Normalized defining polynomial
\( x^{32} + 352 x^{30} + 56144 x^{28} + 5366592 x^{26} + 342599400 x^{24} + 15409359680 x^{22} + \cdots + 91\!\cdots\!22 \)
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: | \(144215564533589000876246801170130951941346253827716612240069988132090544128\) \(\medspace = 2^{191}\cdot 11^{16}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(207.72\) | 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: | $2^{191/32}11^{1/2}\approx 207.71560705723326$ | ||
Ramified primes: | \(2\), \(11\) | 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: | \(1408=2^{7}\cdot 11\) | ||
Dirichlet character group: | $\lbrace$$\chi_{1408}(1,·)$, $\chi_{1408}(901,·)$, $\chi_{1408}(265,·)$, $\chi_{1408}(1165,·)$, $\chi_{1408}(529,·)$, $\chi_{1408}(21,·)$, $\chi_{1408}(793,·)$, $\chi_{1408}(285,·)$, $\chi_{1408}(1057,·)$, $\chi_{1408}(549,·)$, $\chi_{1408}(1321,·)$, $\chi_{1408}(813,·)$, $\chi_{1408}(177,·)$, $\chi_{1408}(1077,·)$, $\chi_{1408}(441,·)$, $\chi_{1408}(1341,·)$, $\chi_{1408}(705,·)$, $\chi_{1408}(197,·)$, $\chi_{1408}(969,·)$, $\chi_{1408}(461,·)$, $\chi_{1408}(1233,·)$, $\chi_{1408}(725,·)$, $\chi_{1408}(89,·)$, $\chi_{1408}(989,·)$, $\chi_{1408}(353,·)$, $\chi_{1408}(1253,·)$, $\chi_{1408}(617,·)$, $\chi_{1408}(109,·)$, $\chi_{1408}(881,·)$, $\chi_{1408}(373,·)$, $\chi_{1408}(1145,·)$, $\chi_{1408}(637,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{32768}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{11}a^{2}$, $\frac{1}{11}a^{3}$, $\frac{1}{121}a^{4}$, $\frac{1}{121}a^{5}$, $\frac{1}{1331}a^{6}$, $\frac{1}{1331}a^{7}$, $\frac{1}{14641}a^{8}$, $\frac{1}{14641}a^{9}$, $\frac{1}{161051}a^{10}$, $\frac{1}{161051}a^{11}$, $\frac{1}{1771561}a^{12}$, $\frac{1}{1771561}a^{13}$, $\frac{1}{19487171}a^{14}$, $\frac{1}{19487171}a^{15}$, $\frac{1}{214358881}a^{16}$, $\frac{1}{214358881}a^{17}$, $\frac{1}{2357947691}a^{18}$, $\frac{1}{2357947691}a^{19}$, $\frac{1}{25937424601}a^{20}$, $\frac{1}{25937424601}a^{21}$, $\frac{1}{285311670611}a^{22}$, $\frac{1}{285311670611}a^{23}$, $\frac{1}{3138428376721}a^{24}$, $\frac{1}{3138428376721}a^{25}$, $\frac{1}{34522712143931}a^{26}$, $\frac{1}{34522712143931}a^{27}$, $\frac{1}{379749833583241}a^{28}$, $\frac{1}{379749833583241}a^{29}$, $\frac{1}{41\!\cdots\!51}a^{30}$, $\frac{1}{41\!\cdots\!51}a^{31}$
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 $C_{32}$ |
Intermediate fields
\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), \(\Q(\zeta_{32})^+\), \(\Q(\zeta_{64})^+\) |
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 | $32$ | $32$ | $16^{2}$ | R | $32$ | ${\href{/padicField/17.8.0.1}{8} }^{4}$ | $32$ | $16^{2}$ | $32$ | ${\href{/padicField/31.4.0.1}{4} }^{8}$ | $32$ | $16^{2}$ | $32$ | ${\href{/padicField/47.8.0.1}{8} }^{4}$ | $32$ | $32$ |
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 $32$ | $32$ | $1$ | $191$ | |||
\(11\) | Deg $32$ | $2$ | $16$ | $16$ |