Normalized defining polynomial
\( x^{32} + 113 x^{24} + 6208 x^{16} + 741393 x^{8} + 43046721 \)
Invariants
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{35} a^{16} + \frac{4}{35} a^{8} + \frac{16}{35}$, $\frac{1}{105} a^{17} - \frac{31}{105} a^{9} + \frac{16}{105} a$, $\frac{1}{315} a^{18} - \frac{31}{315} a^{10} + \frac{16}{315} a^{2}$, $\frac{1}{945} a^{19} - \frac{346}{945} a^{11} - \frac{299}{945} a^{3}$, $\frac{1}{2835} a^{20} + \frac{599}{2835} a^{12} - \frac{1244}{2835} a^{4}$, $\frac{1}{8505} a^{21} + \frac{599}{8505} a^{13} + \frac{1591}{8505} a^{5}$, $\frac{1}{25515} a^{22} - \frac{7906}{25515} a^{14} - \frac{6914}{25515} a^{6}$, $\frac{1}{76545} a^{23} + \frac{17609}{76545} a^{15} - \frac{6914}{76545} a^{7}$, $\frac{1}{1425574080} a^{24} + \frac{929}{229635} a^{16} - \frac{98414}{229635} a^{8} + \frac{62193}{217280}$, $\frac{1}{4276722240} a^{25} + \frac{929}{688905} a^{17} - \frac{98414}{688905} a^{9} + \frac{279473}{651840} a$, $\frac{1}{12830166720} a^{26} + \frac{929}{2066715} a^{18} - \frac{98414}{2066715} a^{10} + \frac{931313}{1955520} a^{2}$, $\frac{1}{38490500160} a^{27} + \frac{929}{6200145} a^{19} + \frac{1968301}{6200145} a^{11} + \frac{931313}{5866560} a^{3}$, $\frac{1}{115471500480} a^{28} + \frac{929}{18600435} a^{20} + \frac{8168446}{18600435} a^{12} + \frac{931313}{17599680} a^{4}$, $\frac{1}{346414501440} a^{29} + \frac{929}{55801305} a^{21} - \frac{10431989}{55801305} a^{13} + \frac{931313}{52799040} a^{5}$, $\frac{1}{1039243504320} a^{30} + \frac{929}{167403915} a^{22} - \frac{66233294}{167403915} a^{14} + \frac{931313}{158397120} a^{6}$, $\frac{1}{3117730512960} a^{31} + \frac{929}{502211745} a^{23} - \frac{66233294}{502211745} a^{15} - \frac{157465807}{475191360} a^{7}$
Class group and class number
Not computed
Unit group
| Rank: | $15$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( \frac{28589}{38490500160} a^{27} + \frac{253}{6200145} a^{19} + \frac{15467}{6200145} a^{11} + \frac{61479}{217280} a^{3} \) (order $48$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Not computed | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | Not computed | 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 | R | ${\href{/LocalNumberField/5.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/7.2.0.1}{2} }^{16}$ | R | ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$ |
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 | ||||||
| $3$ | 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $11$ | 11.8.4.1 | $x^{8} + 484 x^{4} - 1331 x^{2} + 58564$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 11.8.4.1 | $x^{8} + 484 x^{4} - 1331 x^{2} + 58564$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 11.8.4.1 | $x^{8} + 484 x^{4} - 1331 x^{2} + 58564$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 11.8.4.1 | $x^{8} + 484 x^{4} - 1331 x^{2} + 58564$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |