Normalized defining polynomial
\( x^{32} + 31 x^{24} + 705 x^{16} + 7936 x^{8} + 65536 \)
Invariants
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{8} + \frac{1}{3}$, $\frac{1}{6} a^{17} - \frac{1}{6} a^{9} + \frac{1}{6} a$, $\frac{1}{12} a^{18} - \frac{1}{12} a^{10} + \frac{1}{12} a^{2}$, $\frac{1}{24} a^{19} - \frac{1}{24} a^{11} + \frac{1}{24} a^{3}$, $\frac{1}{48} a^{20} - \frac{1}{48} a^{12} + \frac{1}{48} a^{4}$, $\frac{1}{96} a^{21} - \frac{1}{96} a^{13} + \frac{1}{96} a^{5}$, $\frac{1}{192} a^{22} + \frac{95}{192} a^{14} + \frac{1}{192} a^{6}$, $\frac{1}{384} a^{23} - \frac{97}{384} a^{15} - \frac{191}{384} a^{7}$, $\frac{1}{541440} a^{24} - \frac{11}{256} a^{16} - \frac{85}{256} a^{8} + \frac{736}{2115}$, $\frac{1}{1082880} a^{25} - \frac{11}{512} a^{17} - \frac{85}{512} a^{9} + \frac{368}{2115} a$, $\frac{1}{2165760} a^{26} - \frac{11}{1024} a^{18} - \frac{85}{1024} a^{10} + \frac{184}{2115} a^{2}$, $\frac{1}{4331520} a^{27} - \frac{11}{2048} a^{19} + \frac{939}{2048} a^{11} - \frac{1931}{4230} a^{3}$, $\frac{1}{8663040} a^{28} - \frac{11}{4096} a^{20} + \frac{939}{4096} a^{12} - \frac{1931}{8460} a^{4}$, $\frac{1}{17326080} a^{29} - \frac{11}{8192} a^{21} - \frac{3157}{8192} a^{13} + \frac{6529}{16920} a^{5}$, $\frac{1}{34652160} a^{30} - \frac{11}{16384} a^{22} + \frac{5035}{16384} a^{14} + \frac{6529}{33840} a^{6}$, $\frac{1}{69304320} a^{31} - \frac{11}{32768} a^{23} + \frac{5035}{32768} a^{15} + \frac{6529}{67680} a^{7}$
Class group and class number
$C_{4}\times C_{4}\times C_{4}$, which has order $64$ (assuming GRH)
Unit group
| Rank: | $15$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{7}{270720} a^{31} + \frac{7193}{270720} a^{7} \) (order $48$) | 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: | \( 374180447387.275 \) (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 | R | ${\href{/LocalNumberField/5.4.0.1}{4} }^{8}$ | 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 | ||||||
| $3$ | 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 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $7$ | 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 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |