Normalized defining polynomial
\( x^{32} - x^{28} - 15x^{24} + 31x^{20} + 209x^{16} + 496x^{12} - 3840x^{8} - 4096x^{4} + 65536 \)
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: | \(36539993586348786372837376000000000000000000000000\) \(\medspace = 2^{64}\cdot 5^{24}\cdot 7^{16}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(35.39\) | 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: | $2^{2}5^{3/4}7^{1/2}\approx 35.38641077306205$ | ||
Ramified primes: | \(2\), \(5\), \(7\) | 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: | \(280=2^{3}\cdot 5\cdot 7\) | ||
Dirichlet character group: | $\lbrace$$\chi_{280}(1,·)$, $\chi_{280}(139,·)$, $\chi_{280}(13,·)$, $\chi_{280}(279,·)$, $\chi_{280}(153,·)$, $\chi_{280}(27,·)$, $\chi_{280}(29,·)$, $\chi_{280}(69,·)$, $\chi_{280}(167,·)$, $\chi_{280}(41,·)$, $\chi_{280}(43,·)$, $\chi_{280}(181,·)$, $\chi_{280}(183,·)$, $\chi_{280}(111,·)$, $\chi_{280}(57,·)$, $\chi_{280}(267,·)$, $\chi_{280}(197,·)$, $\chi_{280}(71,·)$, $\chi_{280}(141,·)$, $\chi_{280}(209,·)$, $\chi_{280}(83,·)$, $\chi_{280}(223,·)$, $\chi_{280}(97,·)$, $\chi_{280}(99,·)$, $\chi_{280}(237,·)$, $\chi_{280}(239,·)$, $\chi_{280}(113,·)$, $\chi_{280}(211,·)$, $\chi_{280}(169,·)$, $\chi_{280}(251,·)$, $\chi_{280}(253,·)$, $\chi_{280}(127,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{32768}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3}a^{16}+\frac{1}{3}a^{12}+\frac{1}{3}a^{8}+\frac{1}{3}a^{4}+\frac{1}{3}$, $\frac{1}{6}a^{17}+\frac{1}{6}a^{13}+\frac{1}{6}a^{9}+\frac{1}{6}a^{5}+\frac{1}{6}a$, $\frac{1}{12}a^{18}-\frac{5}{12}a^{14}+\frac{1}{12}a^{10}-\frac{5}{12}a^{6}+\frac{1}{12}a^{2}$, $\frac{1}{24}a^{19}+\frac{7}{24}a^{15}+\frac{1}{24}a^{11}+\frac{7}{24}a^{7}+\frac{1}{24}a^{3}$, $\frac{1}{10032}a^{20}-\frac{1}{48}a^{16}+\frac{17}{48}a^{12}-\frac{1}{48}a^{8}+\frac{17}{48}a^{4}+\frac{80}{209}$, $\frac{1}{20064}a^{21}-\frac{1}{96}a^{17}+\frac{17}{96}a^{13}-\frac{1}{96}a^{9}+\frac{17}{96}a^{5}+\frac{40}{209}a$, $\frac{1}{40128}a^{22}-\frac{1}{192}a^{18}-\frac{79}{192}a^{14}+\frac{95}{192}a^{10}+\frac{17}{192}a^{6}-\frac{169}{418}a^{2}$, $\frac{1}{80256}a^{23}-\frac{1}{384}a^{19}+\frac{113}{384}a^{15}-\frac{97}{384}a^{11}-\frac{175}{384}a^{7}-\frac{169}{836}a^{3}$, $\frac{1}{160512}a^{24}-\frac{1}{160512}a^{20}+\frac{11}{256}a^{16}-\frac{91}{256}a^{12}-\frac{85}{256}a^{8}-\frac{3313}{10032}a^{4}-\frac{224}{627}$, $\frac{1}{321024}a^{25}-\frac{1}{321024}a^{21}+\frac{11}{512}a^{17}-\frac{91}{512}a^{13}-\frac{85}{512}a^{9}-\frac{3313}{20064}a^{5}-\frac{112}{627}a$, $\frac{1}{642048}a^{26}-\frac{1}{642048}a^{22}+\frac{11}{1024}a^{18}-\frac{91}{1024}a^{14}-\frac{85}{1024}a^{10}+\frac{16751}{40128}a^{6}-\frac{56}{627}a^{2}$, $\frac{1}{1284096}a^{27}-\frac{1}{1284096}a^{23}+\frac{11}{2048}a^{19}+\frac{933}{2048}a^{15}+\frac{939}{2048}a^{11}+\frac{16751}{80256}a^{7}+\frac{571}{1254}a^{3}$, $\frac{1}{2568192}a^{28}-\frac{1}{2568192}a^{24}-\frac{5}{856064}a^{20}-\frac{529}{12288}a^{16}+\frac{4097}{12288}a^{12}+\frac{17845}{53504}a^{8}+\frac{3329}{10032}a^{4}+\frac{208}{627}$, $\frac{1}{5136384}a^{29}-\frac{1}{5136384}a^{25}-\frac{5}{1712128}a^{21}-\frac{529}{24576}a^{17}+\frac{4097}{24576}a^{13}+\frac{17845}{107008}a^{9}+\frac{3329}{20064}a^{5}+\frac{104}{627}a$, $\frac{1}{10272768}a^{30}-\frac{1}{10272768}a^{26}-\frac{5}{3424256}a^{22}-\frac{529}{49152}a^{18}-\frac{20479}{49152}a^{14}+\frac{17845}{214016}a^{10}-\frac{16735}{40128}a^{6}+\frac{52}{627}a^{2}$, $\frac{1}{20545536}a^{31}-\frac{1}{20545536}a^{27}-\frac{5}{6848512}a^{23}-\frac{529}{98304}a^{19}-\frac{20479}{98304}a^{15}-\frac{196171}{428032}a^{11}-\frac{16735}{80256}a^{7}-\frac{575}{1254}a^{3}$
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: | \( \frac{17}{321024} a^{29} + \frac{8641}{321024} a^{9} \) (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 $C_2^3\times C_4$ |
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 | R | ${\href{/padicField/11.2.0.1}{2} }^{16}$ | ${\href{/padicField/13.4.0.1}{4} }^{8}$ | ${\href{/padicField/17.4.0.1}{4} }^{8}$ | ${\href{/padicField/19.2.0.1}{2} }^{16}$ | ${\href{/padicField/23.4.0.1}{4} }^{8}$ | ${\href{/padicField/29.2.0.1}{2} }^{16}$ | ${\href{/padicField/31.2.0.1}{2} }^{16}$ | ${\href{/padicField/37.4.0.1}{4} }^{8}$ | ${\href{/padicField/41.2.0.1}{2} }^{16}$ | ${\href{/padicField/43.4.0.1}{4} }^{8}$ | ${\href{/padicField/47.4.0.1}{4} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{8}$ | ${\href{/padicField/59.2.0.1}{2} }^{16}$ |
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 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
\(7\) | 7.8.4.1 | $x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
7.8.4.1 | $x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
7.8.4.1 | $x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
7.8.4.1 | $x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |