Normalized defining polynomial
\( x^{32} - 1457 x^{24} + 565953 x^{16} - 372992 x^{8} + 65536 \)
Invariants
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3}$, $\frac{1}{3} a^{9} + \frac{1}{3} a$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{5}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{6}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{7}$, $\frac{1}{63} a^{16} - \frac{4}{63} a^{8} + \frac{4}{63}$, $\frac{1}{126} a^{17} + \frac{17}{126} a^{9} + \frac{25}{126} a$, $\frac{1}{252} a^{18} - \frac{25}{252} a^{10} + \frac{109}{252} a^{2}$, $\frac{1}{504} a^{19} - \frac{25}{504} a^{11} - \frac{143}{504} a^{3}$, $\frac{1}{1008} a^{20} + \frac{143}{1008} a^{12} - \frac{479}{1008} a^{4}$, $\frac{1}{2016} a^{21} + \frac{143}{2016} a^{13} - \frac{479}{2016} a^{5}$, $\frac{1}{4032} a^{22} + \frac{143}{4032} a^{14} + \frac{1537}{4032} a^{6}$, $\frac{1}{8064} a^{23} - \frac{1201}{8064} a^{15} + \frac{193}{8064} a^{7}$, $\frac{1}{3034983168} a^{24} + \frac{3625669}{1011661056} a^{16} - \frac{112406549}{1011661056} a^{8} + \frac{788758}{11855403}$, $\frac{1}{6069966336} a^{25} + \frac{3625669}{2023322112} a^{17} + \frac{224813803}{2023322112} a^{9} + \frac{4740559}{23710806} a$, $\frac{1}{12139932672} a^{26} + \frac{3625669}{4046644224} a^{18} + \frac{224813803}{4046644224} a^{10} - \frac{18970247}{47421612} a^{2}$, $\frac{1}{24279865344} a^{27} + \frac{3625669}{8093288448} a^{19} - \frac{1124067605}{8093288448} a^{11} - \frac{34777451}{94843224} a^{3}$, $\frac{1}{48559730688} a^{28} + \frac{3625669}{16186576896} a^{20} - \frac{1124067605}{16186576896} a^{12} + \frac{60065773}{189686448} a^{4}$, $\frac{1}{97119461376} a^{29} + \frac{3625669}{32373153792} a^{21} - \frac{1124067605}{32373153792} a^{13} + \frac{60065773}{379372896} a^{5}$, $\frac{1}{194238922752} a^{30} + \frac{3625669}{64746307584} a^{22} - \frac{1124067605}{64746307584} a^{14} - \frac{319307123}{758745792} a^{6}$, $\frac{1}{388477845504} a^{31} + \frac{3625669}{129492615168} a^{23} + \frac{20458034923}{129492615168} a^{15} - \frac{66391859}{1517491584} a^{7}$
Class group and class number
$C_{2}\times C_{8}\times C_{8}$, which has order $128$ (assuming GRH)
Unit group
| Rank: | $15$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{136187}{6069966336} a^{25} + \frac{66134233}{2023322112} a^{17} - \frac{25676808809}{2023322112} a^{9} + \frac{54395189}{11855403} a \) (order $16$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 3469532895544.2065 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^3\times C_4$ (as 32T34):
| An abelian group of order 32 |
| The 32 conjugacy class representatives for $C_2^3\times C_4$ |
| Character table for $C_2^3\times C_4$ is not computed |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/LocalNumberField/3.4.0.1}{4} }^{8}$ | R | R | ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$ |
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 | Data not computed | ||||||
| $5$ | 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $7$ | 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |