Normalized defining polynomial
\( x^{27} - 4x - 5 \)
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: | \(-660646059101739963744377659248723289832910846176228525291\) \(\medspace = -\,75879359\cdot 6952540213\cdot 1353159565227731\cdot 925449390832864291185083\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(127.19\) | 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: | $75879359^{1/2}6952540213^{1/2}1353159565227731^{1/2}925449390832864291185083^{1/2}\approx 2.570303598997091e+28$ | ||
Ramified primes: | \(75879359\), \(6952540213\), \(1353159565227731\), \(925449390832864291185083\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-66064\!\cdots\!25291}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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: | $a^{26}-a^{25}+a^{24}-a^{23}+a^{22}-a^{21}+a^{20}-a^{19}+a^{18}-a^{17}+a^{16}-a^{15}+a^{14}-6$, $6a^{26}-6a^{25}+6a^{24}-6a^{23}+6a^{22}-6a^{21}+6a^{20}-6a^{19}+6a^{18}-6a^{17}+6a^{16}-6a^{15}+6a^{14}-6a^{13}+5a^{12}-4a^{11}+3a^{10}-2a^{9}+a^{8}-a^{6}+2a^{5}-3a^{4}+4a^{3}-5a^{2}+6a-31$, $5a^{26}-a^{25}-3a^{24}+6a^{23}-9a^{22}+10a^{21}-10a^{20}+10a^{19}-9a^{18}+8a^{17}-7a^{16}+5a^{15}-3a^{14}-2a^{13}+6a^{12}-10a^{11}+15a^{10}-18a^{9}+19a^{8}-18a^{7}+13a^{6}-8a^{5}-3a^{4}+11a^{3}-17a^{2}+23a-44$, $12a^{26}-7a^{25}-a^{24}+9a^{23}-15a^{22}+16a^{21}-14a^{20}+7a^{19}+4a^{18}-13a^{17}+18a^{16}-20a^{15}+17a^{14}-5a^{13}-7a^{12}+19a^{11}-23a^{10}+25a^{9}-18a^{8}+4a^{7}+16a^{6}-24a^{5}+30a^{4}-30a^{3}+17a^{2}+3a-69$, $3a^{26}+27a^{25}-35a^{24}-3a^{23}+33a^{22}-34a^{21}-9a^{20}+33a^{19}-29a^{18}-10a^{17}+25a^{16}-23a^{15}-4a^{14}+11a^{13}-22a^{12}+14a^{11}-10a^{10}-30a^{9}+42a^{8}-27a^{7}-57a^{6}+77a^{5}-35a^{4}-96a^{3}+105a^{2}-27a-156$, $115a^{26}-12a^{25}-111a^{24}+191a^{23}-166a^{22}+31a^{21}+150a^{20}-255a^{19}+202a^{18}-9a^{17}-193a^{16}+279a^{15}-204a^{14}+30a^{13}+168a^{12}-307a^{11}+311a^{10}-111a^{9}-217a^{8}+465a^{7}-408a^{6}+63a^{5}+340a^{4}-515a^{3}+390a^{2}-54a-756$, $4a^{26}+a^{25}+8a^{24}-4a^{23}-a^{22}-7a^{21}+14a^{20}-3a^{19}+15a^{18}-17a^{17}+6a^{16}-5a^{15}+19a^{14}-2a^{13}+8a^{12}-24a^{11}+18a^{10}+2a^{9}+28a^{8}-8a^{7}-12a^{6}-12a^{5}+28a^{4}+17a^{3}+21a^{2}-29a-26$, $5a^{25}-5a^{24}-15a^{23}-7a^{22}+8a^{21}+7a^{20}+a^{19}-5a^{18}-8a^{17}+8a^{16}+29a^{15}+20a^{14}-9a^{13}-6a^{12}+15a^{11}+30a^{10}+28a^{9}+5a^{8}-34a^{7}-24a^{6}+26a^{5}+28a^{4}-27a^{3}-62a^{2}-57a-26$, $8a^{26}-7a^{25}-11a^{24}+10a^{23}+20a^{22}-17a^{21}-19a^{20}+23a^{19}+18a^{18}-43a^{17}-10a^{16}+48a^{15}-15a^{14}-35a^{13}+14a^{12}+37a^{11}-48a^{9}+19a^{8}+62a^{7}-38a^{6}-79a^{5}+59a^{4}+61a^{3}-111a^{2}-30a+71$, $9a^{26}-a^{25}+3a^{24}-7a^{23}+10a^{22}+4a^{21}-24a^{20}+2a^{19}+12a^{18}-20a^{17}-4a^{16}+7a^{15}+10a^{14}+13a^{13}-23a^{12}+33a^{11}+28a^{10}-44a^{9}+14a^{8}+a^{7}-10a^{6}-19a^{5}-50a^{4}+56a^{3}-3a^{2}-55a+29$, $17a^{26}-10a^{25}-4a^