Normalized defining polynomial
\( x^{28} - x + 3 \)
Invariants
Degree: | $28$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 14]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(252754417691002970292836280287462187156691663543501629\) \(\medspace = 3^{27}\cdot 11\cdot 23\cdot 83\cdot 3121\cdot 20446009\cdot 38719859\cdot 223913377\cdot 2853054979\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(80.77\) | 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: | \(3\), \(11\), \(23\), \(83\), \(3121\), \(20446009\), \(38719859\), \(223913377\), \(2853054979\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{99436\!\cdots\!67701}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$
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: | $a^{19}-a^{10}+1$, $a^{14}+2a^{7}+2$, $a^{25}-a^{22}+a^{16}-a^{13}+a^{7}-a^{4}+1$, $a^{27}-a^{26}+a^{24}-a^{23}+a^{21}-a^{20}+a^{18}-a^{17}+a^{15}-a^{14}+a^{12}-a^{11}+a^{9}-a^{8}+a^{6}-a^{5}+a^{3}-a^{2}+1$, $a^{22}+a^{21}-a^{18}-a^{17}+a^{14}-a^{12}-a^{7}+a^{6}+2a^{5}-a^{3}-3a^{2}-a+2$, $2a^{27}-2a^{24}-a^{23}-a^{22}+a^{21}+2a^{20}+a^{19}-3a^{17}+4a^{14}+a^{13}-2a^{12}-4a^{11}-3a^{10}+3a^{9}+3a^{8}+4a^{7}-3a^{6}-2a^{5}-3a^{4}+3a^{3}+3a^{2}-2$, $3a^{26}+a^{25}-3a^{24}-3a^{23}+3a^{21}+4a^{20}-5a^{18}-3a^{17}+3a^{16}+3a^{15}-a^{14}-2a^{13}-a^{12}+2a^{11}+5a^{10}-8a^{8}-5a^{7}+5a^{6}+7a^{5}+2a^{4}-3a^{3}-6a^{2}-a+7$, $a^{27}+a^{26}-a^{25}-a^{24}+a^{23}-3a^{21}-a^{20}+2a^{19}+a^{18}-a^{17}+a^{16}+2a^{15}-2a^{14}-4a^{13}+2a^{11}-3a^{10}-4a^{9}+a^{8}+2a^{7}-2a^{6}+4a^{4}-5a^{2}-3a+1$, $62a^{27}+33a^{26}-31a^{25}-70a^{24}-44a^{23}+23a^{22}+76a^{21}+58a^{20}-15a^{19}-78a^{18}-74a^{17}+4a^{16}+80a^{15}+88a^{14}+13a^{13}-82a^{12}-102a^{11}-31a^{10}+78a^{9}+119a^{8}+49a^{7}-69a^{6}-135a^{5}-72a^{4}+59a^{3}+145a^{2}+102a-109$, $a^{27}+4a^{26}+3a^{25}-a^{24}-3a^{23}-2a^{20}-2a^{19}+a^{18}+a^{17}+a^{16}+4a^{15}+3a^{14}-4a^{13}-10a^{12}-4a^{11}+3a^{10}+6a^{9}+7a^{8}+6a^{7}-3a^{6}-14a^{5}-10a^{4}+3a^{3}+10a^{2}+6a+4$, $a^{25}+a^{24}+a^{23}+a^{22}-a^{21}-a^{20}-a^{18}-a^{17}-a^{16}-a^{15}-a^{14}-2a^{13}+2a^{11}+2a^{10}+4a^{9}+3a^{8}+a^{7}+a^{6}-2a^{5}-3a^{4}-2a^{3}-2a^{2}+a+1$, $4a^{27}+2a^{25}-2a^{24}-2a^{23}-2a^{22}-6a^{21}-3a^{20}-9a^{19}-7a^{18}-10a^{17}-13a^{16}-11a^{15}-17a^{14}-13a^{13}-17a^{12}-17a^{11}-14a^{10}-19a^{9}-13a^{8}-18a^{7}-17a^{6}-13a^{5}-20a^{4}-10a^{3}-18a^{2}-11a-11$, $39a^{27}-131a^{26}+28a^{25}+10a^{24}-149a^{23}+66a^{22}+46a^{21}-113a^{20}+140a^{19}+84a^{18}-106a^{17}+153a^{16}+21a^{15}-208a^{14}+90a^{13}-62a^{12}-242a^{11}+178a^{10}+34a^{9}-126a^{8}+322a^{7}+49a^{6}-190a^{5}+244a^{4}-130a^{3}-327a^{2}+238a-185$ (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: | \( 5281401797704566.0 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{14}\cdot 5281401797704566.0 \cdot 1}{2\cdot\sqrt{252754417691002970292836280287462187156691663543501629}}\cr\approx \mathstrut & 0.785033143619367 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 304888344611713860501504000000 |
The 3718 conjugacy class representatives for $S_{28}$ |
Character table for $S_{28}$ |
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 | $28$ | R | $22{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | R | $20{,}\,{\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.14.0.1}{14} }{,}\,{\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | $20{,}\,{\href{/padicField/19.8.0.1}{8} }$ | R | ${\href{/padicField/29.14.0.1}{14} }{,}\,{\href{/padicField/29.11.0.1}{11} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.11.0.1}{11} }^{2}{,}\,{\href{/padicField/31.6.0.1}{6} }$ | $25{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.12.0.1}{12} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | $27{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | $19{,}\,{\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.14.0.1}{14} }{,}\,{\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 $27$ | $27$ | $1$ | $27$ | ||||
\(11\) | 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Deg $26$ | $1$ | $26$ | $0$ | $C_{26}$ | $[\ ]^{26}$ | ||
\(23\) | $\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
23.10.0.1 | $x^{10} + 17 x^{5} + 5 x^{4} + 15 x^{3} + 6 x^{2} + x + 5$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
23.14.0.1 | $x^{14} + x^{8} + 5 x^{7} + 16 x^{6} + x^{5} + 18 x^{4} + 19 x^{3} + x^{2} + 22 x + 5$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | |
\(83\) | $\Q_{83}$ | $x + 81$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{83}$ | $x + 81$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
83.2.1.2 | $x^{2} + 83$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
83.6.0.1 | $x^{6} + x^{4} + 76 x^{3} + 32 x^{2} + 17 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
83.18.0.1 | $x^{18} - x + 35$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | |
\(3121\) | $\Q_{3121}$ | $x$ | $1$ | $1$ | $0$ | 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 $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | ||
\(20446009\) | $\Q_{20446009}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | ||
\(38719859\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $21$ | $1$ | $21$ | $0$ | $C_{21}$ | $[\ ]^{21}$ | ||
\(223913377\) | $\Q_{223913377}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{223913377}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{223913377}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
\(2853054979\) | $\Q_{2853054979}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{2853054979}$ | $x$ | $1$ | $1$ | $0$ | 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 $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ |