Normalized defining polynomial
\( x^{27} + x - 4 \)
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)
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Discriminant: | \(-1997015367217552532430060947581195242227227631565144064\) \(\medspace = -\,2^{26}\cdot 97\cdot 739\cdot 7753\cdot 53\!\cdots\!99\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(102.59\) | 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: | not computed | ||
Ramified primes: | \(2\), \(97\), \(739\), \(7753\), \(53544\!\cdots\!89899\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-29757\!\cdots\!21801}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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)
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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)
| |
Fundamental units: | $5a^{26}+a^{25}-2a^{24}-9a^{23}+12a^{22}+a^{21}-11a^{20}+3a^{19}+3a^{18}+8a^{17}-15a^{16}-a^{15}+18a^{14}-9a^{13}-4a^{12}-5a^{11}+17a^{10}+2a^{9}-27a^{8}+19a^{7}+6a^{6}-2a^{5}-15a^{4}-3a^{3}+39a^{2}-33a-5$, $4a^{25}+7a^{24}+8a^{23}+4a^{22}-3a^{21}-8a^{20}-12a^{19}-13a^{18}-6a^{17}+7a^{16}+15a^{15}+15a^{14}+9a^{13}-4a^{12}-19a^{11}-23a^{10}-16a^{9}-4a^{8}+12a^{7}+25a^{6}+20a^{5}+3a^{4}-9a^{3}-19a^{2}-27a-21$, $2a^{26}-a^{25}-4a^{23}-a^{22}-3a^{21}+3a^{20}+2a^{19}+6a^{18}-a^{17}-2a^{16}-9a^{15}-6a^{14}-7a^{13}+a^{11}+3a^{10}-2a^{9}-6a^{8}-8a^{7}-9a^{6}-5a^{5}-6a^{4}+a^{3}-4a^{2}+4a-1$, $13a^{26}+15a^{25}-27a^{24}-3a^{23}+31a^{22}-17a^{21}-25a^{20}+35a^{19}+9a^{18}-39a^{17}+20a^{16}+35a^{15}-42a^{14}-16a^{13}+50a^{12}-16a^{11}-49a^{10}+42a^{9}+27a^{8}-64a^{7}+3a^{6}+71a^{5}-37a^{4}-48a^{3}+82a^{2}+16a-91$, $2a^{24}+2a^{23}+2a^{22}+5a^{21}+3a^{20}+7a^{19}+5a^{18}+9a^{17}+4a^{16}+4a^{15}+3a^{14}+a^{13}+3a^{12}-2a^{11}+3a^{10}-6a^{9}-5a^{8}-13a^{7}-9a^{6}-8a^{5}-11a^{4}-5a^{3}-12a^{2}-4a-15$, $2a^{26}+22a^{25}+6a^{24}-22a^{23}-16a^{22}+17a^{21}+25a^{20}-9a^{19}-31a^{18}-a^{17}+34a^{16}+16a^{15}-30a^{14}-31a^{13}+23a^{12}+44a^{11}-12a^{10}-53a^{9}-7a^{8}+54a^{7}+29a^{6}-50a^{5}-51a^{4}+39a^{3}+70a^{2}-16a-79$, $a^{26}+3a^{25}+2a^{24}-a^{23}-5a^{22}-8a^{21}-9a^{20}-8a^{19}-4a^{18}-2a^{17}+a^{15}-2a^{14}-3a^{13}-5a^{12}-6a^{11}-3a^{10}+2a^{9}+10a^{8}+17a^{7}+23a^{6}+21a^{5}+13a^{4}+5a^{3}-7a^{2}-9a-3$, $3a^{26}-3a^{24}+7a^{23}-5a^{22}+a^{21}+7a^{20}-11a^{19}+12a^{18}-9a^{17}+5a^{16}-4a^{15}-a^{14}+3a^{13}-12a^{12}+12a^{11}-11a^{10}-2a^{9}+11a^{8}-15a^{7}+9a^{6}+4a^{5}-12a^{4}+17a^{3}-14a^{2}+16a-11$, $3a^{26}+10a^{25}+5a^{24}+3a^{23}+12a^{22}+9a^{21}+7a^{19}+11a^{18}-3a^{17}-5a^{16}+6a^{15}-4a^{14}-16a^{13}-5a^{12}-5a^{11}-22a^{10}-15a^{9}-2a^{8}-20a^{7}-23a^{6}+3a^{5}-2a^{4}-22a^{3}+3a^{2}+20a-1$, $4a^{26}+a^{25}+4a^{24}+a^{23}-3a^{21}-3a^{20}-6a^{19}-4a^{18}-2a^{17}-3a^{16}+2a^{15}+4a^{14}+8a^{13}+2a^{12}+7a^{11}+2a^{10}+5a^{9}-7a^{8}+a^{7}-13a^{6}-5a^{5}-11a^{4}+a^{3}-9a^{2}+12a+9$, $31a^{26}-35a^{25}+41a^{24}-44a^{23}+44a^{22}-48a^{21}+53a^{20}-51a^{19}+50a^{18}-55a^{17}+53a^{16}-46a^{15}+45a^{14}-43a^{13}+33a^{12}-23a^{11}+18a^{10}-10a^{9}-8a^{8}+22a^{7}-29a^{6}+49a^{5}-74a^{4}+84a^{3}-99a^{2}+128a-117$, $a^{26}+4a^{25}+3a^{24}+7a^{23}+a^{22}+6a^{21}-3a^{20}-2a^{19}-5a^{18}-8a^{17}-3a^{16}-8a^{15}+2a^{14}-2a^{13}+9a^{12}+3a^{11}+10a^{10}+4a^{9}+5a^{8}+5a^{7}-8a^{6}+3a^{5}-16a^{4}-3a^{3}-12a^{2}-9a-1$, $47a^{26}+46a^{25}+21a^{24}-10a^{23}-36a^{22}-49a^{21}-43a^{20}-18a^{19}+17a^{18}+52a^{17}+73a^{16}+55a^{15}-3a^{14}-59a^{13}-89a^{12}-71a^{11}-22a^{10}+42a^{9}+77a^{8}+83a^{7}+66a^{6}+20a^{5}-59a^{4}-117a^{3}-128a^{2}-65a+83$ (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: | \( 78800537748117780 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{13}\cdot 78800537748117780 \cdot 1}{2\cdot\sqrt{1997015367217552532430060947581195242227227631565144064}}\cr\approx \mathstrut & 1.32640688174117 \end{aligned}\] (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 | ${\href{/padicField/3.6.0.1}{6} }^{4}{,}\,{\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | $27$ | $18{,}\,{\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | $18{,}\,{\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/17.9.0.1}{9} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/19.5.0.1}{5} }$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | $17{,}\,{\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | $16{,}\,{\href{/padicField/31.9.0.1}{9} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | $17{,}\,{\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.4.0.1}{4} }$ | ${\href{/padicField/41.11.0.1}{11} }{,}\,{\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | $20{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $27$ | ${\href{/padicField/59.11.0.1}{11} }{,}\,{\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
2.2.2.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
Deg $24$ | $2$ | $12$ | $24$ | ||||
\(97\) | $\Q_{97}$ | $x + 92$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
97.2.1.2 | $x^{2} + 485$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
97.3.0.1 | $x^{3} + 9 x + 92$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
97.3.0.1 | $x^{3} + 9 x + 92$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
97.18.0.1 | $x^{18} + x^{2} - x + 60$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | |
\(739\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | ||
\(7753\) | $\Q_{7753}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
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 $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
\(535\!\cdots\!899\) | $\Q_{53\!\cdots\!99}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{53\!\cdots\!99}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $19$ | $1$ | $19$ | $0$ | $C_{19}$ | $[\ ]^{19}$ |