Normalized defining polynomial
\( x^{32} - 799 x^{24} + 631840 x^{16} - 5242239 x^{8} + 43046721 \)
Invariants
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{217} a^{16} - \frac{74}{217} a^{8} + \frac{51}{217}$, $\frac{1}{651} a^{17} + \frac{143}{651} a^{9} + \frac{268}{651} a$, $\frac{1}{1953} a^{18} + \frac{794}{1953} a^{10} - \frac{383}{1953} a^{2}$, $\frac{1}{5859} a^{19} + \frac{794}{5859} a^{11} - \frac{2336}{5859} a^{3}$, $\frac{1}{17577} a^{20} + \frac{6653}{17577} a^{12} + \frac{3523}{17577} a^{4}$, $\frac{1}{52731} a^{21} + \frac{24230}{52731} a^{13} - \frac{14054}{52731} a^{5}$, $\frac{1}{158193} a^{22} - \frac{28501}{158193} a^{14} - \frac{14054}{158193} a^{6}$, $\frac{1}{474579} a^{23} - \frac{186694}{474579} a^{15} + \frac{144139}{474579} a^{7}$, $\frac{1}{899573986080} a^{24} - \frac{61}{203391} a^{16} - \frac{69359}{203391} a^{8} + \frac{32223041}{137109280}$, $\frac{1}{2698721958240} a^{25} - \frac{61}{610173} a^{17} - \frac{272750}{610173} a^{9} + \frac{32223041}{411327840} a$, $\frac{1}{8096165874720} a^{26} - \frac{61}{1830519} a^{18} + \frac{337423}{1830519} a^{10} + \frac{443550881}{1233983520} a^{2}$, $\frac{1}{24288497624160} a^{27} - \frac{61}{5491557} a^{19} - \frac{1493096}{5491557} a^{11} + \frac{443550881}{3701950560} a^{3}$, $\frac{1}{72865492872480} a^{28} - \frac{61}{16474671} a^{20} + \frac{3998461}{16474671} a^{12} - \frac{3258399679}{11105851680} a^{4}$, $\frac{1}{218596478617440} a^{29} - \frac{61}{49424013} a^{21} + \frac{20473132}{49424013} a^{13} - \frac{3258399679}{33317555040} a^{5}$, $\frac{1}{655789435852320} a^{30} - \frac{61}{148272039} a^{22} + \frac{20473132}{148272039} a^{14} - \frac{36575954719}{99952665120} a^{6}$, $\frac{1}{1967368307556960} a^{31} - \frac{61}{444816117} a^{23} - \frac{127798907}{444816117} a^{15} - \frac{136528619839}{299857995360} a^{7}$
Class group and class number
Not computed
Unit group
| Rank: | $15$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{2143717}{24288497624160} a^{27} + \frac{2683}{38440899} a^{19} - \frac{2117473}{38440899} a^{11} + \frac{651969}{137109280} a^{3} \) (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 | ${\href{/LocalNumberField/5.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/7.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ | R | ${\href{/LocalNumberField/17.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$ |
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 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $13$ | 13.8.4.1 | $x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 13.8.4.1 | $x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 13.8.4.1 | $x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 13.8.4.1 | $x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |