Normalized defining polynomial
\( x^{29} - 4x - 4 \)
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: | \(26146730757746268604771485057902487410941420896256\) \(\medspace = 2^{28}\cdot 53\cdot 179\cdot 10\!\cdots\!23\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(50.59\) | 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))
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Galois root discriminant: | $2^{28/29}53^{1/2}179^{1/2}10267119471732676863546062197607495323^{1/2}\approx 6.094504598928962e+20$ | ||
Ramified primes: | \(2\), \(53\), \(179\), \(10267\!\cdots\!95323\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{97404\!\cdots\!29301}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2}a^{15}$, $\frac{1}{2}a^{16}$, $\frac{1}{2}a^{17}$, $\frac{1}{2}a^{18}$, $\frac{1}{2}a^{19}$, $\frac{1}{2}a^{20}$, $\frac{1}{2}a^{21}$, $\frac{1}{2}a^{22}$, $\frac{1}{2}a^{23}$, $\frac{1}{2}a^{24}$, $\frac{1}{2}a^{25}$, $\frac{1}{2}a^{26}$, $\frac{1}{2}a^{27}$, $\frac{1}{10}a^{28}+\frac{1}{5}a^{27}-\frac{1}{10}a^{26}-\frac{1}{5}a^{25}+\frac{1}{10}a^{24}+\frac{1}{5}a^{23}-\frac{1}{10}a^{22}-\frac{1}{5}a^{21}+\frac{1}{10}a^{20}+\frac{1}{5}a^{19}-\frac{1}{10}a^{18}-\frac{1}{5}a^{17}+\frac{1}{10}a^{16}+\frac{1}{5}a^{15}+\frac{2}{5}a^{14}-\frac{1}{5}a^{13}-\frac{2}{5}a^{12}+\frac{1}{5}a^{11}+\frac{2}{5}a^{10}-\frac{1}{5}a^{9}-\frac{2}{5}a^{8}+\frac{1}{5}a^{7}+\frac{2}{5}a^{6}-\frac{1}{5}a^{5}-\frac{2}{5}a^{4}+\frac{1}{5}a^{3}+\frac{2}{5}a^{2}-\frac{1}{5}a+\frac{1}{5}$
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)
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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)
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Fundamental units: | $\frac{1}{2}a^{28}-\frac{1}{2}a^{27}+\frac{1}{2}a^{26}-\frac{1}{2}a^{25}+\frac{1}{2}a^{24}-\frac{1}{2}a^{23}+\frac{1}{2}a^{22}-\frac{1}{2}a^{21}+\frac{1}{2}a^{20}-\frac{1}{2}a^{19}+\frac{1}{2}a^{18}-\frac{1}{2}a^{17}+\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-3$, $\frac{1}{2}a^{28}-\frac{1}{2}a^{27}+\frac{1}{2}a^{26}-\frac{1}{2}a^{25}+\frac{1}{2}a^{24}-\frac{1}{2}a^{23}+\frac{1}{2}a^{22}-\frac{1}{2}a^{21}+\frac{1}{2}a^{20}-\frac{1}{2}a^{19}+\frac{1}{2}a^{18}-\frac{1}{2}a^{17}+\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-1$, $\frac{1}{2}a^{22}+\frac{1}{2}a^{15}+a^{8}+2a+1$, $\frac{1}{10}a^{28}-\frac{3}{10}a^{27}-\frac{1}{10}a^{26}-\frac{1}{5}a^{25}+\frac{1}{10}a^{24}+\frac{1}{5}a^{23}-\frac{1}{10}a^{22}+\frac{3}{10}a^{21}+\frac{1}{10}a^{20}+\frac{1}{5}a^{19}+\frac{2}{5}a^{18}+\frac{3}{10}a^{17}-\frac{2}{5}a^{16}-\frac{3}{10}a^{15}-\frac{3}{5}a^{14}-\frac{1}{5}a^{13}-\frac{2}{5}a^{12}+\frac{1}{5}a^{11}-\frac{3}{5}a^{10}-\frac{1}{5}a^{9}+\frac{3}{5}a^{8}+\frac{1}{5}a^{7}+\frac{7}{5}a^{6}+\frac{4}{5}a^{5}+\frac{3}{5}a^{4}-\frac{4}{5}a^{3}+\frac{2}{5}a^{2}-\frac{1}{5}a-\frac{9}{5}$, $\frac{1}{10}a^{28}-\frac{3}{10}a^{27}+\frac{2}{5}a^{26}-\frac{7}{10}a^{25}+\frac{3}{5}a^{24}-\frac{3}{10}a^{23}+\frac{2}{5}a^{22}-\frac{7}{10}a^{21}+\frac{3}{5}a^{20}-\frac{3}{10}a^{19}-\frac{1}{10}a^{18}+\frac{3}{10}a^{17}+\frac{1}{10}a^{16}+\frac{1}{5}a^{15}-\frac{3}{5}a^{14}+\frac{4}{5}a^{13}-\frac{2}{5}a^{12}+\frac{1}{5}a^{11}-\frac{3}{5}a^{10}+\frac{4}{5}a^{9}-\frac{2}{5}a^{8}+\frac{1}{5}a^{7}+\frac{2}{5}a^{6}-\frac{1}{5}a^{5}-\frac{2}{5}a^{4}-\frac{4}{5}a^{3}+\frac{2}{5}a^{2}-\frac{1}{5}a+\frac{1}{5}$, $\frac{3}{10}a^{28}-\frac{2}{5}a^{27}-\frac{3}{10}a^{26}-\frac{1}{10}a^{25}-\frac{1}{5}a^{24}+\frac{3}{5}a^{23}-\frac{3}{10}a^{22}+\frac{2}{5}a^{21}-\frac{1}{5}a^{20}+\frac{3}{5}a^{19}-\frac{3}{10}a^{18}-\frac{1}{10}a^{17}-\frac{7}{10}a^{16}+\frac{1}{10}a^{15}+\frac{1}{5}a^{14}+\frac{2}{5}a^{13}-\frac{1}{5}a^{12}-\frac{2}{5}a^{11}+\frac{1}{5}a^{10}-\frac{3}{5}a^{9}-\frac{1}{5}a^{8}-\frac{7}{5}a^{7}+\frac{1}{5}a^{6}+\frac{2}{5}a^{5}+\frac{9}{5}a^{4}+\frac{3}{5}a^{3}+\frac{1}{5}a^{2}+\frac{2}{5}a-\frac{7}{5}$, $\frac{2}{5}a^{28}-\frac{1}{5}a^{27}+\frac{1}{10}a^{26}+\frac{1}{5}a^{25}+\frac{2}{5}a^{24}+\frac{3}{10}a^{23}-\frac{2}{5}a^{22}-\frac{3}{10}a^{21}-\frac{3}{5}a^{20}+\frac{3}{10}a^{19}+\frac{1}{10}a^{18}+\frac{1}{5}a^{17}-\frac{3}{5}a^{16}-\frac{7}{10}a^{15}-\frac{2}{5}a^{14}+\frac{1}{5}a^{13}+\frac{2}{5}a^{12}-\frac{1}{5}a^{11}-\frac{2}{5}a^{10}-\frac{4}{5}a^{9}+\frac{2}{5}a^{8}+\frac{4}{5}a^{7}+\frac{3}{5}a^{6}-\frac{4}{5}a^{5}-\frac{8}{5}a^{4}-\frac{6}{5}a^{3}+\frac{3}{5}a^{2}+\frac{11}{5}a-\frac{1}{5}$, $\frac{1}{5}a^{28}-\frac{3}{5}a^{27}-\frac{1}{5}a^{26}+\frac{3}{5}a^{25}+\frac{1}{5}a^{24}-\frac{3}{5}a^{23}-\frac{1}{5}a^{22}+\frac{3}{5}a^{21}+\frac{1}{5}a^{20}-\frac{3}{5}a^{19}-\frac{1}{5}a^{18}+\frac{3}{5}a^{17}+\frac{1}{5}a^{16}-\frac{3}{5}a^{15}-\frac{1}{5}a^{14}+\frac{3}{5}a^{13}+\frac{1}{5}a^{12}-\frac{3}{5}a^{11}-\frac{6}{5}a^{10}+\frac{3}{5}a^{9}+\frac{6}{5}a^{8}-\frac{3}{5}a^{7}-\frac{6}{5}a^{6}+\frac{3}{5}a^{5}+\frac{11}{5}a^{4}-\frac{3}{5}a^{3}-\frac{11}{5}a^{2}+\frac{3}{5}a+\frac{7}{5}$, $\frac{9}{10}a^{28}-\frac{6}{5}a^{27}+\frac{3}{5}a^{26}-\frac{4}{5}a^{25}+\frac{7}{5}a^{24}-\frac{7}{10}a^{23}+\frac{11}{10}a^{22}-\frac{9}{5}a^{21}+\frac{9}{10}a^{20}-\frac{7}{10}a^{19}+\frac{8}{5}a^{18}-\frac{3}{10}a^{17}+\frac{2}{5}a^{16}-\frac{17}{10}a^{15}+\frac{3}{5}a^{14}+\frac{1}{5}a^{13}+\frac{7}{5}a^{12}-\frac{1}{5}a^{11}-\frac{2}{5}a^{10}-\frac{9}{5}a^{9}+\frac{7}{5}a^{8}+\frac{4}{5}a^{7}+\frac{8}{5}a^{6}-\frac{4}{5}a^{5}-\frac{8}{5}a^{4}-\frac{6}{5}a^{3}+\frac{8}{5}a^{2}+\frac{11}{5}a-\frac{11}{5}$, $\frac{1}{10}a^{28}+\frac{1}{5}a^{27}-\frac{1}{10}a^{26}-\frac{1}{5}a^{25}+\frac{3}{5}a^{24}-\frac{4}{5}a^{23}+\frac{2}{5}a^{22}-\frac{1}{5}a^{21}-\frac{2}{5}a^{20}-\frac{3}{10}a^{19}+\frac{2}{5}a^{18}-\frac{7}{10}a^{17}+\frac{3}{5}a^{16}+\frac{1}{5}a^{15}+\frac{2}{5}a^{14}+\frac{4}{5}a^{13}+\frac{3}{5}a^{12}+\frac{1}{5}a^{11}+\frac{7}{5}a^{10}-\frac{1}{5}a^{9}+\frac{3}{5}a^{8}+\frac{1}{5}a^{7}-\frac{3}{5}a^{6}-\frac{1}{5}a^{5}-\frac{2}{5}a^{4}-\frac{9}{5}a^{3}+\frac{2}{5}a^{2}-\frac{1}{5}a-\frac{9}{5}$, $\frac{1}{2}a^{27}+\frac{1}{2}a^{25}-\frac{1}{2}a^{24}-\frac{1}{2}a^{22}-\frac{1}{2}a^{21}-\frac{1}{2}a^{19}-\frac{1}{2}a^{17}-a^{16}-\frac{1}{2}a^{15}-a^{14}+a^{7}+a^{6}+2a^{5}+2a^{4}+a^{3}+a^{2}+1$, $\frac{3}{5}a^{28}-\frac{3}{10}a^{27}-\frac{1}{10}a^{26}-\frac{7}{10}a^{25}+\frac{3}{5}a^{24}-\frac{3}{10}a^{23}-\frac{3}{5}a^{22}+\frac{3}{10}a^{21}+\frac{3}{5}a^{20}-\frac{3}{10}a^{19}-\frac{1}{10}a^{18}+\frac{4}{5}a^{17}+\frac{3}{5}a^{16}-\frac{4}{5}a^{15}+\frac{2}{5}a^{14}+\frac{4}{5}a^{13}-\frac{2}{5}a^{12}-\frac{4}{5}a^{11}+\frac{2}{5}a^{10}+\frac{4}{5}a^{9}-\frac{7}{5}a^{8}+\frac{1}{5}a^{7}+\frac{7}{5}a^{6}-\frac{1}{5}a^{5}-\frac{7}{5}a^{4}+\frac{6}{5}a^{3}+\frac{7}{5}a^{2}-\frac{11}{5}a-\frac{19}{5}$, $\frac{1}{5}a^{28}-\frac{1}{10}a^{27}+\frac{3}{10}a^{26}-\frac{2}{5}a^{25}+\frac{1}{5}a^{24}-\frac{3}{5}a^{23}+\frac{3}{10}a^{22}+\frac{1}{10}a^{21}+\frac{1}{5}a^{20}-\frac{1}{10}a^{19}-\frac{1}{5}a^{18}+\frac{1}{10}a^{17}+\frac{1}{5}a^{16}-\frac{3}{5}a^{15}+\frac{4}{5}a^{14}-\frac{2}{5}a^{13}+\frac{1}{5}a^{12}-\frac{3}{5}a^{11}+\frac{4}{5}a^{10}+\frac{3}{5}a^{9}-\frac{4}{5}a^{8}-\frac{3}{5}a^{7}-\frac{1}{5}a^{6}-\frac{2}{5}a^{5}+\frac{1}{5}a^{4}+\frac{2}{5}a^{3}+\frac{4}{5}a^{2}-\frac{2}{5}a-\frac{3}{5}$, $\frac{1}{2}a^{28}-\frac{1}{2}a^{26}+\frac{1}{2}a^{25}+\frac{1}{2}a^{24}-a^{23}+\frac{1}{2}a^{21}-\frac{1}{2}a^{20}-\frac{1}{2}a^{19}+\frac{1}{2}a^{18}+\frac{1}{2}a^{17}-\frac{1}{2}a^{16}+a^{14}-a^{12}+a^{10}-a^{9}-a^{8}+2a^{7}-2a^{5}+a^{4}+a^{3}-a^{2}-a-1$ (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)]
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Regulator: | \( 28349352931361.156 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{14}\cdot 28349352931361.156 \cdot 1}{2\cdot\sqrt{26146730757746268604771485057902487410941420896256}}\cr\approx \mathstrut & 0.828614655435449 \end{aligned}\] (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 | $22{,}\,{\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ | $23{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | $17{,}\,{\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | $29$ | ${\href{/padicField/13.11.0.1}{11} }{,}\,{\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.14.0.1}{14} }{,}\,{\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | $29$ | $21{,}\,{\href{/padicField/23.8.0.1}{8} }$ | ${\href{/padicField/29.14.0.1}{14} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $27{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | $21{,}\,{\href{/padicField/37.8.0.1}{8} }$ | $18{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.13.0.1}{13} }{,}\,{\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.6.0.1}{6} }$ | $23{,}\,{\href{/padicField/47.6.0.1}{6} }$ | R | $20{,}\,{\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 $29$ | $29$ | $1$ | $28$ | |||
\(53\) | $\Q_{53}$ | $x + 51$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{53}$ | $x + 51$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
53.2.1.2 | $x^{2} + 106$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
53.8.0.1 | $x^{8} + 8 x^{4} + 29 x^{3} + 18 x^{2} + x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
53.17.0.1 | $x^{17} + 12 x + 51$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ | |
\(179\) | $\Q_{179}$ | $x + 177$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
179.2.1.1 | $x^{2} + 358$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
179.4.0.1 | $x^{4} + x^{2} + 109 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
179.5.0.1 | $x^{5} + 2 x + 177$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
179.17.0.1 | $x^{17} + 4 x + 177$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ | |
\(102\!\cdots\!323\) | $\Q_{10\!\cdots\!23}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{10\!\cdots\!23}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{10\!\cdots\!23}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ |