Normalized defining polynomial
\( x^{28} - x + 2 \)
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: | \(4448716805188928174143786050979756117181869491005\) \(\medspace = 5\cdot 6249889\cdot 26339207\cdot 398948659\cdot 13547923849744568758845893\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(54.63\) | 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: | $5^{1/2}6249889^{1/2}26339207^{1/2}398948659^{1/2}13547923849744568758845893^{1/2}\approx 2.1091981427046934e+24$ | ||
Ramified primes: | \(5\), \(6249889\), \(26339207\), \(398948659\), \(13547923849744568758845893\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{44487\!\cdots\!91005}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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: | $a^{24}-a^{20}+a^{16}-a^{12}+a^{8}-a^{4}+1$, $a^{19}-a^{10}+1$, $a^{25}-a^{22}+a^{16}-a^{13}+a^{7}-a^{4}+1$, $a^{24}+a^{22}-a^{18}-a^{16}+a^{12}+a^{10}-a^{6}-a^{4}+1$, $a^{24}+a^{23}+a^{22}+a^{21}+a^{16}-a^{14}-a^{13}-2a^{12}-a^{11}+a^{8}+a^{5}+a^{3}+a^{2}+1$, $a^{23}-a^{21}-a^{20}-a^{17}-a^{13}-a^{10}+a^{8}+2a^{7}-a^{6}+a^{3}-a^{2}+1$, $a^{27}-2a^{25}-a^{24}-a^{22}-a^{21}-a^{20}+2a^{18}+a^{17}-a^{16}+a^{15}+2a^{14}+a^{10}+2a^{9}-a^{8}-3a^{7}-2a^{4}-a^{3}-a^{2}+a+1$, $2a^{26}-a^{25}-a^{24}+2a^{23}-a^{21}+2a^{20}+2a^{19}-2a^{18}+2a^{16}-2a^{15}-a^{14}+3a^{13}-2a^{11}+2a^{10}+2a^{9}-3a^{8}+a^{7}+3a^{6}-3a^{5}-2a^{4}+3a^{3}-a^{2}-3a+3$, $3a^{27}-a^{26}-4a^{25}-a^{24}+2a^{23}-4a^{21}-2a^{20}+3a^{19}+3a^{18}-a^{17}-3a^{16}+2a^{15}+5a^{14}-4a^{12}-a^{11}+5a^{10}+2a^{9}-5a^{8}-3a^{7}+3a^{6}+4a^{5}-3a^{4}-6a^{3}+a^{2}+5a-3$, $a^{25}+a^{24}+a^{23}+a^{20}-a^{19}+2a^{17}+a^{15}+a^{14}-a^{13}+2a^{7}-2a^{5}+a^{4}-a^{3}-2a^{2}+2a-1$, $2a^{27}+a^{26}-a^{25}-a^{24}-a^{23}-a^{22}+a^{21}+a^{20}-a^{19}+3a^{17}+a^{16}-a^{15}-a^{14}-a^{13}-a^{12}+3a^{11}+2a^{10}-2a^{9}-2a^{8}+a^{7}-a^{6}-a^{5}-a^{3}-a^{2}+4a+1$, $a^{27}-a^{26}-a^{22}-a^{21}+3a^{20}-2a^{19}+a^{18}-a^{17}+3a^{16}-3a^{15}+a^{14}-a^{13}+3a^{12}-3a^{11}+3a^{10}-2a^{9}+5a^{8}-6a^{7}+3a^{6}-a^{5}+2a^{4}-3a^{3}+3a^{2}+a-1$, $7a^{27}-a^{25}-3a^{24}-4a^{23}+8a^{22}-a^{21}+a^{20}-4a^{19}-4a^{18}+8a^{17}-2a^{16}+4a^{15}-7a^{14}-3a^{13}+8a^{12}-2a^{11}+6a^{10}-10a^{9}-2a^{8}+6a^{7}+9a^{5}-13a^{4}-2a^{3}+4a^{2}+a+5$ (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: | \( 35776227834950.36 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{14}\cdot 35776227834950.36 \cdot 1}{2\cdot\sqrt{4448716805188928174143786050979756117181869491005}}\cr\approx \mathstrut & 1.26755066874265 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 304888344611713860501504000000 |
The 3718 conjugacy class representatives for $S_{28}$ are not computed |
Character table for $S_{28}$ is not computed |
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 | $18{,}\,{\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.2.0.1}{2} }{,}\,{\href{/padicField/2.1.0.1}{1} }^{2}$ | ${\href{/padicField/3.12.0.1}{12} }^{2}{,}\,{\href{/padicField/3.4.0.1}{4} }$ | R | $25{,}\,{\href{/padicField/7.3.0.1}{3} }$ | $27{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/13.8.0.1}{8} }$ | $25{,}\,{\href{/padicField/17.3.0.1}{3} }$ | $20{,}\,{\href{/padicField/19.8.0.1}{8} }$ | ${\href{/padicField/23.11.0.1}{11} }{,}\,{\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ | ${\href{/padicField/29.9.0.1}{9} }{,}\,{\href{/padicField/29.5.0.1}{5} }^{3}{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }$ | ${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | $26{,}\,{\href{/padicField/43.2.0.1}{2} }$ | $16{,}\,{\href{/padicField/47.11.0.1}{11} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | $19{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.4.0.1 | $x^{4} + 4 x^{2} + 4 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
5.22.0.1 | $x^{22} + x^{12} + 3 x^{11} + 4 x^{9} + 3 x^{8} + 2 x^{6} + 2 x^{5} + 4 x^{3} + 3 x^{2} + 3 x + 2$ | $1$ | $22$ | $0$ | 22T1 | $[\ ]^{22}$ | |
\(6249889\) | 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 $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $17$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ | ||
\(26339207\) | $\Q_{26339207}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
Deg $16$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ | ||
\(398948659\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $17$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ | ||
\(135\!\cdots\!893\) | $\Q_{13\!\cdots\!93}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ |