Normalized defining polynomial
\( x^{32} + 113x^{24} + 6208x^{16} + 741393x^{8} + 43046721 \)
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: | \(156712132992285501115475016655592096377279059345801216\) \(\medspace = 2^{96}\cdot 3^{16}\cdot 11^{16}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(45.96\) | 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: | $2^{3}3^{1/2}11^{1/2}\approx 45.95650117230423$ | ||
Ramified primes: | \(2\), \(3\), \(11\) | 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: | \(528=2^{4}\cdot 3\cdot 11\) | ||
Dirichlet character group: | $\lbrace$$\chi_{528}(1,·)$, $\chi_{528}(131,·)$, $\chi_{528}(133,·)$, $\chi_{528}(263,·)$, $\chi_{528}(265,·)$, $\chi_{528}(395,·)$, $\chi_{528}(397,·)$, $\chi_{528}(527,·)$, $\chi_{528}(23,·)$, $\chi_{528}(155,·)$, $\chi_{528}(287,·)$, $\chi_{528}(419,·)$, $\chi_{528}(43,·)$, $\chi_{528}(175,·)$, $\chi_{528}(307,·)$, $\chi_{528}(439,·)$, $\chi_{528}(65,·)$, $\chi_{528}(67,·)$, $\chi_{528}(197,·)$, $\chi_{528}(199,·)$, $\chi_{528}(329,·)$, $\chi_{528}(331,·)$, $\chi_{528}(461,·)$, $\chi_{528}(463,·)$, $\chi_{528}(89,·)$, $\chi_{528}(221,·)$, $\chi_{528}(353,·)$, $\chi_{528}(485,·)$, $\chi_{528}(109,·)$, $\chi_{528}(241,·)$, $\chi_{528}(373,·)$, $\chi_{528}(505,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{32768}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{35}a^{16}+\frac{4}{35}a^{8}+\frac{16}{35}$, $\frac{1}{105}a^{17}-\frac{31}{105}a^{9}+\frac{16}{105}a$, $\frac{1}{315}a^{18}-\frac{31}{315}a^{10}+\frac{16}{315}a^{2}$, $\frac{1}{945}a^{19}-\frac{346}{945}a^{11}-\frac{299}{945}a^{3}$, $\frac{1}{2835}a^{20}+\frac{599}{2835}a^{12}-\frac{1244}{2835}a^{4}$, $\frac{1}{8505}a^{21}+\frac{599}{8505}a^{13}+\frac{1591}{8505}a^{5}$, $\frac{1}{25515}a^{22}-\frac{7906}{25515}a^{14}-\frac{6914}{25515}a^{6}$, $\frac{1}{76545}a^{23}+\frac{17609}{76545}a^{15}-\frac{6914}{76545}a^{7}$, $\frac{1}{1425574080}a^{24}+\frac{929}{229635}a^{16}-\frac{98414}{229635}a^{8}+\frac{62193}{217280}$, $\frac{1}{4276722240}a^{25}+\frac{929}{688905}a^{17}-\frac{98414}{688905}a^{9}+\frac{279473}{651840}a$, $\frac{1}{12830166720}a^{26}+\frac{929}{2066715}a^{18}-\frac{98414}{2066715}a^{10}+\frac{931313}{1955520}a^{2}$, $\frac{1}{38490500160}a^{27}+\frac{929}{6200145}a^{19}+\frac{1968301}{6200145}a^{11}+\frac{931313}{5866560}a^{3}$, $\frac{1}{115471500480}a^{28}+\frac{929}{18600435}a^{20}+\frac{8168446}{18600435}a^{12}+\frac{931313}{17599680}a^{4}$, $\frac{1}{346414501440}a^{29}+\frac{929}{55801305}a^{21}-\frac{10431989}{55801305}a^{13}+\frac{931313}{52799040}a^{5}$, $\frac{1}{1039243504320}a^{30}+\frac{929}{167403915}a^{22}-\frac{66233294}{167403915}a^{14}+\frac{931313}{158397120}a^{6}$, $\frac{1}{3117730512960}a^{31}+\frac{929}{502211745}a^{23}-\frac{66233294}{502211745}a^{15}-\frac{157465807}{475191360}a^{7}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
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: | \( \frac{28589}{38490500160} a^{27} + \frac{253}{6200145} a^{19} + \frac{15467}{6200145} a^{11} + \frac{61479}{217280} a^{3} \) (order $48$) | 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 | R | ${\href{/padicField/5.4.0.1}{4} }^{8}$ | ${\href{/padicField/7.2.0.1}{2} }^{16}$ | R | ${\href{/padicField/13.4.0.1}{4} }^{8}$ | ${\href{/padicField/17.2.0.1}{2} }^{16}$ | ${\href{/padicField/19.4.0.1}{4} }^{8}$ | ${\href{/padicField/23.2.0.1}{2} }^{16}$ | ${\href{/padicField/29.4.0.1}{4} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{16}$ | ${\href{/padicField/37.4.0.1}{4} }^{8}$ | ${\href{/padicField/41.2.0.1}{2} }^{16}$ | ${\href{/padicField/43.4.0.1}{4} }^{8}$ | ${\href{/padicField/47.2.0.1}{2} }^{16}$ | ${\href{/padicField/53.4.0.1}{4} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.16.48.1 | $x^{16} - 8 x^{15} + 64 x^{14} + 8 x^{13} + 76 x^{12} + 48 x^{11} + 64 x^{10} + 256 x^{9} + 56 x^{8} + 144 x^{7} + 160 x^{6} + 432 x^{5} + 456 x^{4} + 256 x^{2} + 288 x + 516$ | $8$ | $2$ | $48$ | $C_4\times C_2^2$ | $[2, 3, 4]^{2}$ |
2.16.48.1 | $x^{16} - 8 x^{15} + 64 x^{14} + 8 x^{13} + 76 x^{12} + 48 x^{11} + 64 x^{10} + 256 x^{9} + 56 x^{8} + 144 x^{7} + 160 x^{6} + 432 x^{5} + 456 x^{4} + 256 x^{2} + 288 x + 516$ | $8$ | $2$ | $48$ | $C_4\times C_2^2$ | $[2, 3, 4]^{2}$ | |
\(3\) | 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
\(11\) | 11.8.4.1 | $x^{8} + 60 x^{6} + 20 x^{5} + 970 x^{4} - 280 x^{3} + 4664 x^{2} - 5460 x + 2325$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
11.8.4.1 | $x^{8} + 60 x^{6} + 20 x^{5} + 970 x^{4} - 280 x^{3} + 4664 x^{2} - 5460 x + 2325$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
11.8.4.1 | $x^{8} + 60 x^{6} + 20 x^{5} + 970 x^{4} - 280 x^{3} + 4664 x^{2} - 5460 x + 2325$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
11.8.4.1 | $x^{8} + 60 x^{6} + 20 x^{5} + 970 x^{4} - 280 x^{3} + 4664 x^{2} - 5460 x + 2325$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |