Normalized defining polynomial
\( x^{27} - 4x - 2 \)
Invariants
Degree: | $27$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[3, 12]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(110898761632028778210869975092030791887124902069665792\) \(\medspace = 2^{26}\cdot 555937897\cdot 70012520507\cdot 42456561322999590418419407\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(92.18\) | 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))
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Galois root discriminant: | $2^{26/27}555937897^{1/2}70012520507^{1/2}42456561322999590418419407^{1/2}\approx 7.924176517052724e+22$ | ||
Ramified primes: | \(2\), \(555937897\), \(70012520507\), \(42456561322999590418419407\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | $\Q(\sqrt{16525\!\cdots\!89053}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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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: | $14$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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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)
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Fundamental units: | $a+1$, $2a^{14}+4a+1$, $a^{13}-a^{12}+a^{11}-a^{10}+a^{9}-a^{8}+a^{7}-a^{6}+a^{5}-a^{4}+a^{3}-a^{2}+a+1$, $16a^{26}-9a^{25}+6a^{24}-3a^{23}+a^{22}-3a^{21}+a^{20}+a^{19}+2a^{18}-2a^{16}+3a^{13}-2a^{12}-a^{11}-4a^{10}+2a^{9}+a^{8}+2a^{7}-2a^{6}-a^{5}+3a^{4}+4a^{3}+4a^{2}-5a-67$, $12a^{26}-10a^{25}+6a^{24}-a^{23}+a^{22}+5a^{21}-6a^{20}+5a^{19}-7a^{18}+a^{17}-a^{16}-3a^{15}+10a^{14}-5a^{13}+9a^{12}-6a^{11}-2a^{10}+a^{9}-11a^{8}+8a^{7}-7a^{6}+13a^{5}+3a^{4}-a^{3}+5a^{2}-17a-41$, $a^{26}-a^{22}+2a^{18}+a^{16}+a^{15}-2a^{14}+a^{13}-2a^{12}+a^{10}-2a^{9}+2a^{8}-2a^{7}-2a^{6}-a^{5}-3a^{4}+a^{3}+a^{2}+2a+1$, $a^{26}-a^{25}-4a^{24}-3a^{23}-2a^{22}+3a^{20}+2a^{19}+a^{18}+a^{17}+4a^{15}+3a^{14}+a^{13}-2a^{12}-8a^{11}-8a^{10}-4a^{9}+7a^{7}+4a^{6}+2a^{5}+a^{4}+7a^{2}+10a+1$, $a^{25}-2a^{24}+2a^{23}-a^{22}-a^{21}+3a^{20}-5a^{19}+4a^{18}-a^{17}-3a^{16}+5a^{15}-5a^{14}+4a^{13}-2a^{12}-2a^{11}+6a^{10}-8a^{9}+6a^{8}-3a^{7}-a^{6}+4a^{5}-7a^{4}+5a^{3}-a^{2}-3a+1$, $12a^{26}-6a^{25}+3a^{24}+2a^{23}-4a^{21}+a^{20}-2a^{19}+6a^{18}-a^{17}-2a^{16}-4a^{15}+2a^{14}+a^{13}+6a^{12}-6a^{11}-2a^{10}-a^{9}+2a^{8}+9a^{7}-7a^{6}-2a^{5}-4a^{4}+6a^{3}+6a^{2}+2a-61$, $11a^{26}-5a^{25}+3a^{24}-2a^{23}-a^{21}+3a^{20}+a^{19}-2a^{18}+a^{17}+a^{16}+a^{15}-a^{14}-4a^{13}+2a^{11}-a^{10}-3a^{9}+5a^{7}+3a^{6}-2a^{5}-a^{4}+2a^{3}+5a^{2}-4a-53$, $a^{26}+a^{24}+a^{23}+a^{22}+a^{21}-2a^{18}-a^{17}-3a^{16}-2a^{14}+a^{13}-a^{12}-a^{11}-3a^{10}-3a^{9}-2a^{8}-2a^{7}+4a^{6}+2a^{5}+6a^{4}+3a^{3}+4a^{2}-1$, $2a^{26}+a^{25}-2a^{24}+3a^{22}-2a^{21}-4a^{20}+a^{19}+7a^{18}-a^{17}-7a^{16}+a^{15}+6a^{14}-3a^{13}-5a^{12}+3a^{11}+8a^{10}-4a^{9}-8a^{8}+2a^{7}+5a^{6}-a^{5}+a^{4}+a^{3}-3a^{2}-7a-3$, $5a^{26}-4a^{25}-5a^{24}-a^{23}+5a^{22}+7a^{21}-3a^{20}-8a^{19}-3a^{18}+6a^{17}+7a^{16}+a^{15}-10a^{14}-8a^{13}+3a^{12}+14a^{11}+4a^{10}-11a^{9}-13a^{8}+a^{7}+14a^{6}+15a^{5}-10a^{4}-22a^{3}-5a^{2}+23a+1$, $5a^{26}+12a^{25}-16a^{24}+a^{23}+18a^{22}-11a^{21}-12a^{20}+24a^{19}-22a^{17}+18a^{16}+11a^{15}-26a^{14}+8a^{13}+30a^{12}-29a^{11}-10a^{10}+37a^{9}-11a^{8}-32a^{7}+35a^{6}+a^{5}-53a^{4}+24a^{3}+30a^{2}-57a-31$ (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)]
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Regulator: | \( 12475074473078126 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{12}\cdot 12475074473078126 \cdot 1}{2\cdot\sqrt{110898761632028778210869975092030791887124902069665792}}\cr\approx \mathstrut & 0.567280963868321 \end{aligned}\] (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 | R | ${\href{/padicField/3.9.0.1}{9} }^{3}$ | $18{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.14.0.1}{14} }{,}\,{\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.6.0.1}{6} }$ | ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.11.0.1}{11} }{,}\,{\href{/padicField/11.4.0.1}{4} }$ | $20{,}\,{\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $23{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $23{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | $26{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.13.0.1}{13} }{,}\,{\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | $22{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.9.0.1}{9} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | $18{,}\,{\href{/padicField/41.9.0.1}{9} }$ | $16{,}\,{\href{/padicField/43.5.0.1}{5} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | $20{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.14.0.1}{14} }{,}\,{\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 $27$ | $27$ | $1$ | $26$ | |||
\(555937897\) | $\Q_{555937897}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $16$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ | ||
\(70012520507\) | 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 $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | ||
\(424\!\cdots\!407\) | 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 $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ |