Normalized defining polynomial
\( x^{32} - 7x^{28} - 32x^{24} + 791x^{20} - 2945x^{16} + 64071x^{12} - 209952x^{8} - 3720087x^{4} + 43046721 \)
Invariants
Degree: | $32$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 16]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(50522262278163705147147943936000000000000000000000000\) \(\medspace = 2^{64}\cdot 5^{24}\cdot 11^{16}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(44.36\) | 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: | $2^{2}5^{3/4}11^{1/2}\approx 44.35921347589264$ | ||
Ramified primes: | \(2\), \(5\), \(11\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $32$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(440=2^{3}\cdot 5\cdot 11\) | ||
Dirichlet character group: | $\lbrace$$\chi_{440}(1,·)$, $\chi_{440}(131,·)$, $\chi_{440}(133,·)$, $\chi_{440}(263,·)$, $\chi_{440}(397,·)$, $\chi_{440}(21,·)$, $\chi_{440}(23,·)$, $\chi_{440}(153,·)$, $\chi_{440}(287,·)$, $\chi_{440}(417,·)$, $\chi_{440}(419,·)$, $\chi_{440}(43,·)$, $\chi_{440}(177,·)$, $\chi_{440}(307,·)$, $\chi_{440}(309,·)$, $\chi_{440}(439,·)$, $\chi_{440}(67,·)$, $\chi_{440}(197,·)$, $\chi_{440}(199,·)$, $\chi_{440}(329,·)$, $\chi_{440}(331,·)$, $\chi_{440}(87,·)$, $\chi_{440}(89,·)$, $\chi_{440}(219,·)$, $\chi_{440}(221,·)$, $\chi_{440}(351,·)$, $\chi_{440}(353,·)$, $\chi_{440}(109,·)$, $\chi_{440}(111,·)$, $\chi_{440}(241,·)$, $\chi_{440}(243,·)$, $\chi_{440}(373,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{32768}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{5}a^{16}-\frac{1}{5}a^{12}+\frac{1}{5}a^{8}-\frac{1}{5}a^{4}+\frac{1}{5}$, $\frac{1}{15}a^{17}-\frac{1}{15}a^{13}+\frac{1}{15}a^{9}-\frac{1}{15}a^{5}+\frac{1}{15}a$, $\frac{1}{45}a^{18}-\frac{16}{45}a^{14}-\frac{14}{45}a^{10}-\frac{1}{45}a^{6}+\frac{16}{45}a^{2}$, $\frac{1}{135}a^{19}-\frac{61}{135}a^{15}-\frac{59}{135}a^{11}-\frac{46}{135}a^{7}-\frac{29}{135}a^{3}$, $\frac{1}{1192725}a^{20}-\frac{17}{405}a^{16}-\frac{43}{405}a^{12}+\frac{58}{405}a^{8}-\frac{163}{405}a^{4}+\frac{6681}{14725}$, $\frac{1}{3578175}a^{21}-\frac{17}{1215}a^{17}-\frac{43}{1215}a^{13}+\frac{463}{1215}a^{9}+\frac{242}{1215}a^{5}-\frac{8044}{44175}a$, $\frac{1}{10734525}a^{22}-\frac{17}{3645}a^{18}+\frac{1172}{3645}a^{14}+\frac{463}{3645}a^{10}+\frac{242}{3645}a^{6}-\frac{8044}{132525}a^{2}$, $\frac{1}{32203575}a^{23}-\frac{17}{10935}a^{19}+\frac{1172}{10935}a^{15}+\frac{4108}{10935}a^{11}-\frac{3403}{10935}a^{7}-\frac{8044}{397575}a^{3}$, $\frac{1}{96610725}a^{24}-\frac{7}{96610725}a^{20}+\frac{929}{32805}a^{16}+\frac{7996}{32805}a^{12}-\frac{1}{32805}a^{8}+\frac{791}{1192725}a^{4}-\frac{32}{14725}$, $\frac{1}{289832175}a^{25}-\frac{7}{289832175}a^{21}+\frac{929}{98415}a^{17}-\frac{24809}{98415}a^{13}-\frac{1}{98415}a^{9}+\frac{791}{3578175}a^{5}-\frac{32}{44175}a$, $\frac{1}{869496525}a^{26}-\frac{7}{869496525}a^{22}+\frac{929}{295245}a^{18}-\frac{24809}{295245}a^{14}+\frac{98414}{295245}a^{10}-\frac{3577384}{10734525}a^{6}+\frac{44143}{132525}a^{2}$, $\frac{1}{2608489575}a^{27}-\frac{7}{2608489575}a^{23}+\frac{929}{885735}a^{19}+\frac{270436}{885735}a^{15}-\frac{196831}{885735}a^{11}-\frac{14311909}{32203575}a^{7}+\frac{44143}{397575}a^{3}$, $\frac{1}{7825468725}a^{28}-\frac{7}{7825468725}a^{24}-\frac{32}{7825468725}a^{20}-\frac{32518}{531441}a^{16}+\frac{106288}{531441}a^{12}-\frac{19321354}{96610725}a^{8}+\frac{238513}{1192725}a^{4}-\frac{2952}{14725}$, $\frac{1}{23476406175}a^{29}-\frac{7}{23476406175}a^{25}-\frac{32}{23476406175}a^{21}-\frac{32518}{1594323}a^{17}-\frac{425153}{1594323}a^{13}+\frac{77289371}{289832175}a^{9}-\frac{954212}{3578175}a^{5}+\frac{11773}{44175}a$, $\frac{1}{70429218525}a^{30}-\frac{7}{70429218525}a^{26}-\frac{32}{70429218525}a^{22}-\frac{32518}{4782969}a^{18}-\frac{2019476}{4782969}a^{14}-\frac{212542804}{869496525}a^{10}-\frac{954212}{10734525}a^{6}+\frac{55948}{132525}a^{2}$, $\frac{1}{211287655575}a^{31}-\frac{7}{211287655575}a^{27}-\frac{32}{211287655575}a^{23}-\frac{32518}{14348907}a^{19}-\frac{2019476}{14348907}a^{15}-\frac{1082039329}{2608489575}a^{11}+\frac{9780313}{32203575}a^{7}+\frac{188473}{397575}a^{3}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
not computed
Unit group
Rank: | $15$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( \frac{1}{44175} a^{21} + \frac{61126}{44175} a \) (order $40$) | 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: | not computed | 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: | not computed | 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 $
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 | ${\href{/padicField/3.4.0.1}{4} }^{8}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{8}$ | R | ${\href{/padicField/13.4.0.1}{4} }^{8}$ | ${\href{/padicField/17.4.0.1}{4} }^{8}$ | ${\href{/padicField/19.2.0.1}{2} }^{16}$ | ${\href{/padicField/23.4.0.1}{4} }^{8}$ | ${\href{/padicField/29.2.0.1}{2} }^{16}$ | ${\href{/padicField/31.2.0.1}{2} }^{16}$ | ${\href{/padicField/37.4.0.1}{4} }^{8}$ | ${\href{/padicField/41.2.0.1}{2} }^{16}$ | ${\href{/padicField/43.4.0.1}{4} }^{8}$ | ${\href{/padicField/47.4.0.1}{4} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{8}$ | ${\href{/padicField/59.2.0.1}{2} }^{16}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | Deg $16$ | $4$ | $4$ | $32$ | |||
Deg $16$ | $4$ | $4$ | $32$ | ||||
\(5\) | 5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
\(11\) | 11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |