Normalized defining polynomial
\( x^{27} - 3x - 3 \)
Invariants
Degree: | $27$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[1, 13]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-43207461914112074238522908041698612863100209996211\) \(\medspace = -\,3^{27}\cdot 71\cdot 337\cdot 14619373\cdot 78563389\cdot 206180539890995287\) | 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: | \(68.92\) | 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: | not computed | ||
Ramified primes: | \(3\), \(71\), \(337\), \(14619373\), \(78563389\), \(206180539890995287\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | $\Q(\sqrt{-16998\!\cdots\!71259}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $\frac{1}{5}a^{26}-\frac{2}{5}a^{25}-\frac{1}{5}a^{24}+\frac{2}{5}a^{23}+\frac{1}{5}a^{22}-\frac{2}{5}a^{21}-\frac{1}{5}a^{20}+\frac{2}{5}a^{19}+\frac{1}{5}a^{18}-\frac{2}{5}a^{17}-\frac{1}{5}a^{16}+\frac{2}{5}a^{15}+\frac{1}{5}a^{14}-\frac{2}{5}a^{13}-\frac{1}{5}a^{12}+\frac{2}{5}a^{11}+\frac{1}{5}a^{10}-\frac{2}{5}a^{9}-\frac{1}{5}a^{8}+\frac{2}{5}a^{7}+\frac{1}{5}a^{6}-\frac{2}{5}a^{5}-\frac{1}{5}a^{4}+\frac{2}{5}a^{3}+\frac{1}{5}a^{2}-\frac{2}{5}a+\frac{1}{5}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $13$ | 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)
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Fundamental units: | $a+1$, $a^{14}+2a+1$, $2a^{26}-2a^{25}+a^{24}-a^{22}+2a^{21}-2a^{20}+2a^{19}-a^{18}+a^{16}-2a^{15}+2a^{14}-2a^{13}+a^{12}-a^{10}+2a^{9}-3a^{8}+2a^{7}-2a^{6}+2a^{4}-2a^{3}+4a^{2}-2a-5$, $\frac{2}{5}a^{26}-\frac{4}{5}a^{25}+\frac{3}{5}a^{24}-\frac{6}{5}a^{23}+\frac{7}{5}a^{22}-\frac{4}{5}a^{21}+\frac{8}{5}a^{20}-\frac{1}{5}a^{19}+\frac{7}{5}a^{18}-\frac{4}{5}a^{17}+\frac{8}{5}a^{16}-\frac{6}{5}a^{15}+\frac{2}{5}a^{14}-\frac{9}{5}a^{13}+\frac{3}{5}a^{12}-\frac{6}{5}a^{11}+\frac{2}{5}a^{10}-\frac{4}{5}a^{9}+\frac{8}{5}a^{8}+\frac{4}{5}a^{7}+\frac{12}{5}a^{6}+\frac{1}{5}a^{5}+\frac{8}{5}a^{4}+\frac{4}{5}a^{3}+\frac{2}{5}a^{2}-\frac{9}{5}a-\frac{13}{5}$, $a^{24}-a^{23}+a^{21}-a^{20}-a^{19}+a^{17}-a^{16}+2a^{15}+a^{14}-a^{13}-a^{12}+a^{11}-a^{10}-2a^{9}+a^{8}+2a^{5}+2a^{4}-3a^{3}-a^{2}+a-1$, $\frac{3}{5}a^{26}-\frac{6}{5}a^{25}+\frac{2}{5}a^{24}-\frac{4}{5}a^{23}-\frac{7}{5}a^{22}-\frac{1}{5}a^{21}-\frac{8}{5}a^{20}+\frac{1}{5}a^{19}-\frac{2}{5}a^{18}-\frac{6}{5}a^{17}+\frac{2}{5}a^{16}-\frac{4}{5}a^{15}+\frac{8}{5}a^{14}+\frac{4}{5}a^{13}+\frac{7}{5}a^{12}+\frac{1}{5}a^{11}+\frac{13}{5}a^{10}+\frac{4}{5}a^{9}+\frac{22}{5}a^{8}+\frac{6}{5}a^{7}+\frac{8}{5}a^{6}+\frac{9}{5}a^{5}+\frac{2}{5}a^{4}+\frac{16}{5}a^{3}-\frac{2}{5}a^{2}-\frac{6}{5}a-\frac{22}{5}$, $\frac{3}{5}a^{26}+\frac{4}{5}a^{25}-\frac{3}{5}a^{24}+\frac{1}{5}a^{23}+\frac{3}{5}a^{22}-\frac{1}{5}a^{21}-\frac{3}{5}a^{20}+\frac{1}{5}a^{19}+\frac{8}{5}a^{18}-\frac{1}{5}a^{17}-\frac{3}{5}a^{16}+\frac{1}{5}a^{15}+\frac{3}{5}a^{14}-\frac{1}{5}a^{13}-\frac{3}{5}a^{12}+\frac{11}{5}a^{11}+\frac{3}{5}a^{10}-\frac{6}{5}a^{9}-\frac{3}{5}a^{8}+\frac{6}{5}a^{7}+\frac{8}{5}a^{6}-\frac{6}{5}a^{5}+\frac{7}{5}a^{4}+\frac{11}{5}a^{3}-\frac{7}{5}a^{2}-\frac{11}{5}a-\frac{7}{5}$, $a^{26}-a^{25}+a^{24}-a^{23}+2a^{22}-a^{21}+a^{20}-2a^{19}+a^{18}-a^{17}+2a^{16}-a^{15}-2a^{13}+a^{12}+2a^{10}-a^{9}-2a^{7}+a^{6}-a^{3}-2$, $\frac{14}{5}a^{26}+\frac{17}{5}a^{25}+\frac{16}{5}a^{24}-\frac{2}{5}a^{23}-\frac{11}{5}a^{22}-\frac{23}{5}a^{21}-\frac{24}{5}a^{20}-\frac{7}{5}a^{19}+\frac{9}{5}a^{18}+\frac{27}{5}a^{17}+\frac{31}{5}a^{16}+\frac{23}{5}a^{15}-\frac{1}{5}a^{14}-\frac{28}{5}a^{13}-\frac{44}{5}a^{12}-\frac{32}{5}a^{11}-\frac{16}{5}a^{10}+\frac{22}{5}a^{9}+\frac{56}{5}a^{8}+\frac{53}{5}a^{7}+\frac{34}{5}a^{6}-\frac{13}{5}a^{5}-\frac{54}{5}a^{4}-\frac{77}{5}a^{3}-\frac{61}{5}a^{2}-\frac{8}{5}a+\frac{14}{5}$, $\frac{12}{5}a^{26}-\frac{19}{5}a^{25}+\frac{8}{5}a^{24}-\frac{6}{5}a^{23}-\frac{3}{5}a^{22}-\frac{9}{5}a^{21}+\frac{13}{5}a^{20}-\frac{26}{5}a^{19}+\frac{12}{5}a^{18}-\frac{9}{5}a^{17}-\frac{7}{5}a^{16}-\frac{6}{5}a^{15}+\frac{7}{5}a^{14}-\frac{29}{5}a^{13}+\frac{13}{5}a^{12}-\frac{11}{5}a^{11}-\frac{13}{5}a^{10}-\frac{4}{5}a^{9}-\frac{2}{5}a^{8}-\frac{31}{5}a^{7}+\frac{12}{5}a^{6}-\frac{14}{5}a^{5}-\frac{22}{5}a^{4}-\frac{1}{5}a^{3}-\frac{13}{5}a^{2}-\frac{34}{5}a-\frac{23}{5}$, $\frac{14}{5}a^{26}-\frac{3}{5}a^{25}-\frac{9}{5}a^{24}+\frac{23}{5}a^{23}-\frac{21}{5}a^{22}+\frac{2}{5}a^{21}+\frac{11}{5}a^{20}-\frac{22}{5}a^{19}+\frac{24}{5}a^{18}-\frac{8}{5}a^{17}-\frac{14}{5}a^{16}+\frac{28}{5}a^{15}-\frac{26}{5}a^{14}+\frac{12}{5}a^{13}+\frac{11}{5}a^{12}-\frac{32}{5}a^{11}+\frac{29}{5}a^{10}-\frac{18}{5}a^{9}-\frac{9}{5}a^{8}+\frac{38}{5}a^{7}-\frac{31}{5}a^{6}+\frac{17}{5}a^{5}+\frac{1}{5}a^{4}-\frac{37}{5}a^{3}+\frac{44}{5}a^{2}-\frac{23}{5}a-\frac{56}{5}$, $\frac{9}{5}a^{26}+\frac{2}{5}a^{25}+\frac{11}{5}a^{24}-\frac{7}{5}a^{23}-\frac{1}{5}a^{22}-\frac{13}{5}a^{21}+\frac{6}{5}a^{20}-\frac{2}{5}a^{19}+\frac{9}{5}a^{18}-\frac{8}{5}a^{17}+\frac{1}{5}a^{16}-\frac{7}{5}a^{15}+\frac{9}{5}a^{14}+\frac{2}{5}a^{13}+\frac{6}{5}a^{12}-\frac{7}{5}a^{11}-\frac{1}{5}a^{10}-\frac{3}{5}a^{9}+\frac{6}{5}a^{8}+\frac{3}{5}a^{7}+\frac{4}{5}a^{6}-\frac{3}{5}a^{5}-\frac{4}{5}a^{4}-\frac{12}{5}a^{3}-\frac{16}{5}a^{2}-\frac{13}{5}a-\frac{16}{5}$, $4a^{26}-3a^{25}-4a^{24}+4a^{23}+a^{22}-6a^{21}+4a^{20}+5a^{19}-5a^{18}+a^{17}+9a^{16}-2a^{15}-3a^{14}+10a^{13}+2a^{12}-8a^{11}+7a^{10}+6a^{9}-10a^{8}+9a^{6}-11a^{5}-11a^{4}+9a^{3}-7a^{2}-18a-5$ (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)]
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Regulator: | \( 123801473306894.75 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{13}\cdot 123801473306894.75 \cdot 1}{2\cdot\sqrt{43207461914112074238522908041698612863100209996211}}\cr\approx \mathstrut & 0.448006921621206 \end{aligned}\] (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 | $23{,}\,{\href{/padicField/2.4.0.1}{4} }$ | R | ${\href{/padicField/5.9.0.1}{9} }{,}\,{\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.7.0.1}{7} }{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.11.0.1}{11} }{,}\,{\href{/padicField/11.4.0.1}{4} }$ | $20{,}\,{\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.13.0.1}{13} }$ | ${\href{/padicField/23.14.0.1}{14} }{,}\,{\href{/padicField/23.4.0.1}{4} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | $21{,}\,{\href{/padicField/31.6.0.1}{6} }$ | $26{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | ${\href{/padicField/43.13.0.1}{13} }{,}\,{\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | $16{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | $15{,}\,{\href{/padicField/59.12.0.1}{12} }$ |
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 $27$ | $27$ | $1$ | $27$ | |||
\(71\) | $\Q_{71}$ | $x + 64$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $24$ | $1$ | $24$ | $0$ | $C_{24}$ | $[\ ]^{24}$ | ||
\(337\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | ||
\(14619373\) | $\Q_{14619373}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | ||
\(78563389\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $23$ | $1$ | $23$ | $0$ | $C_{23}$ | $[\ ]^{23}$ | ||
\(206180539890995287\) | 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 $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ |