Normalized defining polynomial
\( x^{36} - 5x^{27} - 487x^{18} - 2560x^{9} + 262144 \)
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: | \(14212734556341031905549296191351828189377245025195450200601\) \(\medspace = 3^{90}\cdot 7^{18}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(41.24\) | 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: | $3^{5/2}7^{1/2}\approx 41.24318125460256$ | ||
Ramified primes: | \(3\), \(7\) | 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: | \(189=3^{3}\cdot 7\) | ||
Dirichlet character group: | $\lbrace$$\chi_{189}(1,·)$, $\chi_{189}(134,·)$, $\chi_{189}(8,·)$, $\chi_{189}(139,·)$, $\chi_{189}(13,·)$, $\chi_{189}(146,·)$, $\chi_{189}(20,·)$, $\chi_{189}(22,·)$, $\chi_{189}(155,·)$, $\chi_{189}(29,·)$, $\chi_{189}(160,·)$, $\chi_{189}(34,·)$, $\chi_{189}(167,·)$, $\chi_{189}(41,·)$, $\chi_{189}(43,·)$, $\chi_{189}(176,·)$, $\chi_{189}(50,·)$, $\chi_{189}(181,·)$, $\chi_{189}(55,·)$, $\chi_{189}(188,·)$, $\chi_{189}(62,·)$, $\chi_{189}(64,·)$, $\chi_{189}(71,·)$, $\chi_{189}(76,·)$, $\chi_{189}(83,·)$, $\chi_{189}(85,·)$, $\chi_{189}(92,·)$, $\chi_{189}(97,·)$, $\chi_{189}(104,·)$, $\chi_{189}(106,·)$, $\chi_{189}(113,·)$, $\chi_{189}(118,·)$, $\chi_{189}(169,·)$, $\chi_{189}(148,·)$, $\chi_{189}(125,·)$, $\chi_{189}(127,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{131072}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{17}a^{18}+\frac{6}{17}a^{9}+\frac{2}{17}$, $\frac{1}{34}a^{19}-\frac{11}{34}a^{10}-\frac{15}{34}a$, $\frac{1}{68}a^{20}+\frac{23}{68}a^{11}-\frac{15}{68}a^{2}$, $\frac{1}{136}a^{21}-\frac{45}{136}a^{12}-\frac{15}{136}a^{3}$, $\frac{1}{272}a^{22}+\frac{91}{272}a^{13}+\frac{121}{272}a^{4}$, $\frac{1}{544}a^{23}+\frac{91}{544}a^{14}+\frac{121}{544}a^{5}$, $\frac{1}{1088}a^{24}-\frac{453}{1088}a^{15}-\frac{423}{1088}a^{6}$, $\frac{1}{2176}a^{25}+\frac{635}{2176}a^{16}+\frac{665}{2176}a^{7}$, $\frac{1}{4352}a^{26}-\frac{1541}{4352}a^{17}-\frac{1511}{4352}a^{8}$, $\frac{1}{4238848}a^{27}+\frac{205}{8704}a^{18}-\frac{2561}{8704}a^{9}+\frac{1943}{8279}$, $\frac{1}{8477696}a^{28}+\frac{205}{17408}a^{19}-\frac{2561}{17408}a^{10}-\frac{3168}{8279}a$, $\frac{1}{16955392}a^{29}+\frac{205}{34816}a^{20}-\frac{2561}{34816}a^{11}+\frac{5111}{16558}a^{2}$, $\frac{1}{33910784}a^{30}+\frac{205}{69632}a^{21}+\frac{32255}{69632}a^{12}+\frac{5111}{33116}a^{3}$, $\frac{1}{67821568}a^{31}+\frac{205}{139264}a^{22}-\frac{37377}{139264}a^{13}-\frac{28005}{66232}a^{4}$, $\frac{1}{135643136}a^{32}+\frac{205}{278528}a^{23}+\frac{101887}{278528}a^{14}+\frac{38227}{132464}a^{5}$, $\frac{1}{271286272}a^{33}+\frac{205}{557056}a^{24}-\frac{176641}{557056}a^{15}-\frac{94237}{264928}a^{6}$, $\frac{1}{542572544}a^{34}+\frac{205}{1114112}a^{25}-\frac{176641}{1114112}a^{16}+\frac{170691}{529856}a^{7}$, $\frac{1}{1085145088}a^{35}+\frac{205}{2228224}a^{26}+\frac{937471}{2228224}a^{17}-\frac{359165}{1059712}a^{8}$
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: | \( -\frac{4049}{542572544} a^{34} - \frac{93}{1114112} a^{25} + \frac{4049}{1114112} a^{16} + \frac{20245}{1059712} a^{7} \) (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 $C_2\times C_{18}$ 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 | $18^{2}$ | R | $18^{2}$ | R | $18^{2}$ | $18^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{6}$ | ${\href{/padicField/19.6.0.1}{6} }^{6}$ | $18^{2}$ | $18^{2}$ | $18^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{12}$ | $18^{2}$ | ${\href{/padicField/43.9.0.1}{9} }^{4}$ | $18^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{18}$ | $18^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | Deg $36$ | $18$ | $2$ | $90$ | |||
\(7\) | 7.18.9.2 | $x^{18} + 63 x^{16} + 1764 x^{14} + 12 x^{13} + 28814 x^{12} - 504 x^{11} + 302370 x^{10} - 17044 x^{9} + 2112804 x^{8} - 150180 x^{7} + 9908221 x^{6} - 209592 x^{5} + 29960739 x^{4} + 1787108 x^{3} + 51556212 x^{2} + 7225224 x + 40408804$ | $2$ | $9$ | $9$ | $C_{18}$ | $[\ ]_{2}^{9}$ |
7.18.9.2 | $x^{18} + 63 x^{16} + 1764 x^{14} + 12 x^{13} + 28814 x^{12} - 504 x^{11} + 302370 x^{10} - 17044 x^{9} + 2112804 x^{8} - 150180 x^{7} + 9908221 x^{6} - 209592 x^{5} + 29960739 x^{4} + 1787108 x^{3} + 51556212 x^{2} + 7225224 x + 40408804$ | $2$ | $9$ | $9$ | $C_{18}$ | $[\ ]_{2}^{9}$ |