Normalized defining polynomial
\( x^{27} + 3x - 5 \)
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: | \(-73417444981913314280334795949102827452496728781547778643\) \(\medspace = -\,3^{25}\cdot 41\cdot 181\cdot 3917\cdot 23593\cdot 985450601\cdot 128213533389611086945001\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(117.25\) | 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: | $3^{25/26}41^{1/2}181^{1/2}3917^{1/2}23593^{1/2}985450601^{1/2}128213533389611086945001^{1/2}\approx 2.6770370785035566e+22$ | ||
Ramified primes: | \(3\), \(41\), \(181\), \(3917\), \(23593\), \(985450601\), \(128213533389611086945001\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-25994\!\cdots\!33203}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $\frac{1}{3}a^{26}-\frac{1}{3}a^{25}+\frac{1}{3}a^{24}-\frac{1}{3}a^{23}+\frac{1}{3}a^{22}-\frac{1}{3}a^{21}+\frac{1}{3}a^{20}-\frac{1}{3}a^{19}+\frac{1}{3}a^{18}-\frac{1}{3}a^{17}+\frac{1}{3}a^{16}-\frac{1}{3}a^{15}+\frac{1}{3}a^{14}-\frac{1}{3}a^{13}+\frac{1}{3}a^{12}-\frac{1}{3}a^{11}+\frac{1}{3}a^{10}-\frac{1}{3}a^{9}+\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$
Monogenic: | Not computed | |
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: | $a-1$, $5a^{26}-5a^{24}-5a^{23}+5a^{21}+5a^{20}-7a^{18}+9a^{17}+2a^{16}-7a^{15}-9a^{14}-2a^{13}+7a^{12}+9a^{11}+2a^{10}-15a^{9}+15a^{8}+6a^{7}-9a^{6}-15a^{5}-6a^{4}+9a^{3}+15a^{2}+7a-16$, $9a^{26}+9a^{25}+4a^{24}-7a^{23}-9a^{22}-9a^{21}+3a^{20}+10a^{19}+15a^{18}+3a^{17}-5a^{16}-18a^{15}-11a^{14}+a^{13}+20a^{12}+18a^{11}+10a^{10}-16a^{9}-25a^{8}-17a^{7}+9a^{6}+27a^{5}+29a^{4}+5a^{3}-27a^{2}-30a+6$, $2a^{26}+3a^{25}+12a^{24}-9a^{23}+7a^{22}-11a^{21}-7a^{20}+8a^{19}-11a^{18}+19a^{17}-3a^{16}+3a^{15}+9a^{14}-18a^{13}+17a^{12}-19a^{11}+3a^{10}+9a^{9}-26a^{8}+37a^{7}-21a^{6}+19a^{5}+20a^{4}-37a^{3}+37a^{2}-60a+26$, $\frac{40}{3}a^{26}+\frac{38}{3}a^{25}+\frac{22}{3}a^{24}+\frac{29}{3}a^{23}+\frac{16}{3}a^{22}+\frac{17}{3}a^{21}+\frac{16}{3}a^{20}+\frac{26}{3}a^{19}+\frac{37}{3}a^{18}+\frac{20}{3}a^{17}+\frac{31}{3}a^{16}+\frac{11}{3}a^{15}+\frac{16}{3}a^{14}-\frac{34}{3}a^{13}-\frac{50}{3}a^{12}-\frac{76}{3}a^{11}-\frac{98}{3}a^{10}-\frac{109}{3}a^{9}-\frac{137}{3}a^{8}-\frac{91}{3}a^{7}-\frac{104}{3}a^{6}-\frac{40}{3}a^{5}-\frac{32}{3}a^{4}+\frac{38}{3}a^{3}+\frac{58}{3}a^{2}+\frac{71}{3}a+\frac{244}{3}$, $\frac{53}{3}a^{26}-\frac{23}{3}a^{25}-\frac{40}{3}a^{24}+\frac{46}{3}a^{23}+\frac{59}{3}a^{22}-\frac{62}{3}a^{21}-\frac{94}{3}a^{20}+\frac{31}{3}a^{19}+\frac{56}{3}a^{18}-\frac{44}{3}a^{17}-\frac{19}{3}a^{16}+\frac{133}{3}a^{15}+\frac{89}{3}a^{14}-\frac{143}{3}a^{13}-\frac{136}{3}a^{12}+\frac{67}{3}a^{11}+\frac{56}{3}a^{10}-\frac{110}{3}a^{9}-\frac{13}{3}a^{8}+\frac{241}{3}a^{7}+\frac{83}{3}a^{6}-\frac{239}{3}a^{5}-\frac{106}{3}a^{4}+\frac{196}{3}a^{3}+\frac{50}{3}a^{2}-\frac{278}{3}a+\frac{131}{3}$, $a^{26}+a^{25}-21a^{24}-2a^{23}+7a^{22}-9a^{21}-6a^{20}+19a^{19}+13a^{18}-15a^{17}+22a^{16}+16a^{15}-14a^{14}-14a^{13}+21a^{12}-15a^{11}-42a^{10}+22a^{9}+a^{8}-29a^{7}-6a^{6}+60a^{5}-20a^{4}-18a^{3}+66a^{2}+9a-41$, $\frac{25}{3}a^{26}-\frac{40}{3}a^{25}+\frac{46}{3}a^{24}-\frac{13}{3}a^{23}+\frac{28}{3}a^{22}-\frac{28}{3}a^{21}+\frac{49}{3}a^{20}-\frac{85}{3}a^{19}+\frac{43}{3}a^{18}-\frac{43}{3}a^{17}+\frac{16}{3}a^{16}-\frac{25}{3}a^{15}+\frac{91}{3}a^{14}-\frac{70}{3}a^{13}+\frac{61}{3}a^{12}-\frac{4}{3}a^{11}-\frac{44}{3}a^{10}-\frac{28}{3}a^{9}+\frac{28}{3}a^{8}-\frac{55}{3}a^{7}-\frac{11}{3}a^{6}+\frac{128}{3}a^{5}-\frac{104}{3}a^{4}+\frac{110}{3}a^{3}-\frac{41}{3}a^{2}+\frac{35}{3}a-\frac{119}{3}$, $\frac{13}{3}a^{26}-\frac{43}{3}a^{25}+\frac{16}{3}a^{24}-\frac{31}{3}a^{23}+\frac{22}{3}a^{22}-\frac{40}{3}a^{21}+\frac{10}{3}a^{20}-\frac{10}{3}a^{19}-\frac{35}{3}a^{18}+\frac{32}{3}a^{17}-\frac{50}{3}a^{16}+\frac{35}{3}a^{15}-\frac{77}{3}a^{14}+\frac{71}{3}a^{13}-\frac{62}{3}a^{12}-\frac{13}{3}a^{11}+\frac{10}{3}a^{10}-\frac{25}{3}a^{9}+\frac{40}{3}a^{8}-\frac{124}{3}a^{7}+\frac{121}{3}a^{6}-\frac{142}{3}a^{5}+\frac{97}{3}a^{4}-\frac{70}{3}a^{3}+\frac{16}{3}a^{2}-\frac{55}{3}a-\frac{26}{3}$, $7a^{26}+7a^{25}-a^{24}-12a^{23}-14a^{22}-7a^{21}-a^{20}-3a^{19}-6a^{18}-4a^{17}+4a^{16}+15a^{15}+22a^{14}+16a^{13}-2a^{12}-12a^{11}-2a^{10}+9a^{9}-2a^{8}-25a^{7}-32a^{6}-14a^{5}+6a^{4}+18a^{3}+16a^{2}+6a+19$, $5a^{26}-3a^{25}-3a^{24}+10a^{23}-2a^{22}-2a^{21}+11a^{20}+2a^{19}-4a^{18}+10a^{17}+7a^{16}-10a^{15}+12a^{14}+4a^{13}-11a^{12}+8a^{11}+2a^{10}-14a^{9}+6a^{8}+a^{7}-24a^{6}+18a^{5}-13a^{4}-23a^{3}+26a^{2}-17a-11$, $\frac{52}{3}a^{26}-\frac{49}{3}a^{25}+\frac{49}{3}a^{24}-\frac{43}{3}a^{23}+\frac{37}{3}a^{22}-\frac{25}{3}a^{21}-\frac{20}{3}a^{20}+\frac{47}{3}a^{19}-\frac{98}{3}a^{18}+\frac{134}{3}a^{17}-\frac{137}{3}a^{16}+\frac{119}{3}a^{15}-\frac{68}{3}a^{14}+\frac{35}{3}a^{13}+\frac{10}{3}a^{12}-\frac{70}{3}a^{11}+\frac{85}{3}a^{10}-\frac{142}{3}a^{9}+\frac{199}{3}a^{8}-\frac{259}{3}a^{7}+\frac{274}{3}a^{6}-\frac{217}{3}a^{5}+\frac{130}{3}a^{4}+\frac{26}{3}a^{3}-\frac{209}{3}a^{2}+\frac{314}{3}a-\frac{239}{3}$, $55a^{26}-6a^{25}-64a^{24}-30a^{23}+60a^{22}+58a^{21}-58a^{20}-91a^{19}+46a^{18}+129a^{17}+6a^{16}-119a^{15}-49a^{14}+87a^{13}+80a^{12}-63a^{11}-139a^{10}+14a^{9}+214a^{8}+98a^{7}-212a^{6}-181a^{5}+165a^{4}+215a^{3}-118a^{2}-255a+189$ (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: | \( 216942989987115230 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{13}\cdot 216942989987115230 \cdot 1}{2\cdot\sqrt{73417444981913314280334795949102827452496728781547778643}}\cr\approx \mathstrut & 0.602260356520453 \end{aligned}\] (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 | $23{,}\,{\href{/padicField/2.4.0.1}{4} }$ | R | ${\href{/padicField/5.4.0.1}{4} }^{6}{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.14.0.1}{14} }{,}\,{\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.6.0.1}{6} }$ | $24{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | $19{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | $16{,}\,{\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | $26{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | R | $27$ | ${\href{/padicField/47.11.0.1}{11} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | $22{,}\,{\href{/padicField/53.5.0.1}{5} }$ | ${\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $26$ | $26$ | $1$ | $25$ | ||||
\(41\) | 41.2.1.2 | $x^{2} + 123$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
41.3.0.1 | $x^{3} + x + 35$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
41.22.0.1 | $x^{22} + x^{2} - 3 x + 11$ | $1$ | $22$ | $0$ | 22T1 | $[\ ]^{22}$ | |
\(181\) | $\Q_{181}$ | $x + 179$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
181.2.1.1 | $x^{2} + 181$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
181.4.0.1 | $x^{4} + 6 x^{2} + 105 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
181.20.0.1 | $x^{20} - x + 103$ | $1$ | $20$ | $0$ | 20T1 | $[\ ]^{20}$ | |
\(3917\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
\(23593\) | $\Q_{23593}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $24$ | $1$ | $24$ | $0$ | $C_{24}$ | $[\ ]^{24}$ | ||
\(985450601\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
Deg $17$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ | ||
\(128\!\cdots\!001\) | $\Q_{12\!\cdots\!01}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{12\!\cdots\!01}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
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 $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ |