Normalized defining polynomial
\( x^{27} - 2 \)
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: | \(-29757847893499620650320371499270765951731105792\) \(\medspace = -\,2^{26}\cdot 3^{81}\) | 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: | $2^{26/27}3^{1579/486}\approx 69.18850536448$ | ||
Ramified primes: | \(2\), \(3\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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: | $a^{9}-1$, $a^{3}-1$, $a^{2}+a+1$, $a^{15}-a^{3}-1$, $a^{14}+a+1$, $a^{21}+a^{18}-a^{12}-a^{9}-1$, $a^{25}-a^{23}+a^{21}-a^{19}+a^{17}+a^{12}-a^{10}+a^{8}-a^{6}+a^{4}+a-1$, $a^{25}+a^{24}-a^{22}-a^{21}+a^{19}+a^{18}-a^{16}-a^{15}+a^{13}+a^{12}-a^{10}-a^{9}+a^{7}+a^{6}-a^{4}-a^{3}-1$, $a^{26}+2a^{25}+a^{24}-2a^{22}-2a^{21}-a^{20}+2a^{19}+2a^{18}+2a^{17}-2a^{16}-2a^{15}-2a^{14}+2a^{12}+3a^{11}+a^{10}-2a^{9}-3a^{8}-2a^{7}+a^{6}+3a^{5}+3a^{4}-2a^{2}-5a-1$, $5a^{26}-a^{25}-a^{24}+a^{23}-5a^{22}+a^{21}+a^{20}-2a^{19}+5a^{18}-a^{17}-a^{16}+4a^{15}-4a^{14}+a^{13}+a^{12}-6a^{11}+3a^{10}-a^{9}-a^{8}+8a^{7}-2a^{6}+a^{4}-9a^{3}+2a^{2}+a-1$, $3a^{26}-5a^{25}+4a^{24}-2a^{23}+3a^{21}-5a^{20}+6a^{19}-4a^{18}+a^{17}+a^{16}-4a^{15}+7a^{14}-5a^{13}+3a^{12}-a^{11}-3a^{10}+6a^{9}-6a^{8}+5a^{7}-a^{6}-2a^{5}+5a^{4}-8a^{3}+6a^{2}-a-1$, $a^{26}-2a^{25}+2a^{24}-2a^{23}+a^{22}+2a^{21}-2a^{20}+4a^{19}-4a^{18}+a^{17}-a^{16}-2a^{15}+5a^{14}-3a^{13}+5a^{12}-2a^{11}-2a^{10}+2a^{9}-5a^{8}+5a^{7}-2a^{6}+2a^{5}+2a^{4}-4a^{3}+4a^{2}-3a+1$, $a^{25}-a^{24}+a^{23}+a^{22}+a^{21}-2a^{20}-a^{19}-a^{18}+2a^{17}-a^{16}+2a^{13}-a^{11}-3a^{10}+a^{9}+a^{8}+2a^{4}+a^{3}-3a^{2}-3a+1$ (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: | \( 6049054674787.264 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{13}\cdot 6049054674787.264 \cdot 1}{2\cdot\sqrt{29757847893499620650320371499270765951731105792}}\cr\approx \mathstrut & 0.834113285967335 \end{aligned}\] (assuming GRH)
Galois group
$C_{27}:C_{18}$ (as 27T176):
A solvable group of order 486 |
The 31 conjugacy class representatives for $C_{27}:C_{18}$ |
Character table for $C_{27}:C_{18}$ |
Intermediate fields
3.1.108.1, 9.1.99179645184.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | R | $18{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $27$ | $18{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $27$ | ${\href{/padicField/17.6.0.1}{6} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $27$ | $18{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.9.0.1}{9} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ | $27$ | $18{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.9.0.1}{9} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | $18{,}\,{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.2.0.1}{2} }^{13}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 $27$ | $27$ | $1$ | $26$ | |||
\(3\) | Deg $27$ | $27$ | $1$ | $81$ |