Normalized defining polynomial
\( x^{27} + 2x - 4 \)
Invariants
Degree: | $27$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[1, 13]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-499253842010953227406414947741939356093911362179170304\) \(\medspace = -\,2^{50}\cdot 61379\cdot 35649361663\cdot 202651613609462963242873\) | 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: | \(97.46\) | 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: | not computed | ||
Ramified primes: | \(2\), \(61379\), \(35649361663\), \(202651613609462963242873\) | 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: | $\Q(\sqrt{-44342\!\cdots\!24821}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $\frac{1}{2}a^{26}$
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)
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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: | $a-1$, $a^{26}-a^{22}+a^{21}+a^{19}-a^{17}-a^{15}+a^{14}+a^{13}+a^{11}-2a^{10}+a^{9}+3a^{6}-a^{5}+2a^{4}-a^{3}+a+1$, $4a^{26}-5a^{25}-6a^{24}+3a^{23}+8a^{22}-10a^{20}-3a^{19}+8a^{18}+9a^{17}-8a^{16}-9a^{15}+a^{14}+15a^{13}-a^{12}-11a^{11}-10a^{10}+15a^{9}+10a^{8}-5a^{7}-21a^{6}+4a^{5}+18a^{4}+11a^{3}-21a^{2}-14a+21$, $3a^{26}-4a^{25}+a^{24}-2a^{23}-a^{22}+8a^{21}+2a^{19}+a^{18}-8a^{17}+2a^{16}+3a^{15}-a^{14}+10a^{13}-4a^{12}-7a^{11}+3a^{10}-13a^{9}+5a^{8}+6a^{7}-8a^{6}+8a^{5}-9a^{4}-14a^{3}+9a^{2}-8a+13$, $4a^{26}-2a^{25}+2a^{23}-3a^{22}+2a^{21}-a^{20}+2a^{19}-3a^{18}+7a^{17}-11a^{16}+12a^{15}-13a^{14}+11a^{13}-8a^{12}+10a^{11}-12a^{10}+17a^{9}-23a^{8}+22a^{7}-20a^{6}+16a^{5}-11a^{4}+13a^{3}-17a^{2}+21a-17$, $7a^{26}+19a^{25}+26a^{24}+13a^{23}-7a^{22}-25a^{21}-27a^{20}-11a^{19}+5a^{18}+13a^{17}-6a^{16}-28a^{15}-49a^{14}-41a^{13}-14a^{12}+16a^{11}+30a^{10}+10a^{9}-14a^{8}-39a^{7}-19a^{6}+15a^{5}+59a^{4}+69a^{3}+40a^{2}-21$, $3a^{26}+a^{24}-2a^{22}+3a^{21}-a^{20}+3a^{19}+2a^{18}-2a^{17}+2a^{16}-3a^{15}+5a^{14}+a^{13}-3a^{12}+3a^{11}-3a^{10}+7a^{9}-a^{7}+5a^{6}-9a^{5}+7a^{4}-a^{3}+4a^{2}+6a-7$, $2a^{26}-a^{25}-3a^{24}+5a^{23}-5a^{22}-3a^{21}+2a^{20}-a^{19}-4a^{18}+10a^{17}-a^{16}-2a^{15}+6a^{14}-a^{13}-7a^{12}+9a^{11}-2a^{10}-7a^{9}+6a^{8}-5a^{7}-10a^{6}+9a^{5}+a^{4}-7a^{3}+19a^{2}-6a-3$, $3a^{26}+2a^{25}+2a^{24}+a^{23}-3a^{22}+7a^{21}-2a^{20}+3a^{19}-3a^{18}+10a^{17}-4a^{16}-2a^{15}+6a^{14}+9a^{12}-13a^{11}+11a^{10}-a^{9}+7a^{8}-7a^{7}+5a^{6}+11a^{5}-12a^{4}+9a^{3}-7a^{2}+28a-19$, $29a^{26}-30a^{25}+31a^{24}-32a^{23}+31a^{22}-35a^{21}+38a^{20}-36a^{19}+32a^{18}-29a^{17}+23a^{16}-22a^{15}+22a^{14}-15a^{13}+3a^{12}+6a^{11}-18a^{10}+27a^{9}-33a^{8}+46a^{7}-67a^{6}+81a^{5}-95a^{4}+109a^{3}-119a^{2}+134a-103$, $2a^{26}-9a^{25}-3a^{24}+4a^{23}+a^{22}-7a^{21}-7a^{20}+6a^{19}+5a^{18}-15a^{17}-8a^{16}+16a^{15}-2a^{14}-20a^{13}+a^{12}+14a^{11}-7a^{10}-14a^{9}-5a^{8}+15a^{7}+a^{6}-26a^{5}-6a^{4}+34a^{3}-14a^{2}-36a+21$, $12a^{26}-15a^{25}-32a^{24}-27a^{23}-3a^{22}+22a^{21}+29a^{20}+14a^{19}-10a^{18}-23a^{17}-15a^{16}+7a^{15}+24a^{14}+18a^{13}-9a^{12}-35a^{11}-34a^{10}-a^{9}+41a^{8}+60a^{7}+36a^{6}-21a^{5}-73a^{4}-79a^{3}-26a^{2}+51a+121$, $3a^{25}-5a^{24}+4a^{23}-2a^{22}-4a^{21}+2a^{20}-6a^{19}-2a^{18}+3a^{17}-4a^{16}+7a^{15}+7a^{14}-3a^{13}+14a^{12}-7a^{11}-2a^{10}-12a^{8}-5a^{7}+7a^{6}-18a^{5}+22a^{4}-6a^{3}+a^{2}+15a-7$ (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: | \( 36474179686808500 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{13}\cdot 36474179686808500 \cdot 1}{2\cdot\sqrt{499253842010953227406414947741939356093911362179170304}}\cr\approx \mathstrut & 1.22790032441356 \end{aligned}\] (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 | ${\href{/padicField/3.9.0.1}{9} }^{3}$ | $15{,}\,{\href{/padicField/5.7.0.1}{7} }{,}\,{\href{/padicField/5.5.0.1}{5} }$ | $18{,}\,{\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $24{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.14.0.1}{14} }{,}\,{\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.5.0.1}{5} }$ | ${\href{/padicField/17.13.0.1}{13} }{,}\,{\href{/padicField/17.11.0.1}{11} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.9.0.1}{9} }^{2}{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | $19{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | $25{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | $22{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$ | $16{,}\,{\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $26$ | $26$ | $1$ | $50$ | ||||
\(61379\) | $\Q_{61379}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{61379}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | ||
\(35649361663\) | $\Q_{35649361663}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $24$ | $1$ | $24$ | $0$ | $C_{24}$ | $[\ ]^{24}$ | ||
\(202\!\cdots\!873\) | $\Q_{20\!\cdots\!73}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{20\!\cdots\!73}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
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 $19$ | $1$ | $19$ | $0$ | $C_{19}$ | $[\ ]^{19}$ |