Normalized defining polynomial
\( x^{27} - x - 4 \)
Invariants
Degree: | $27$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[1, 13]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-29757847893499620558587041306002149293331489783\) \(\medspace = -\,167\cdot 359\cdot 10245289151\cdot 48\!\cdots\!61\) | 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: | \(52.63\) | 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: | $167^{1/2}359^{1/2}10245289151^{1/2}48446943180119740684317685671961^{1/2}\approx 1.7250463151318464e+23$ | ||
Ramified primes: | \(167\), \(359\), \(10245289151\), \(48446\!\cdots\!71961\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-29757\!\cdots\!89783}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2}a^{14}-\frac{1}{2}a$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{20}-\frac{1}{2}a^{7}$, $\frac{1}{2}a^{21}-\frac{1}{2}a^{8}$, $\frac{1}{2}a^{22}-\frac{1}{2}a^{9}$, $\frac{1}{2}a^{23}-\frac{1}{2}a^{10}$, $\frac{1}{2}a^{24}-\frac{1}{2}a^{11}$, $\frac{1}{2}a^{25}-\frac{1}{2}a^{12}$, $\frac{1}{2}a^{26}-\frac{1}{2}a^{13}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $13$ | 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: | $\frac{1}{2}a^{14}-\frac{1}{2}a-1$, $\frac{1}{2}a^{26}-\frac{1}{2}a^{25}+\frac{1}{2}a^{24}-\frac{1}{2}a^{23}+\frac{1}{2}a^{22}-\frac{1}{2}a^{21}+\frac{1}{2}a^{20}-\frac{1}{2}a^{19}+\frac{1}{2}a^{18}-\frac{1}{2}a^{17}+\frac{1}{2}a^{16}-\frac{1}{2}a^{15}+\frac{1}{2}a^{14}-\frac{1}{2}a^{13}+\frac{1}{2}a^{12}-\frac{1}{2}a^{11}+\frac{1}{2}a^{10}-\frac{1}{2}a^{9}+\frac{1}{2}a^{8}-\frac{1}{2}a^{7}+\frac{1}{2}a^{6}-\frac{1}{2}a^{5}+\frac{1}{2}a^{4}-\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{1}{2}a-1$, $\frac{1}{2}a^{26}+\frac{1}{2}a^{25}+\frac{1}{2}a^{24}+\frac{1}{2}a^{23}+\frac{1}{2}a^{22}+\frac{1}{2}a^{21}+\frac{1}{2}a^{20}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-a^{15}-\frac{3}{2}a^{14}-\frac{3}{2}a^{13}-\frac{3}{2}a^{12}-\frac{3}{2}a^{11}-\frac{3}{2}a^{10}-\frac{3}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}+\frac{1}{2}a^{4}+\frac{3}{2}a^{3}+a^{2}+\frac{3}{2}a+1$, $\frac{1}{2}a^{22}+\frac{1}{2}a^{21}-\frac{1}{2}a^{19}-\frac{1}{2}a^{18}+\frac{1}{2}a^{16}+a^{15}-a^{13}-a^{12}-\frac{1}{2}a^{9}+\frac{1}{2}a^{8}+a^{7}-\frac{1}{2}a^{6}-\frac{5}{2}a^{5}-a^{4}+\frac{3}{2}a^{3}+2a^{2}-1$, $\frac{1}{2}a^{26}-\frac{1}{2}a^{25}+\frac{1}{2}a^{23}-a^{22}+a^{21}-\frac{1}{2}a^{20}-\frac{1}{2}a^{19}+a^{18}-a^{17}+\frac{1}{2}a^{16}+\frac{1}{2}a^{15}-a^{14}+\frac{1}{2}a^{13}+\frac{1}{2}a^{12}-a^{11}+\frac{3}{2}a^{10}-a^{9}+\frac{3}{2}a^{7}-\frac{5}{2}a^{6}+2a^{5}-\frac{3}{2}a^{3}+\frac{3}{2}a^{2}-a-1$, $\frac{1}{2}a^{26}+\frac{1}{2}a^{25}-\frac{1}{2}a^{22}-\frac{1}{2}a^{21}-\frac{1}{2}a^{20}+\frac{1}{2}a^{18}+\frac{1}{2}a^{17}+a^{16}-\frac{3}{2}a^{13}-\frac{1}{2}a^{12}-a^{11}+\frac{1}{2}a^{9}+\frac{3}{2}a^{8}+\frac{3}{2}a^{7}-\frac{1}{2}a^{5}-\frac{3}{2}a^{4}-a^{3}-2a^{2}+1$, $\frac{1}{2}a^{25}-\frac{1}{2}a^{24}+\frac{1}{2}a^{23}-\frac{1}{2}a^{22}+\frac{1}{2}a^{21}-\frac{1}{2}a^{20}+\frac{1}{2}a^{19}-a^{18}+\frac{3}{2}a^{17}-a^{16}+\frac{1}{2}a^{15}-\frac{1}{2}a^{14}+a^{13}-\frac{3}{2}a^{12}+\frac{3}{2}a^{11}-\frac{1}{2}a^{10}+\frac{1}{2}a^{9}-\frac{1}{2}a^{8}+\frac{1}{2}a^{7}-\frac{1}{2}a^{6}+a^{5}-\frac{1}{2}a^{4}+\frac{3}{2}a^{2}-\frac{1}{2}a-1$, $\frac{1}{2}a^{25}-\frac{1}{2}a^{23}-a^{22}-a^{21}+\frac{1}{2}a^{18}-\frac{1}{2}a^{17}+\frac{1}{2}a^{16}+a^{15}+\frac{3}{2}a^{14}+a^{13}-\frac{1}{2}a^{12}-a^{11}-\frac{3}{2}a^{10}-a^{8}-a^{7}-2a^{6}-\frac{1}{2}a^{5}+\frac{3}{2}a^{4}+\frac{5}{2}a^{3}+3a^{2}+\frac{1}{2}a+1$, $\frac{1}{2}a^{26}-a^{19}+a^{18}-\frac{3}{2}a^{16}+\frac{3}{2}a^{15}+\frac{1}{2}a^{14}-\frac{3}{2}a^{13}+a^{12}+2a^{11}-3a^{10}+a^{9}+2a^{8}-3a^{7}+2a^{5}-2a^{4}-\frac{3}{2}a^{3}+\frac{7}{2}a^{2}-\frac{3}{2}a-1$, $\frac{1}{2}a^{26}-\frac{1}{2}a^{25}-\frac{1}{2}a^{23}+\frac{1}{2}a^{22}+\frac{1}{2}a^{20}+\frac{1}{2}a^{19}-a^{18}-a^{16}+\frac{1}{2}a^{15}+\frac{3}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{3}{2}a^{9}+a^{8}-\frac{1}{2}a^{7}+\frac{3}{2}a^{6}-a^{5}+a^{4}-2a^{3}+\frac{1}{2}a^{2}+a-1$, $\frac{3}{2}a^{26}+2a^{25}-2a^{24}-\frac{5}{2}a^{23}+2a^{22}+\frac{5}{2}a^{21}-a^{20}-2a^{19}-a^{18}+\frac{5}{2}a^{17}+3a^{16}-\frac{7}{2}a^{15}-4a^{14}+\frac{7}{2}a^{13}+4a^{12}-2a^{11}-\frac{7}{2}a^{10}-a^{9}+\frac{9}{2}a^{8}+4a^{7}-6a^{6}-6a^{5}+\frac{11}{2}a^{4}+6a^{3}-\frac{7}{2}a^{2}-6a-3$, $\frac{1}{2}a^{25}+\frac{1}{2}a^{24}+\frac{1}{2}a^{22}+\frac{1}{2}a^{21}+\frac{1}{2}a^{20}+\frac{1}{2}a^{18}-\frac{1}{2}a^{16}-a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-a^{10}-\frac{3}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}+2a+1$, $\frac{1}{2}a^{26}+a^{24}-\frac{3}{2}a^{23}+\frac{1}{2}a^{22}-a^{21}+\frac{3}{2}a^{20}-\frac{1}{2}a^{17}-a^{16}+\frac{3}{2}a^{15}-\frac{1}{2}a^{14}+\frac{3}{2}a^{13}-2a^{12}+a^{11}-\frac{1}{2}a^{10}+\frac{3}{2}a^{9}-\frac{3}{2}a^{7}-a^{5}+\frac{7}{2}a^{4}-a^{3}+\frac{1}{2}a^{2}-\frac{7}{2}a-1$ (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: | \( 6097282211234.32 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{13}\cdot 6097282211234.32 \cdot 1}{2\cdot\sqrt{29757847893499620558587041306002149293331489783}}\cr\approx \mathstrut & 0.840763453813830 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 10888869450418352160768000000 |
The 3010 conjugacy class representatives for $S_{27}$ |
Character table for $S_{27}$ |
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 | $24{,}\,{\href{/padicField/2.2.0.1}{2} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | ${\href{/padicField/3.9.0.1}{9} }^{3}$ | $18{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.9.0.1}{9} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.11.0.1}{11} }{,}\,{\href{/padicField/11.4.0.1}{4} }$ | $19{,}\,{\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | $23{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.10.0.1}{10} }^{2}{,}\,{\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | $22{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | $15{,}\,{\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | $18{,}\,{\href{/padicField/41.9.0.1}{9} }$ | ${\href{/padicField/43.14.0.1}{14} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(167\) | $\Q_{167}$ | $x + 162$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{167}$ | $x + 162$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{167}$ | $x + 162$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
167.2.1.2 | $x^{2} + 167$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
167.4.0.1 | $x^{4} + 3 x^{2} + 120 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
167.8.0.1 | $x^{8} + 2 x^{4} + 149 x^{3} + 56 x^{2} + 113 x + 5$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
167.10.0.1 | $x^{10} + 85 x^{5} + 68 x^{4} + 109 x^{3} + 143 x^{2} + 148 x + 5$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
\(359\) | 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 $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
\(10245289151\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | ||
\(484\!\cdots\!961\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $19$ | $1$ | $19$ | $0$ | $C_{19}$ | $[\ ]^{19}$ |