Normalized defining polynomial
\( x^{27} - 2x - 2 \)
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: | \(-28931587510147902226726652006325673846179364864\) \(\medspace = -\,2^{26}\cdot 239\cdot 911\cdot 1492637\cdot 10924843\cdot 125481509\cdot 967669291801\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(52.58\) | 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: | $2^{26/27}239^{1/2}911^{1/2}1492637^{1/2}10924843^{1/2}125481509^{1/2}967669291801^{1/2}\approx 4.0474074356440924e+19$ | ||
Ramified primes: | \(2\), \(239\), \(911\), \(1492637\), \(10924843\), \(125481509\), \(967669291801\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-43111\!\cdots\!85251}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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: | $a+1$, $a^{4}-a^{2}+1$, $a^{18}+a^{9}+a+1$, $a^{14}-a-1$, $a^{12}-a^{11}+a^{9}-a^{8}+a^{6}-a^{5}+a^{3}-a^{2}+1$, $a^{26}-a^{25}+a^{24}-a^{23}+a^{22}-a^{21}+a^{20}-a^{12}+a^{8}+a^{4}+a^{2}-a-3$, $a^{26}-a^{24}+a^{23}-a^{22}+a^{21}-a^{18}+a^{17}-a^{16}+2a^{15}-a^{14}-a^{13}+a^{12}-a^{11}+2a^{10}-a^{9}-a^{8}+a^{6}+a^{4}-2a^{3}+a-3$, $a^{25}-2a^{24}+2a^{23}-2a^{22}+2a^{21}-2a^{20}+a^{19}-a^{14}+2a^{13}-2a^{12}+2a^{11}-2a^{10}+2a^{9}-a^{8}+a^{4}-a^{3}-a^{2}+a-1$, $a^{24}-a^{23}-a^{22}-2a^{20}-a^{19}-a^{17}+2a^{15}+2a^{13}+2a^{12}+a^{10}+a^{9}-2a^{8}-2a^{5}+a^{4}+2a^{3}+3a+3$, $a^{26}-2a^{25}+a^{24}-a^{23}-a^{20}-a^{17}+a^{16}-a^{14}+2a^{13}-a^{12}+a^{11}+a^{10}+2a^{8}+a^{6}+a^{5}-a^{4}+a^{3}+a^{2}-2a-1$, $a^{26}-2a^{25}+a^{22}+a^{21}-a^{20}-a^{19}-a^{18}+2a^{17}+2a^{16}-2a^{15}-a^{14}-a^{13}+2a^{12}+a^{11}-a^{9}-3a^{8}+2a^{7}+2a^{6}+a^{5}-2a^{4}-3a^{3}+a^{2}+2a+1$, $2a^{24}-2a^{21}+a^{20}+a^{19}+a^{18}-2a^{16}+2a^{14}+2a^{13}-2a^{11}-a^{10}+a^{9}+3a^{8}+a^{7}-2a^{6}-a^{5}+4a^{3}+2a^{2}-2a-3$, $2a^{25}-3a^{24}+4a^{23}-4a^{22}+4a^{21}-4a^{20}+3a^{19}-3a^{18}+2a^{17}-2a^{16}-2a^{13}+2a^{12}-4a^{11}+4a^{10}-5a^{9}+4a^{8}-3a^{7}+2a^{6}-a^{4}+3a^{3}-2a^{2}+4a-3$ (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: | \( 4511852311652.407 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{13}\cdot 4511852311652.407 \cdot 1}{2\cdot\sqrt{28931587510147902226726652006325673846179364864}}\cr\approx \mathstrut & 0.630967560369905 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 10888869450418352160768000000 |
The 3010 conjugacy class representatives for $S_{27}$ are not computed |
Character table for $S_{27}$ 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 | ${\href{/padicField/3.6.0.1}{6} }^{4}{,}\,{\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/5.7.0.1}{7} }{,}\,{\href{/padicField/5.5.0.1}{5} }$ | ${\href{/padicField/7.14.0.1}{14} }{,}\,{\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.6.0.1}{6} }$ | $20{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.14.0.1}{14} }{,}\,{\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $24{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | $27$ | ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.9.0.1}{9} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | $20{,}\,{\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.13.0.1}{13} }{,}\,{\href{/padicField/47.9.0.1}{9} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 $27$ | $27$ | $1$ | $26$ | |||
\(239\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $23$ | $1$ | $23$ | $0$ | $C_{23}$ | $[\ ]^{23}$ | ||
\(911\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $25$ | $1$ | $25$ | $0$ | $C_{25}$ | $[\ ]^{25}$ | ||
\(1492637\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $19$ | $1$ | $19$ | $0$ | $C_{19}$ | $[\ ]^{19}$ | ||
\(10924843\) | $\Q_{10924843}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | ||
\(125481509\) | $\Q_{125481509}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{125481509}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $23$ | $1$ | $23$ | $0$ | $C_{23}$ | $[\ ]^{23}$ | ||
\(967669291801\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ |