Normalized defining polynomial
\( x^{27} - x - 2 \)
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)
| |
Discriminant: | \(-29757847887343501070113214188474091663330902016\) \(\medspace = -\,2^{27}\cdot 47\cdot 47\!\cdots\!51\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
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: | not computed | ||
Ramified primes: | \(2\), \(47\), \(47173\!\cdots\!88051\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-44342\!\cdots\!76794}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$
Monogenic: | Yes | |
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)
| |
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: | $a^{14}+a+1$, $a^{2}+a+1$, $a^{6}+a^{3}+1$, $a^{18}+a^{9}+1$, $a^{25}+a^{23}+a^{21}+a^{19}+a^{17}+a^{15}+a^{13}+a^{11}+a^{9}+a^{7}+a^{5}+a^{3}+a^{2}+a+1$, $a^{25}-a^{23}+a^{21}-a^{19}+a^{17}-a^{15}+a^{13}-a^{11}+a^{9}-a^{7}+a^{2}-a-1$, $2a^{26}-a^{22}+a^{21}-a^{20}-a^{18}-a^{17}+2a^{16}+a^{14}-a^{13}-a^{12}+a^{11}+2a^{9}-a^{8}-a^{7}-a^{5}+a^{4}-2a^{3}-a^{2}-1$, $2a^{22}-a^{21}-a^{20}-a^{17}+2a^{16}+a^{15}-a^{14}-a^{13}-a^{11}+2a^{9}+a^{8}-a^{7}-a^{6}-a^{4}+a^{3}+2a^{2}-3$, $2a^{26}-2a^{25}+2a^{24}-a^{23}+a^{22}-a^{21}-a^{18}+a^{17}-2a^{16}+2a^{15}-2a^{14}+3a^{13}-3a^{12}+3a^{11}-3a^{10}+3a^{9}-4a^{8}+4a^{7}-3a^{6}+5a^{5}-3a^{4}+4a^{3}-4a^{2}+3a-5$, $a^{25}-a^{24}+a^{23}+a^{19}+a^{16}+a^{12}+a^{9}+a^{8}-a^{7}+2a^{6}-a^{5}+a^{4}+a^{2}+1$, $a^{26}-2a^{25}-2a^{24}+a^{23}-a^{20}+a^{19}+3a^{18}-a^{17}-a^{16}-3a^{15}+a^{14}+a^{13}-2a^{12}+a^{10}+3a^{9}-4a^{7}-a^{6}+2a^{4}-2a^{3}-a^{2}+4a+1$, $a^{25}+a^{23}+2a^{21}+a^{19}+a^{18}+a^{17}+a^{16}-a^{12}-a^{10}-a^{9}-2a^{8}-2a^{7}-a^{6}-3a^{5}-a^{4}-a^{3}-a-1$, $a^{26}-4a^{25}-a^{24}+a^{23}-2a^{22}-2a^{21}+2a^{20}+a^{19}-3a^{18}+2a^{17}+a^{16}-3a^{15}-2a^{14}+2a^{13}-2a^{12}-2a^{11}+5a^{10}+3a^{9}+a^{8}+5a^{7}+7a^{6}-a^{5}+a^{4}+4a^{3}-3a^{2}-5a-1$ (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: | \( 6055111116377.756 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{13}\cdot 6055111116377.756 \cdot 1}{2\cdot\sqrt{29757847887343501070113214188474091663330902016}}\cr\approx \mathstrut & 0.834948417932631 \end{aligned}\] (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 | R | ${\href{/padicField/3.9.0.1}{9} }^{3}$ | $27$ | ${\href{/padicField/7.14.0.1}{14} }{,}\,{\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.6.0.1}{6} }$ | $15{,}\,{\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | $17{,}\,{\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | $23{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | $27$ | ${\href{/padicField/29.11.0.1}{11} }{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.13.0.1}{13} }{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $24{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.4.0.1}{4} }$ | R | $27$ | $21{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
2.2.3.4 | $x^{2} + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
Deg $24$ | $2$ | $12$ | $24$ | ||||
\(47\) | $\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
47.2.1.1 | $x^{2} + 235$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
47.3.0.1 | $x^{3} + 3 x + 42$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
47.7.0.1 | $x^{7} + 12 x + 42$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
47.14.0.1 | $x^{14} + 36 x^{7} + 20 x^{6} + 30 x^{5} + 17 x^{4} + 24 x^{3} + 9 x^{2} + 32 x + 5$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | |
\(471\!\cdots\!051\) | $\Q_{47\!\cdots\!51}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{47\!\cdots\!51}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
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 $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |