Normalized defining polynomial
\( x^{32} + 160 x^{30} + 11600 x^{28} + 504000 x^{26} + 14625000 x^{24} + 299000000 x^{22} + \cdots + 28500781250 \)
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: | \(187072209578355573530071658587684226515959365500928000000000000000000000000\) \(\medspace = 2^{191}\cdot 5^{24}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(209.41\) | 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: | $2^{191/32}5^{3/4}\approx 209.41138535741914$ | ||
Ramified primes: | \(2\), \(5\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{2}) \) | ||
$\card{ \Gal(K/\Q) }$: | $32$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(640=2^{7}\cdot 5\) | ||
Dirichlet character group: | $\lbrace$$\chi_{640}(1,·)$, $\chi_{640}(517,·)$, $\chi_{640}(9,·)$, $\chi_{640}(13,·)$, $\chi_{640}(401,·)$, $\chi_{640}(277,·)$, $\chi_{640}(409,·)$, $\chi_{640}(413,·)$, $\chi_{640}(161,·)$, $\chi_{640}(37,·)$, $\chi_{640}(169,·)$, $\chi_{640}(173,·)$, $\chi_{640}(561,·)$, $\chi_{640}(437,·)$, $\chi_{640}(569,·)$, $\chi_{640}(573,·)$, $\chi_{640}(321,·)$, $\chi_{640}(197,·)$, $\chi_{640}(329,·)$, $\chi_{640}(333,·)$, $\chi_{640}(81,·)$, $\chi_{640}(597,·)$, $\chi_{640}(89,·)$, $\chi_{640}(93,·)$, $\chi_{640}(481,·)$, $\chi_{640}(357,·)$, $\chi_{640}(489,·)$, $\chi_{640}(493,·)$, $\chi_{640}(241,·)$, $\chi_{640}(117,·)$, $\chi_{640}(249,·)$, $\chi_{640}(253,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{32768}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5}a^{4}$, $\frac{1}{5}a^{5}$, $\frac{1}{5}a^{6}$, $\frac{1}{5}a^{7}$, $\frac{1}{25}a^{8}$, $\frac{1}{25}a^{9}$, $\frac{1}{25}a^{10}$, $\frac{1}{25}a^{11}$, $\frac{1}{125}a^{12}$, $\frac{1}{125}a^{13}$, $\frac{1}{125}a^{14}$, $\frac{1}{125}a^{15}$, $\frac{1}{539375}a^{16}+\frac{16}{107875}a^{14}-\frac{343}{107875}a^{12}+\frac{34}{21575}a^{10}+\frac{103}{21575}a^{8}+\frac{403}{4315}a^{6}-\frac{287}{4315}a^{4}+\frac{233}{863}a^{2}+\frac{387}{863}$, $\frac{1}{103020625}a^{17}-\frac{46586}{20604125}a^{15}-\frac{33137}{20604125}a^{13}-\frac{35349}{4120825}a^{11}+\frac{63102}{4120825}a^{9}+\frac{78073}{824165}a^{7}-\frac{63286}{824165}a^{5}+\frac{66684}{164833}a^{3}+\frac{79783}{164833}a$, $\frac{1}{103020625}a^{18}+\frac{18}{20604125}a^{16}+\frac{68857}{20604125}a^{14}+\frac{6233}{20604125}a^{12}+\frac{73798}{4120825}a^{10}-\frac{3859}{4120825}a^{8}+\frac{53797}{824165}a^{6}+\frac{49212}{824165}a^{4}-\frac{21447}{164833}a^{2}+\frac{79}{863}$, $\frac{1}{103020625}a^{19}-\frac{24061}{20604125}a^{15}+\frac{21569}{20604125}a^{13}-\frac{41452}{4120825}a^{11}-\frac{78717}{4120825}a^{9}-\frac{49787}{824165}a^{7}-\frac{24203}{824165}a^{5}+\frac{75814}{164833}a^{3}-\frac{77562}{164833}a$, $\frac{1}{515103125}a^{20}+\frac{1}{20604125}a^{16}-\frac{39196}{20604125}a^{14}-\frac{10888}{4120825}a^{12}-\frac{54631}{4120825}a^{10}-\frac{43484}{4120825}a^{8}+\frac{1577}{824165}a^{6}-\frac{14595}{164833}a^{4}-\frac{45423}{164833}a^{2}-\frac{429}{863}$, $\frac{1}{515103125}a^{21}+\frac{28901}{20604125}a^{15}-\frac{53588}{20604125}a^{13}-\frac{42719}{4120825}a^{11}-\frac{29328}{4120825}a^{9}-\frac{59122}{824165}a^{7}+\frac{78622}{824165}a^{5}-\frac{49177}{164833}a^{3}+\frac{13645}{164833}a$, $\frac{1}{515103125}a^{22}-\frac{82}{103020625}a^{16}-\frac{59318}{20604125}a^{14}-\frac{70154}{20604125}a^{12}-\frac{296}{4120825}a^{10}-\frac{4687}{824165}a^{8}-\frac{778}{164833}a^{6}+\frac{42334}{824165}a^{4}-\frac{49194}{164833}a^{2}-\frac{402}{863}$, $\frac{1}{515103125}a^{23}+\frac{76622}{20604125}a^{15}+\frac{14773}{20604125}a^{13}+\frac{13616}{824165}a^{11}+\frac{41106}{4120825}a^{9}-\frac{30391}{824165}a^{7}-\frac{7459}{164833}a^{5}-\frac{20595}{164833}a^{3}+\frac{36937}{164833}a$, $\frac{1}{2575515625}a^{24}+\frac{31}{103020625}a^{16}+\frac{63196}{20604125}a^{14}-\frac{34487}{20604125}a^{12}-\frac{24478}{4120825}a^{10}-\frac{1456}{824165}a^{8}-\frac{49861}{824165}a^{6}+\frac{38233}{824165}a^{4}-\frac{3385}{164833}a^{2}+\frac{153}{863}$, $\frac{1}{2575515625}a^{25}+\frac{4773}{4120825}a^{15}+\frac{3762}{20604125}a^{13}+\frac{82343}{4120825}a^{11}+\frac{14554}{4120825}a^{9}+\frac{2371}{824165}a^{7}+\frac{22103}{824165}a^{5}+\frac{72240}{164833}a^{3}+\frac{28445}{164833}a$, $\frac{1}{2575515625}a^{26}-\frac{2}{4120825}a^{16}+\frac{71758}{20604125}a^{14}-\frac{15743}{20604125}a^{12}+\frac{76629}{4120825}a^{10}+\frac{15741}{824165}a^{8}+\frac{45214}{824165}a^{6}+\frac{1379}{164833}a^{4}+\frac{70847}{164833}a^{2}-\frac{235}{863}$, $\frac{1}{2575515625}a^{27}+\frac{10024}{4120825}a^{15}-\frac{24263}{20604125}a^{13}-\frac{42491}{4120825}a^{11}-\frac{12571}{824165}a^{9}-\frac{7128}{824165}a^{7}-\frac{25578}{824165}a^{5}-\frac{56446}{164833}a^{3}-\frac{11727}{164833}a$, $\frac{1}{12877578125}a^{28}+\frac{78}{103020625}a^{16}+\frac{51607}{20604125}a^{14}-\frac{20717}{20604125}a^{12}-\frac{65669}{4120825}a^{10}-\frac{51631}{4120825}a^{8}-\frac{29449}{824165}a^{6}-\frac{34863}{824165}a^{4}+\frac{8045}{164833}a^{2}-\frac{423}{863}$, $\frac{1}{12877578125}a^{29}+\frac{58989}{20604125}a^{15}-\frac{73359}{20604125}a^{13}+\frac{2169}{164833}a^{11}-\frac{28597}{4120825}a^{9}-\frac{20322}{824165}a^{7}-\frac{8709}{164833}a^{5}+\frac{81349}{164833}a^{3}-\frac{40213}{164833}a$, $\frac{1}{12877578125}a^{30}+\frac{41}{103020625}a^{16}-\frac{11666}{20604125}a^{14}+\frac{50902}{20604125}a^{12}-\frac{104}{824165}a^{10}+\frac{17383}{4120825}a^{8}-\frac{45264}{824165}a^{6}-\frac{1927}{164833}a^{4}-\frac{17484}{164833}a^{2}-\frac{332}{863}$, $\frac{1}{12877578125}a^{31}-\frac{79636}{20604125}a^{15}-\frac{73978}{20604125}a^{13}-\frac{34708}{4120825}a^{11}+\frac{67529}{4120825}a^{9}+\frac{50403}{824165}a^{7}-\frac{52237}{824165}a^{5}+\frac{50633}{164833}a^{3}-\frac{37855}{164833}a$
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: | \( -1 \) (order $2$) | 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
A cyclic group of order 32 |
The 32 conjugacy class representatives for $C_{32}$ |
Character table for $C_{32}$ is not computed |
Intermediate fields
\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), \(\Q(\zeta_{32})^+\), 16.16.236118324143482260684800000000.1 |
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 | $32$ | R | $16^{2}$ | $32$ | $32$ | ${\href{/padicField/17.8.0.1}{8} }^{4}$ | $32$ | $16^{2}$ | $32$ | ${\href{/padicField/31.4.0.1}{4} }^{8}$ | $32$ | $16^{2}$ | $32$ | ${\href{/padicField/47.8.0.1}{8} }^{4}$ | $32$ | $32$ |
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 $32$ | $32$ | $1$ | $191$ | |||
\(5\) | Deg $32$ | $4$ | $8$ | $24$ |