Normalized defining polynomial
\( x^{32} + 480 x^{30} + 104400 x^{28} + 13608000 x^{26} + 1184625000 x^{24} + 72657000000 x^{22} + 3227647500000 x^{20} + 105129090000000 x^{18} + \cdots + 12\!\cdots\!50 \)
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: | \(805\!\cdots\!000\) \(\medspace = 2^{191}\cdot 3^{16}\cdot 5^{24}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(362.71\) | 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: | $2^{191/32}3^{1/2}5^{3/4}\approx 362.71115912243516$ | ||
Ramified primes: | \(2\), \(3\), \(5\) | 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: | \(1920=2^{7}\cdot 3\cdot 5\) | ||
Dirichlet character group: | $\lbrace$$\chi_{1920}(1,·)$, $\chi_{1920}(1283,·)$, $\chi_{1920}(649,·)$, $\chi_{1920}(1547,·)$, $\chi_{1920}(1681,·)$, $\chi_{1920}(1043,·)$, $\chi_{1920}(409,·)$, $\chi_{1920}(1307,·)$, $\chi_{1920}(1441,·)$, $\chi_{1920}(803,·)$, $\chi_{1920}(169,·)$, $\chi_{1920}(1067,·)$, $\chi_{1920}(1201,·)$, $\chi_{1920}(563,·)$, $\chi_{1920}(1849,·)$, $\chi_{1920}(827,·)$, $\chi_{1920}(961,·)$, $\chi_{1920}(323,·)$, $\chi_{1920}(1609,·)$, $\chi_{1920}(587,·)$, $\chi_{1920}(721,·)$, $\chi_{1920}(83,·)$, $\chi_{1920}(1369,·)$, $\chi_{1920}(347,·)$, $\chi_{1920}(481,·)$, $\chi_{1920}(1763,·)$, $\chi_{1920}(1129,·)$, $\chi_{1920}(107,·)$, $\chi_{1920}(241,·)$, $\chi_{1920}(1523,·)$, $\chi_{1920}(889,·)$, $\chi_{1920}(1787,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{32768}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{3}a^{2}$, $\frac{1}{3}a^{3}$, $\frac{1}{45}a^{4}$, $\frac{1}{45}a^{5}$, $\frac{1}{135}a^{6}$, $\frac{1}{135}a^{7}$, $\frac{1}{2025}a^{8}$, $\frac{1}{2025}a^{9}$, $\frac{1}{6075}a^{10}$, $\frac{1}{6075}a^{11}$, $\frac{1}{91125}a^{12}$, $\frac{1}{91125}a^{13}$, $\frac{1}{273375}a^{14}$, $\frac{1}{273375}a^{15}$, $\frac{1}{3538839375}a^{16}+\frac{16}{235922625}a^{14}-\frac{343}{78640875}a^{12}+\frac{34}{5242725}a^{10}+\frac{103}{1747575}a^{8}+\frac{403}{116505}a^{6}-\frac{287}{38835}a^{4}+\frac{233}{2589}a^{2}+\frac{387}{863}$, $\frac{1}{675918320625}a^{17}-\frac{46586}{45061221375}a^{15}-\frac{33137}{15020407125}a^{13}-\frac{11783}{333786825}a^{11}+\frac{21034}{111262275}a^{9}+\frac{78073}{22252455}a^{7}-\frac{63286}{7417485}a^{5}+\frac{22228}{164833}a^{3}+\frac{79783}{164833}a$, $\frac{1}{2027754961875}a^{18}+\frac{2}{15020407125}a^{16}+\frac{68857}{45061221375}a^{14}+\frac{6233}{15020407125}a^{12}+\frac{73798}{1001360475}a^{10}-\frac{3859}{333786825}a^{8}+\frac{53797}{22252455}a^{6}+\frac{5468}{824165}a^{4}-\frac{7149}{164833}a^{2}+\frac{79}{863}$, $\frac{1}{2027754961875}a^{19}-\frac{24061}{45061221375}a^{15}+\frac{21569}{15020407125}a^{13}-\frac{41452}{1001360475}a^{11}-\frac{26239}{111262275}a^{9}-\frac{49787}{22252455}a^{7}-\frac{24203}{7417485}a^{5}+\frac{75814}{494499}a^{3}-\frac{77562}{164833}a$, $\frac{1}{30416324428125}a^{20}+\frac{1}{135183664125}a^{16}-\frac{39196}{45061221375}a^{14}-\frac{10888}{3004081425}a^{12}-\frac{54631}{1001360475}a^{10}-\frac{43484}{333786825}a^{8}+\frac{1577}{22252455}a^{6}-\frac{4865}{494499}a^{4}-\frac{15141}{164833}a^{2}-\frac{429}{863}$, $\frac{1}{30416324428125}a^{21}+\frac{28901}{45061221375}a^{15}-\frac{53588}{15020407125}a^{13}-\frac{42719}{1001360475}a^{11}-\frac{9776}{111262275}a^{9}-\frac{59122}{22252455}a^{7}+\frac{78622}{7417485}a^{5}-\frac{49177}{494499}a^{3}+\frac{13645}{164833}a$, $\frac{1}{91248973284375}a^{22}-\frac{82}{675918320625}a^{16}-\frac{59318}{45061221375}a^{14}-\frac{70154}{15020407125}a^{12}-\frac{296}{1001360475}a^{10}-\frac{4687}{66757365}a^{8}-\frac{778}{4450491}a^{6}+\frac{42334}{7417485}a^{4}-\frac{16398}{164833}a^{2}-\frac{402}{863}$, $\frac{1}{91248973284375}a^{23}+\frac{76622}{45061221375}a^{15}+\frac{14773}{15020407125}a^{13}+\frac{13616}{200272095}a^{11}+\frac{13702}{111262275}a^{9}-\frac{30391}{22252455}a^{7}-\frac{7459}{1483497}a^{5}-\frac{6865}{164833}a^{3}+\frac{36937}{164833}a$, $\frac{1}{13\!\cdots\!25}a^{24}+\frac{31}{675918320625}a^{16}+\frac{63196}{45061221375}a^{14}-\frac{34487}{15020407125}a^{12}-\frac{24478}{1001360475}a^{10}-\frac{1456}{66757365}a^{8}-\frac{49861}{22252455}a^{6}+\frac{38233}{7417485}a^{4}-\frac{3385}{494499}a^{2}+\frac{153}{863}$, $\frac{1}{13\!\cdots\!25}a^{25}+\frac{1591}{3004081425}a^{15}+\frac{418}{1668934125}a^{13}+\frac{82343}{1001360475}a^{11}+\frac{14554}{333786825}a^{9}+\frac{2371}{22252455}a^{7}+\frac{22103}{7417485}a^{5}+\frac{24080}{164833}a^{3}+\frac{28445}{164833}a$, $\frac{1}{41\!\cdots\!75}a^{26}-\frac{2}{27036732825}a^{16}+\frac{71758}{45061221375}a^{14}-\frac{15743}{15020407125}a^{12}+\frac{25543}{333786825}a^{10}+\frac{583}{2472495}a^{8}+\frac{45214}{22252455}a^{6}+\frac{1379}{1483497}a^{4}+\frac{70847}{494499}a^{2}-\frac{235}{863}$, $\frac{1}{41\!\cdots\!75}a^{27}+\frac{10024}{9012244275}a^{15}-\frac{24263}{15020407125}a^{13}-\frac{42491}{1001360475}a^{11}-\frac{12571}{66757365}a^{9}-\frac{264}{824165}a^{7}-\frac{2842}{824165}a^{5}-\frac{56446}{494499}a^{3}-\frac{11727}{164833}a$, $\frac{1}{61\!\cdots\!25}a^{28}+\frac{26}{225306106875}a^{16}+\frac{51607}{45061221375}a^{14}-\frac{20717}{15020407125}a^{12}-\frac{65669}{1001360475}a^{10}-\frac{51631}{333786825}a^{8}-\frac{29449}{22252455}a^{6}-\frac{11621}{2472495}a^{4}+\frac{8045}{494499}a^{2}-\frac{423}{863}$, $\frac{1}{61\!\cdots\!25}a^{29}+\frac{19663}{15020407125}a^{15}-\frac{2717}{556311375}a^{13}+\frac{241}{4450491}a^{11}-\frac{28597}{333786825}a^{9}-\frac{2258}{2472495}a^{7}-\frac{2903}{494499}a^{5}+\frac{81349}{494499}a^{3}-\frac{40213}{164833}a$, $\frac{1}{18\!\cdots\!75}a^{30}+\frac{41}{675918320625}a^{16}-\frac{11666}{45061221375}a^{14}+\frac{50902}{15020407125}a^{12}-\frac{104}{200272095}a^{10}+\frac{17383}{333786825}a^{8}-\frac{15088}{7417485}a^{6}-\frac{1927}{1483497}a^{4}-\frac{5828}{164833}a^{2}-\frac{332}{863}$, $\frac{1}{18\!\cdots\!75}a^{31}-\frac{79636}{45061221375}a^{15}-\frac{73978}{15020407125}a^{13}-\frac{34708}{1001360475}a^{11}+\frac{67529}{333786825}a^{9}+\frac{16801}{7417485}a^{7}-\frac{52237}{7417485}a^{5}+\frac{50633}{494499}a^{3}-\frac{37855}{164833}a$
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 | R | 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 $32$ | $32$ | $1$ | $191$ | |||
\(3\) | Deg $32$ | $2$ | $16$ | $16$ | |||
\(5\) | Deg $32$ | $4$ | $8$ | $24$ |