac{7}{9}$, $\frac{1}{2}a^{28}-\frac{1}{2}a^{22}-\frac{1}{2}a^{21}-\frac{1}{2}a^{20}-\frac{1}{2}a^{19}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}+\frac{1}{2}a^{16}+a^{11}-a^{5}+1$, $\frac{5}{9}a^{28}+\frac{11}{18}a^{27}+\frac{13}{18}a^{26}+\frac{17}{18}a^{25}+\frac{8}{9}a^{24}+\frac{7}{9}a^{23}+\frac{5}{9}a^{22}+\frac{11}{18}a^{21}+\frac{2}{9}a^{20}+\frac{4}{9}a^{19}-\frac{1}{9}a^{18}+\frac{5}{18}a^{17}+\frac{1}{18}a^{16}+\frac{1}{9}a^{15}+\frac{2}{9}a^{14}+\frac{4}{9}a^{13}+\frac{8}{9}a^{12}+\frac{7}{9}a^{11}+\frac{5}{9}a^{10}+\frac{1}{9}a^{9}+\frac{2}{9}a^{8}-\frac{5}{9}a^{7}-\frac{1}{9}a^{6}-\frac{2}{9}a^{5}+\frac{5}{9}a^{4}+\frac{10}{9}a^{3}+\frac{2}{9}a^{2}+\frac{4}{9}a+\frac{19}{9}$, $\frac{1}{3}a^{28}+\frac{1}{6}a^{27}+\frac{1}{3}a^{26}+\frac{2}{3}a^{25}+\frac{1}{3}a^{24}+\frac{1}{6}a^{23}-\frac{1}{6}a^{22}+\frac{1}{6}a^{21}+\frac{5}{6}a^{20}+\frac{2}{3}a^{19}-\frac{1}{6}a^{18}-\frac{5}{6}a^{17}+\frac{1}{3}a^{16}+\frac{5}{3}a^{15}+\frac{1}{3}a^{14}-\frac{4}{3}a^{13}-\frac{2}{3}a^{12}+\frac{5}{3}a^{11}+\frac{4}{3}a^{10}-\frac{4}{3}a^{9}-\frac{2}{3}a^{8}+\frac{2}{3}a^{7}+\frac{4}{3}a^{6}-\frac{1}{3}a^{5}-\frac{2}{3}a^{4}+\frac{2}{3}a^{3}-\frac{2}{3}a^{2}+\frac{2}{3}a+\frac{5}{3}$, $\frac{4}{9}a^{28}+\frac{7}{18}a^{27}+\frac{7}{9}a^{26}+\frac{19}{18}a^{25}+\frac{10}{9}a^{24}+\frac{2}{9}a^{23}-\frac{5}{9}a^{22}-\frac{11}{18}a^{21}-\frac{2}{9}a^{20}-\frac{4}{9}a^{19}-\frac{25}{18}a^{18}-\frac{23}{18}a^{17}-\frac{19}{18}a^{16}+\frac{7}{18}a^{15}+\frac{7}{9}a^{14}+\frac{5}{9}a^{13}+\frac{1}{9}a^{12}+\frac{11}{9}a^{11}+\frac{22}{9}a^{10}+\frac{17}{9}a^{9}+\frac{7}{9}a^{8}-\frac{4}{9}a^{7}+\frac{1}{9}a^{6}+\frac{2}{9}a^{5}+\frac{4}{9}a^{4}-\frac{19}{9}a^{3}-\frac{29}{9}a^{2}-\frac{13}{9}a+\frac{17}{9}$, $\frac{1}{3}a^{28}+\frac{1}{6}a^{27}-\frac{2}{3}a^{26}+\frac{1}{6}a^{25}+\frac{1}{3}a^{24}-\frac{5}{6}a^{23}-\frac{1}{6}a^{22}+\frac{2}{3}a^{21}-\frac{2}{3}a^{20}-\frac{1}{3}a^{19}+\frac{4}{3}a^{18}-\frac{1}{3}a^{17}-\frac{7}{6}a^{16}+\frac{7}{6}a^{15}+\frac{1}{3}a^{14}-\frac{7}{3}a^{13}+\frac{1}{3}a^{12}+\frac{5}{3}a^{11}-\frac{5}{3}a^{10}-\frac{1}{3}a^{9}+\frac{7}{3}a^{8}-\frac{4}{3}a^{7}-\frac{5}{3}a^{6}+\frac{8}{3}a^{5}-\frac{2}{3}a^{4}-\frac{10}{3}a^{3}+\frac{7}{3}a^{2}+\frac{5}{3}a-\frac{7}{3}$, $\frac{1}{9}a^{28}+\frac{2}{9}a^{27}+\frac{4}{9}a^{26}+\frac{7}{18}a^{25}-\frac{2}{9}a^{24}+\frac{1}{18}a^{23}+\frac{1}{9}a^{22}+\frac{2}{9}a^{21}-\frac{1}{18}a^{20}-\frac{1}{9}a^{19}-\frac{2}{9}a^{18}+\frac{5}{9}a^{17}-\frac{7}{18}a^{16}-\frac{7}{9}a^{15}+\frac{4}{9}a^{14}+\frac{8}{9}a^{13}-\frac{11}{9}a^{12}-\frac{4}{9}a^{11}+\frac{10}{9}a^{10}+\frac{2}{9}a^{9}-\frac{5}{9}a^{8}+\frac{8}{9}a^{7}+\frac{7}{9}a^{6}-\frac{4}{9}a^{5}+\frac{1}{9}a^{4}+\frac{2}{9}a^{3}-\frac{5}{9}a^{2}-\frac{10}{9}a+\frac{11}{9}$, $\frac{1}{3}a^{28}+\frac{1}{6}a^{27}-\frac{1}{6}a^{26}-\frac{1}{3}a^{25}-\frac{1}{6}a^{24}-\frac{1}{3}a^{23}-\frac{1}{6}a^{22}+\frac{2}{3}a^{21}+\frac{1}{3}a^{20}+\frac{1}{6}a^{19}+\frac{5}{6}a^{18}+\frac{1}{6}a^{17}-\frac{1}{6}a^{16}+\frac{1}{6}a^{15}-\frac{2}{3}a^{14}-\frac{1}{3}a^{13}+\frac{1}{3}a^{12}-\frac{1}{3}a^{11}+\frac{1}{3}a^{10}+\frac{2}{3}a^{9}+\frac{1}{3}a^{8}+\frac{2}{3}a^{7}-\frac{2}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{4}{3}a^{3}-\frac{2}{3}a^{2}+\frac{2}{3}a+\frac{5}{3}$ (assuming GRH) | 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: | \( 20818517115126.277 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{14}\cdot 20818517115126.277 \cdot 1}{2\cdot\sqrt{8948737504120671477993534661511237990596936204288}}\cr\approx \mathstrut & 1.04012866279047 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 8841761993739701954543616000000 |
The 4565 conjugacy class representatives for $S_{29}$ are not computed |
Character table for $S_{29}$ 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 | R | ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.7.0.1}{7} }{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | $15{,}\,{\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ | $22{,}\,{\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.5.0.1}{5} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/13.7.0.1}{7} }$ | $15{,}\,{\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | $22{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $24{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.7.0.1}{7} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/31.9.0.1}{9} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.13.0.1}{13} }{,}\,{\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | $18{,}\,{\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.13.0.1}{13} }{,}\,{\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | $20{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 $29$ | $29$ | $1$ | $28$ | |||
\(27325532669\) | $\Q_{27325532669}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $20$ | $1$ | $20$ | $0$ | 20T1 | $[\ ]^{20}$ | ||
\(121\!\cdots\!817\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ |