Normalized defining polynomial
\( x^{36} + 512x^{18} + 262144 \)
Invariants
Degree: | $36$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 18]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(157229013891772345498599951736092754417390325814062078754816\) \(\medspace = 2^{54}\cdot 3^{90}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(44.09\) | 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: | $2^{3/2}3^{5/2}\approx 44.090815370097204$ | ||
Ramified primes: | \(2\), \(3\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $36$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(216=2^{3}\cdot 3^{3}\) | ||
Dirichlet character group: | $\lbrace$$\chi_{216}(1,·)$, $\chi_{216}(5,·)$, $\chi_{216}(65,·)$, $\chi_{216}(137,·)$, $\chi_{216}(13,·)$, $\chi_{216}(17,·)$, $\chi_{216}(149,·)$, $\chi_{216}(25,·)$, $\chi_{216}(157,·)$, $\chi_{216}(133,·)$, $\chi_{216}(161,·)$, $\chi_{216}(37,·)$, $\chi_{216}(41,·)$, $\chi_{216}(173,·)$, $\chi_{216}(29,·)$, $\chi_{216}(49,·)$, $\chi_{216}(181,·)$, $\chi_{216}(185,·)$, $\chi_{216}(61,·)$, $\chi_{216}(53,·)$, $\chi_{216}(193,·)$, $\chi_{216}(197,·)$, $\chi_{216}(73,·)$, $\chi_{216}(77,·)$, $\chi_{216}(205,·)$, $\chi_{216}(209,·)$, $\chi_{216}(85,·)$, $\chi_{216}(89,·)$, $\chi_{216}(97,·)$, $\chi_{216}(101,·)$, $\chi_{216}(145,·)$, $\chi_{216}(109,·)$, $\chi_{216}(113,·)$, $\chi_{216}(169,·)$, $\chi_{216}(121,·)$, $\chi_{216}(125,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{131072}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{4}a^{5}$, $\frac{1}{8}a^{6}$, $\frac{1}{8}a^{7}$, $\frac{1}{16}a^{8}$, $\frac{1}{16}a^{9}$, $\frac{1}{32}a^{10}$, $\frac{1}{32}a^{11}$, $\frac{1}{64}a^{12}$, $\frac{1}{64}a^{13}$, $\frac{1}{128}a^{14}$, $\frac{1}{128}a^{15}$, $\frac{1}{256}a^{16}$, $\frac{1}{256}a^{17}$, $\frac{1}{512}a^{18}$, $\frac{1}{512}a^{19}$, $\frac{1}{1024}a^{20}$, $\frac{1}{1024}a^{21}$, $\frac{1}{2048}a^{22}$, $\frac{1}{2048}a^{23}$, $\frac{1}{4096}a^{24}$, $\frac{1}{4096}a^{25}$, $\frac{1}{8192}a^{26}$, $\frac{1}{8192}a^{27}$, $\frac{1}{16384}a^{28}$, $\frac{1}{16384}a^{29}$, $\frac{1}{32768}a^{30}$, $\frac{1}{32768}a^{31}$, $\frac{1}{65536}a^{32}$, $\frac{1}{65536}a^{33}$, $\frac{1}{131072}a^{34}$, $\frac{1}{131072}a^{35}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
not computed
Unit group
Rank: | $17$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -\frac{1}{2} a^{2} \) (order $54$) | 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
$C_2\times C_{18}$ (as 36T2):
An abelian group of order 36 |
The 36 conjugacy class representatives for $C_2\times C_{18}$ |
Character table for $C_2\times C_{18}$ |
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 | $18^{2}$ | ${\href{/padicField/7.9.0.1}{9} }^{4}$ | $18^{2}$ | $18^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{6}$ | ${\href{/padicField/19.6.0.1}{6} }^{6}$ | $18^{2}$ | $18^{2}$ | ${\href{/padicField/31.9.0.1}{9} }^{4}$ | ${\href{/padicField/37.6.0.1}{6} }^{6}$ | $18^{2}$ | $18^{2}$ | $18^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{18}$ | $18^{2}$ |
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 $36$ | $2$ | $18$ | $54$ | |||
\(3\) | Deg $36$ | $18$ | $2$ | $90$ |