Normalized defining polynomial
\( x^{27} - x - 3 \)
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: | \(-1127130637840902624857160283640102926725220750412411\) \(\medspace = -\,5388993289\cdot 58684642644166161743\cdot 3564036539687671224493\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(77.77\) | 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: | $5388993289^{1/2}58684642644166161743^{1/2}3564036539687671224493^{1/2}\approx 3.357276631201103e+25$ | ||
Ramified primes: | \(5388993289\), \(58684642644166161743\), \(3564036539687671224493\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-11271\!\cdots\!12411}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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: | $a^{14}-a-1$, $a^{12}-a^{11}+a^{9}-a^{8}+a^{6}-a^{5}+a^{3}-a^{2}+1$, $2a^{26}-a^{24}-2a^{23}-2a^{22}+a^{19}+3a^{18}+2a^{17}+a^{16}-2a^{15}-4a^{14}-2a^{13}-a^{12}+3a^{10}+3a^{9}+4a^{8}+a^{7}-4a^{6}-4a^{5}-4a^{4}-2a^{3}+2a^{2}+3a+4$, $a^{26}+a^{24}-a^{21}-a^{19}+a^{18}+a^{16}-a^{13}-a^{12}-a^{11}+a^{9}+2a^{8}+a^{7}+a^{6}-2a^{4}-2a^{3}-3a^{2}-1$, $a^{24}-2a^{23}+a^{21}+a^{20}-3a^{18}-2a^{13}-2a^{11}+2a^{10}+3a^{9}-2a^{8}-2a^{7}-2a^{6}+3a^{5}+a^{4}-4a^{3}-2a^{2}-3a+1$, $a^{26}-a^{22}+a^{20}+a^{19}-3a^{17}-a^{16}+5a^{15}+a^{14}-6a^{13}+6a^{11}-6a^{9}-2a^{8}+6a^{7}+5a^{6}-6a^{5}-6a^{4}+6a^{3}+4a^{2}-5a-2$, $2a^{26}+5a^{25}-2a^{24}-5a^{23}+2a^{22}+6a^{21}-3a^{20}-7a^{19}+5a^{18}+7a^{17}-7a^{16}-6a^{15}+8a^{14}+5a^{13}-9a^{12}-5a^{11}+11a^{10}+5a^{9}-14a^{8}-3a^{7}+17a^{6}-a^{5}-18a^{4}+6a^{3}+17a^{2}-10a-17$, $2a^{26}+a^{25}-5a^{24}-a^{23}+2a^{22}+4a^{21}-a^{20}-4a^{19}-a^{18}+2a^{17}+3a^{16}-a^{15}-2a^{14}-a^{13}+a^{11}+a^{10}+2a^{9}-2a^{8}-4a^{7}-2a^{6}+5a^{5}+7a^{4}-4a^{3}-8a^{2}-6a+10$, $3a^{26}-5a^{25}+3a^{24}+2a^{23}-5a^{22}+3a^{21}+2a^{20}-4a^{19}+2a^{18}+2a^{17}-3a^{16}+a^{15}+2a^{14}-4a^{13}+3a^{11}-5a^{10}+a^{9}+4a^{8}-7a^{7}+4a^{6}+5a^{5}-9a^{4}+8a^{3}+5a^{2}-11a+8$, $14a^{26}-15a^{25}+17a^{24}-15a^{23}+12a^{22}-10a^{21}+11a^{20}-11a^{19}+8a^{18}-4a^{17}+a^{16}-3a^{15}+2a^{14}+a^{13}-8a^{12}+10a^{11}-11a^{10}+10a^{9}-16a^{8}+22a^{7}-24a^{6}+22a^{5}-20a^{4}+25a^{3}-30a^{2}+33a-40$, $a^{26}+3a^{24}+2a^{23}+a^{22}-2a^{21}-2a^{20}-2a^{19}-3a^{18}+a^{16}+4a^{15}-3a^{12}-5a^{11}-6a^{10}-5a^{9}+a^{8}+a^{7}+4a^{6}+2a^{5}+2a^{4}-5a^{3}-7a^{2}-6a-4$, $20a^{26}-12a^{25}+14a^{24}+4a^{23}-a^{22}+18a^{21}-20a^{20}+23a^{19}-32a^{18}+18a^{17}-28a^{16}+6a^{15}-7a^{14}-7a^{13}+22a^{12}-17a^{11}+42a^{10}-23a^{9}+40a^{8}-23a^{7}+15a^{6}-15a^{5}-20a^{4}+3a^{3}-45a^{2}+28a-65$, $17a^{26}-6a^{25}-9a^{24}+16a^{23}+a^{22}-16a^{21}+23a^{20}-10a^{19}-13a^{18}+17a^{17}-5a^{16}-20a^{15}+27a^{14}-11a^{13}-14a^{12}+26a^{11}-9a^{10}-19a^{9}+32a^{8}-11a^{7}-17a^{6}+37a^{5}-18a^{4}-20a^{3}+31a^{2}-20a-46$ (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: | \( 1649261800749849.0 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{13}\cdot 1649261800749849.0 \cdot 1}{2\cdot\sqrt{1127130637840902624857160283640102926725220750412411}}\cr\approx \mathstrut & 1.16853240740492 \end{aligned}\] (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 | $23{,}\,{\href{/padicField/2.4.0.1}{4} }$ | ${\href{/padicField/3.3.0.1}{3} }^{8}{,}\,{\href{/padicField/3.1.0.1}{1} }^{3}$ | $27$ | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.9.0.1}{9} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $25{,}\,{\href{/padicField/11.2.0.1}{2} }$ | $20{,}\,{\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.10.0.1}{10} }^{2}{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.13.0.1}{13} }{,}\,{\href{/padicField/19.9.0.1}{9} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | $27$ | $25{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | $22{,}\,{\href{/padicField/31.5.0.1}{5} }$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $23{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | $27$ | ${\href{/padicField/59.14.0.1}{14} }{,}\,{\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(5388993289\) | $\Q_{5388993289}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{5388993289}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | ||
\(58684642644166161743\) | $\Q_{58684642644166161743}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{58684642644166161743}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{58684642644166161743}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | ||
\(356\!\cdots\!493\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ |