Normalized defining polynomial
\( x^{36} - 2 x^{34} + 4 x^{32} - 8 x^{30} + 16 x^{28} - 32 x^{26} + 64 x^{24} - 128 x^{22} + 256 x^{20} + \cdots + 262144 \)
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: | \(541055976048072725104260184584057273870769716344028569534464\) \(\medspace = 2^{54}\cdot 19^{34}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(45.63\) | 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: | $2^{3/2}19^{17/18}\approx 45.630657614261125$ | ||
Ramified primes: | \(2\), \(19\) | 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: | \(152=2^{3}\cdot 19\) | ||
Dirichlet character group: | $\lbrace$$\chi_{152}(1,·)$, $\chi_{152}(3,·)$, $\chi_{152}(129,·)$, $\chi_{152}(9,·)$, $\chi_{152}(11,·)$, $\chi_{152}(115,·)$, $\chi_{152}(17,·)$, $\chi_{152}(131,·)$, $\chi_{152}(25,·)$, $\chi_{152}(27,·)$, $\chi_{152}(33,·)$, $\chi_{152}(35,·)$, $\chi_{152}(41,·)$, $\chi_{152}(43,·)$, $\chi_{152}(49,·)$, $\chi_{152}(51,·)$, $\chi_{152}(137,·)$, $\chi_{152}(59,·)$, $\chi_{152}(65,·)$, $\chi_{152}(67,·)$, $\chi_{152}(73,·)$, $\chi_{152}(75,·)$, $\chi_{152}(81,·)$, $\chi_{152}(83,·)$, $\chi_{152}(139,·)$, $\chi_{152}(89,·)$, $\chi_{152}(91,·)$, $\chi_{152}(97,·)$, $\chi_{152}(99,·)$, $\chi_{152}(145,·)$, $\chi_{152}(105,·)$, $\chi_{152}(107,·)$, $\chi_{152}(113,·)$, $\chi_{152}(147,·)$, $\chi_{152}(121,·)$, $\chi_{152}(123,·)$$\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: | \( -\frac{1}{131072} a^{34} + \frac{1}{65536} a^{32} - \frac{1}{32768} a^{30} + \frac{1}{16384} a^{28} - \frac{1}{8192} a^{26} + \frac{1}{4096} a^{24} - \frac{1}{2048} a^{22} + \frac{1}{1024} a^{20} - \frac{1}{512} a^{18} + \frac{1}{256} a^{16} - \frac{1}{128} a^{14} + \frac{1}{64} a^{12} - \frac{1}{32} a^{10} + \frac{1}{16} a^{8} - \frac{1}{8} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{2} + 1 \) (order $38$) | 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 $C_2\times C_{18}$ is not computed |
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 | $18^{2}$ | $18^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{6}$ | ${\href{/padicField/11.3.0.1}{3} }^{12}$ | $18^{2}$ | ${\href{/padicField/17.9.0.1}{9} }^{4}$ | R | $18^{2}$ | $18^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{6}$ | ${\href{/padicField/37.2.0.1}{2} }^{18}$ | $18^{2}$ | ${\href{/padicField/43.9.0.1}{9} }^{4}$ | $18^{2}$ | $18^{2}$ | $18^{2}$ |
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 $36$ | $2$ | $18$ | $54$ | |||
\(19\) | 19.18.17.14 | $x^{18} + 19$ | $18$ | $1$ | $17$ | $C_{18}$ | $[\ ]_{18}$ |
19.18.17.14 | $x^{18} + 19$ | $18$ | $1$ | $17$ | $C_{18}$ | $[\ ]_{18}$ |