{24}-4a^{23}+23a^{22}-10a^{21}-9a^{20}-3a^{19}+39a^{18}-22a^{17}-30a^{16}+43a^{15}+11a^{14}-32a^{13}-10a^{12}+53a^{11}-14a^{10}-14a^{9}-15a^{8}+46a^{7}+23a^{6}-75a^{5}+17a^{4}+88a^{3}-28a^{2}-95a+36$, $47a^{26}-121a^{25}+175a^{24}-206a^{23}+249a^{22}-264a^{21}+206a^{20}-150a^{19}+120a^{18}-35a^{17}-83a^{16}+155a^{15}-213a^{14}+270a^{13}-287a^{12}+302a^{11}-290a^{10}+171a^{9}-75a^{8}+63a^{7}+60a^{6}-253a^{5}+289a^{4}-281a^{3}+365a^{2}-380a+119$, $26a^{26}-18a^{25}-99a^{24}-69a^{23}+54a^{22}+62a^{21}-33a^{20}-23a^{19}+97a^{18}+155a^{17}+51a^{16}-109a^{15}-74a^{14}+84a^{13}-11a^{12}-254a^{11}-217a^{10}+44a^{9}+173a^{8}+28a^{7}-112a^{6}+153a^{5}+480a^{4}+237a^{3}-224a^{2}-223a-21$ (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: | \( 1118787466303882500 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{13}\cdot 1118787466303882500 \cdot 1}{2\cdot\sqrt{660646059101739963744377659248723289832910846176228525291}}\cr\approx \mathstrut & 1.03538401721225 \end{aligned}\] (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 | $18{,}\,{\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.2.0.1}{2} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | ${\href{/padicField/3.9.0.1}{9} }^{3}$ | ${\href{/padicField/5.4.0.1}{4} }^{6}{,}\,{\href{/padicField/5.1.0.1}{1} }^{3}$ | ${\href{/padicField/7.14.0.1}{14} }{,}\,{\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.6.0.1}{6} }$ | $24{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | $26{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.14.0.1}{14} }{,}\,{\href{/padicField/23.4.0.1}{4} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.11.0.1}{11} }{,}\,{\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | $25{,}\,{\href{/padicField/41.2.0.1}{2} }$ | $24{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.14.0.1}{14} }{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | $20{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.14.0.1}{14} }{,}\,{\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(75879359\) | 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 $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | ||
\(6952540213\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $21$ | $1$ | $21$ | $0$ | $C_{21}$ | $[\ ]^{21}$ | ||
\(1353159565227731\) | $\Q_{1353159565227731}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{1353159565227731}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{1353159565227731}$ | $x$ | $1$ | $1$ | $0$ | 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 $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
\(925\!\cdots\!083\) | $\Q_{92\!\cdots\!83}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{92\!\cdots\!83}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
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 $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ |