Normalized defining polynomial
\( x^{29} - 2 \)
Invariants
Degree: | $29$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[1, 14]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(689258003388715552418381717014986231255943677476864\) \(\medspace = 2^{28}\cdot 29^{29}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(56.63\) | 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: | $2^{28/29}29^{839/812}\approx 63.33943474262775$ | ||
Ramified primes: | \(2\), \(29\) | 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)
| |
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: | $a-1$, $a^{15}+a+1$, $a^{16}+a^{15}+a^{2}+a+1$, $a^{27}+a^{25}+a^{23}+a^{21}+a^{19}+a^{17}+a^{15}+a^{13}+a^{11}+a^{9}+a^{7}+a^{5}+a^{3}+a+1$, $a^{27}-a^{25}+a^{23}-a^{21}+a^{19}-a^{17}+a^{16}+a^{15}-a^{14}+a^{12}-a^{10}+a^{8}-a^{6}+a^{4}+a-1$, $a^{26}-a^{25}+a^{24}-a^{23}-a^{21}+2a^{20}-2a^{19}+a^{18}-a^{17}+a^{16}-2a^{15}+3a^{14}-a^{13}-a^{11}+2a^{10}-2a^{9}+a^{8}-2a^{5}+a^{4}+a-1$, $a^{28}-2a^{26}+a^{25}+a^{24}-a^{22}-a^{21}+3a^{20}-a^{19}-a^{18}+a^{16}+2a^{15}-3a^{14}+2a^{12}-3a^{9}+3a^{8}+a^{7}-2a^{6}-a^{5}+3a^{3}-2a^{2}-2a+1$, $a^{28}+a^{26}+a^{25}+a^{24}+a^{23}+a^{22}+a^{21}+2a^{19}+a^{18}-a^{16}+a^{15}-2a^{13}-a^{12}-a^{11}-a^{10}-2a^{9}-2a^{7}-2a^{6}-a^{5}-a^{3}-2a^{2}+a+1$, $a^{24}+a^{23}+a^{21}+a^{20}+a^{19}-a^{15}-a^{14}-a^{11}-a^{7}-2a^{4}-a^{3}-1$, $a^{28}-a^{26}+2a^{25}-a^{24}+a^{23}-2a^{21}+3a^{20}-2a^{19}-a^{18}+3a^{17}-3a^{16}+2a^{15}+2a^{14}-3a^{13}+2a^{12}-a^{11}+a^{9}-3a^{8}+3a^{7}-a^{6}-a^{5}+5a^{4}-5a^{3}+4a-5$, $2a^{28}-a^{27}+a^{25}-2a^{24}+a^{22}-2a^{21}+a^{20}+a^{19}-2a^{18}+a^{17}+a^{16}-2a^{15}+2a^{14}-2a^{12}+2a^{11}-2a^{9}+3a^{8}-a^{7}-2a^{6}+2a^{5}-2a^{4}-2a^{3}+3a^{2}-a+1$, $2a^{28}-3a^{26}-a^{25}+2a^{24}+a^{23}+a^{21}-a^{20}-4a^{19}-3a^{18}+a^{16}-3a^{12}-4a^{11}+3a^{9}+2a^{8}+a^{7}+2a^{6}-4a^{4}+6a^{2}+3a+1$, $a^{28}-a^{27}+a^{26}-a^{25}-a^{24}-a^{23}+a^{22}+a^{21}+a^{20}+2a^{19}+a^{15}+3a^{13}+a^{12}-a^{10}-a^{9}-a^{8}-a^{7}+3a^{6}-2a^{5}-2a^{3}-2a^{2}-3a-1$, $a^{28}-a^{27}-2a^{26}-2a^{25}+a^{24}+4a^{23}+2a^{22}-2a^{21}-4a^{20}-2a^{19}+3a^{18}+6a^{17}+a^{16}-5a^{15}-5a^{14}+a^{13}+6a^{12}+5a^{11}-2a^{10}-6a^{9}-3a^{8}+4a^{7}+6a^{6}+2a^{5}-3a^{4}-4a^{3}-2a^{2}+3a+5$ (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: | \( 170509022606467.78 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{14}\cdot 170509022606467.78 \cdot 1}{2\cdot\sqrt{689258003388715552418381717014986231255943677476864}}\cr\approx \mathstrut & 0.970676718152966 \end{aligned}\] (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 | R | $28{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.14.0.1}{14} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.7.0.1}{7} }^{4}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $28{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.14.0.1}{14} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.4.0.1}{4} }^{7}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $28{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.7.0.1}{7} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | 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 |
---|---|---|---|---|---|---|---|
\(2\) | Deg $29$ | $29$ | $1$ | $28$ | |||
\(29\) | Deg $29$ | $29$ | $1$ | $29$ |