Normalized defining polynomial
\( x^{29} - 3 \)
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)
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Discriminant: | \(58740423215346229521832734199516718276445839672493539709\) \(\medspace = 3^{28}\cdot 29^{29}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(83.77\) | 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: | $3^{28/29}29^{839/812}\approx 93.69001963618167$ | ||
Ramified primes: | \(3\), \(29\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{29}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$
Monogenic: | Yes | |
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)
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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: | $a^{15}+a+1$, $a^{28}-a^{26}+2a^{25}-a^{24}+a^{23}+a^{22}-a^{21}+2a^{20}+2a^{17}-a^{16}+a^{15}+a^{14}-2a^{13}+a^{12}-a^{11}-2a^{10}+2a^{9}-3a^{8}+a^{6}-4a^{5}+3a^{4}-3a^{3}-2a^{2}+3a-5$, $a^{28}-a^{27}+a^{26}-a^{25}+a^{24}-a^{23}+a^{22}-a^{21}+a^{20}-a^{19}+a^{18}-a^{17}+a^{16}-a^{15}+a^{14}-a^{13}+a^{12}-a^{11}+a^{10}-a^{9}+a^{8}-a^{7}+a^{6}-a^{5}+a^{4}-a^{3}+a^{2}-2$, $3a^{28}-3a^{27}+2a^{26}-a^{25}+a^{24}-2a^{23}+3a^{22}-3a^{21}+2a^{20}-a^{19}+a^{16}-2a^{15}+a^{14}+a^{13}-2a^{12}+2a^{11}-2a^{10}+a^{9}-a^{8}+3a^{7}-5a^{6}+6a^{5}-5a^{4}+3a^{3}-3a^{2}+5a-7$, $a^{28}+a^{27}+2a^{25}+2a^{24}+a^{23}+a^{22}-a^{20}+a^{19}+2a^{18}+2a^{17}+3a^{16}+a^{13}+a^{12}+4a^{11}+5a^{10}+2a^{9}+3a^{8}+3a^{7}+5a^{5}+5a^{4}+3a^{3}+6a^{2}+5a+1$, $4a^{28}-4a^{27}+8a^{26}-a^{25}-a^{24}+9a^{23}-6a^{22}+4a^{21}+3a^{20}-7a^{19}+8a^{18}-7a^{17}-3a^{16}+6a^{15}-14a^{14}+4a^{13}-2a^{12}-12a^{11}+8a^{10}-12a^{9}-a^{8}+6a^{7}-16a^{6}+12a^{5}-2a^{4}-9a^{3}+18a^{2}-10a+8$, $3a^{28}+a^{27}+2a^{26}+3a^{25}+4a^{23}+a^{22}+3a^{21}+3a^{20}+5a^{18}+a^{17}+4a^{16}+3a^{15}+a^{14}+5a^{13}+2a^{12}+5a^{11}+2a^{10}+3a^{9}+5a^{8}+3a^{7}+6a^{6}+2a^{5}+5a^{4}+4a^{3}+5a^{2}+7a+1$, $3a^{28}+a^{27}+4a^{25}+6a^{24}+3a^{23}-a^{22}+2a^{21}+5a^{20}+a^{19}-4a^{18}-a^{17}+5a^{16}+a^{15}-4a^{14}-a^{13}+6a^{12}+2a^{11}-6a^{10}-4a^{9}+2a^{8}-2a^{7}-12a^{6}-10a^{5}-3a^{4}-2a^{3}-10a^{2}-8a+4$, $2a^{28}-a^{27}-a^{26}+a^{25}-a^{24}+2a^{23}-3a^{22}+2a^{21}+a^{18}-3a^{17}+4a^{16}-2a^{15}-a^{14}+a^{13}-3a^{12}+4a^{11}-4a^{10}-a^{9}+4a^{8}-4a^{7}+5a^{6}-5a^{5}+3a^{4}+4a^{3}-5a^{2}+5a-5$, $6a^{28}+a^{27}-6a^{26}+6a^{24}-6a^{22}-2a^{21}+4a^{20}+2a^{19}-5a^{18}-5a^{17}+5a^{16}+7a^{15}-5a^{14}-7a^{13}+7a^{12}+9a^{11}-7a^{10}-9a^{9}+6a^{8}+7a^{7}-6a^{6}-10a^{5}+a^{4}+10a^{3}-13a-1$, $5a^{28}+7a^{27}+2a^{26}-3a^{25}-4a^{24}-a^{22}-2a^{21}+6a^{20}+9a^{19}+a^{18}-8a^{17}-7a^{16}+2a^{15}+4a^{14}+2a^{13}+5a^{12}+7a^{11}-13a^{9}-10a^{8}+8a^{7}+12a^{6}+5a^{5}-13a-11$, $11a^{28}+9a^{27}-19a^{26}-7a^{25}+19a^{24}+a^{23}-18a^{22}+10a^{21}+23a^{20}-15a^{19}-18a^{18}+19a^{17}+7a^{16}-28a^{15}-3a^{14}+33a^{13}-3a^{12}-26a^{11}+21a^{10}+25a^{9}-30a^{8}-24a^{7}+32a^{6}+7a^{5}-41a^{4}+11a^{3}+49a^{2}-14a-38$, $4a^{28}-8a^{27}+3a^{26}+3a^{25}-3a^{24}+5a^{23}-4a^{22}+3a^{21}+3a^{20}-10a^{19}+8a^{18}-6a^{16}+9a^{15}-5a^{14}+2a^{13}-10a^{11}+10a^{10}-9a^{9}-5a^{8}+16a^{7}-8a^{6}+6a^{5}+7a^{4}-5a^{3}+10a^{2}-17a-2$, $3a^{28}+3a^{27}-4a^{25}-3a^{24}+a^{23}+4a^{22}+4a^{21}-2a^{20}-5a^{19}-3a^{18}+a^{17}+8a^{16}+a^{15}-a^{14}-9a^{13}-a^{12}+3a^{11}+8a^{10}+2a^{9}-4a^{8}-9a^{7}+3a^{5}+12a^{4}-2a^{3}-4a^{2}-12a+1$ (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: | \( 49661461097293670 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{14}\cdot 49661461097293670 \cdot 1}{2\cdot\sqrt{58740423215346229521832734199516718276445839672493539709}}\cr\approx \mathstrut & 0.968431961664099 \end{aligned}\] (assuming GRH)
Galois group
A solvable group of order 812 |
The 29 conjugacy class representatives for $F_{29}$ |
Character table for $F_{29}$ |
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 | $28{,}\,{\href{/padicField/2.1.0.1}{1} }$ | R | ${\href{/padicField/5.14.0.1}{14} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.7.0.1}{7} }^{4}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $28{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.14.0.1}{14} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.4.0.1}{4} }^{7}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $28{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.7.0.1}{7} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | R | $28{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $28{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.4.0.1}{4} }^{7}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $28{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $28{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.7.0.1}{7} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $29$ |
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 $29$ | $29$ | $1$ | $28$ | |||
\(29\) | Deg $29$ | $29$ | $1$ | $29$ |