Normalized defining polynomial
\( x^{36} - 12 x^{34} + 108 x^{32} - 888 x^{30} + 7056 x^{28} - 55296 x^{26} + 430848 x^{24} - 1651968 x^{22} + 5640192 x^{20} - 18551808 x^{18} + 59222016 x^{16} + \cdots + 191102976 \)
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: | \(23610692285332399309092778573219694177406932269512586359440570056704\) \(\medspace = 2^{54}\cdot 3^{54}\cdot 7^{30}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(74.38\) | 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: | $2^{3/2}3^{3/2}7^{5/6}\approx 74.38326581667904$ | ||
Ramified primes: | \(2\), \(3\), \(7\) | 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: | \(504=2^{3}\cdot 3^{2}\cdot 7\) | ||
Dirichlet character group: | $\lbrace$$\chi_{504}(1,·)$, $\chi_{504}(5,·)$, $\chi_{504}(265,·)$, $\chi_{504}(269,·)$, $\chi_{504}(145,·)$, $\chi_{504}(149,·)$, $\chi_{504}(25,·)$, $\chi_{504}(409,·)$, $\chi_{504}(29,·)$, $\chi_{504}(389,·)$, $\chi_{504}(289,·)$, $\chi_{504}(293,·)$, $\chi_{504}(169,·)$, $\chi_{504}(173,·)$, $\chi_{504}(433,·)$, $\chi_{504}(53,·)$, $\chi_{504}(73,·)$, $\chi_{504}(313,·)$, $\chi_{504}(317,·)$, $\chi_{504}(437,·)$, $\chi_{504}(193,·)$, $\chi_{504}(197,·)$, $\chi_{504}(97,·)$, $\chi_{504}(457,·)$, $\chi_{504}(461,·)$, $\chi_{504}(337,·)$, $\chi_{504}(341,·)$, $\chi_{504}(221,·)$, $\chi_{504}(101,·)$, $\chi_{504}(481,·)$, $\chi_{504}(485,·)$, $\chi_{504}(361,·)$, $\chi_{504}(365,·)$, $\chi_{504}(241,·)$, $\chi_{504}(121,·)$, $\chi_{504}(125,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{131072}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{4}a^{5}$, $\frac{1}{24}a^{6}$, $\frac{1}{24}a^{7}$, $\frac{1}{48}a^{8}$, $\frac{1}{48}a^{9}$, $\frac{1}{96}a^{10}$, $\frac{1}{96}a^{11}$, $\frac{1}{576}a^{12}$, $\frac{1}{576}a^{13}$, $\frac{1}{4608}a^{14}-\frac{1}{4}$, $\frac{1}{4608}a^{15}-\frac{1}{4}a$, $\frac{1}{9216}a^{16}-\frac{1}{8}a^{2}$, $\frac{1}{9216}a^{17}-\frac{1}{8}a^{3}$, $\frac{1}{55296}a^{18}+\frac{1}{16}a^{4}$, $\frac{1}{55296}a^{19}+\frac{1}{16}a^{5}$, $\frac{1}{110592}a^{20}-\frac{1}{96}a^{6}$, $\frac{1}{110592}a^{21}-\frac{1}{96}a^{7}$, $\frac{1}{221184}a^{22}-\frac{1}{192}a^{8}$, $\frac{1}{221184}a^{23}-\frac{1}{192}a^{9}$, $\frac{1}{1327104}a^{24}+\frac{1}{384}a^{10}$, $\frac{1}{1327104}a^{25}+\frac{1}{384}a^{11}$, $\frac{1}{37124407296}a^{26}-\frac{3353}{9281101824}a^{24}+\frac{901}{515616768}a^{22}+\frac{139}{128904192}a^{20}+\frac{499}{96678144}a^{18}-\frac{1469}{42968064}a^{16}-\frac{1189}{64452096}a^{14}+\frac{22139}{32226048}a^{12}+\frac{3643}{2685504}a^{10}+\frac{1777}{447584}a^{8}+\frac{181}{111896}a^{6}+\frac{3083}{55948}a^{4}+\frac{6739}{111896}a^{2}-\frac{27711}{55948}$, $\frac{1}{37124407296}a^{27}-\frac{3353}{9281101824}a^{25}+\frac{901}{515616768}a^{23}+\frac{139}{128904192}a^{21}+\frac{499}{96678144}a^{19}-\frac{1469}{42968064}a^{17}-\frac{1189}{64452096}a^{15}+\frac{22139}{32226048}a^{13}+\frac{3643}{2685504}a^{11}+\frac{1777}{447584}a^{9}+\frac{181}{111896}a^{7}+\frac{3083}{55948}a^{5}+\frac{6739}{111896}a^{3}-\frac{27711}{55948}a$, $\frac{1}{296995258368}a^{28}-\frac{10333}{128904192}a^{14}+\frac{99225}{223792}$, $\frac{1}{296995258368}a^{29}-\frac{10333}{128904192}a^{15}+\frac{99225}{223792}a$, $\frac{1}{1781971550208}a^{30}-\frac{12769}{257808384}a^{16}-\frac{22873}{447584}a^{2}$, $\frac{1}{1781971550208}a^{31}-\frac{12769}{257808384}a^{17}-\frac{22873}{447584}a^{3}$, $\frac{1}{3563943100416}a^{32}-\frac{10333}{1546850304}a^{18}+\frac{33075}{895168}a^{4}$, $\frac{1}{3563943100416}a^{33}-\frac{10333}{1546850304}a^{19}+\frac{33075}{895168}a^{5}$, $\frac{1}{7127886200832}a^{34}-\frac{10333}{3093700608}a^{20}+\frac{33075}{1790336}a^{6}$, $\frac{1}{7127886200832}a^{35}-\frac{10333}{3093700608}a^{21}+\frac{33075}{1790336}a^{7}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
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: | \( \frac{1165}{7127886200832} a^{34} + \frac{95341}{343744512} a^{20} + \frac{6554473}{5371008} a^{6} \) (order $14$) | 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
An abelian group of order 36 |
The 36 conjugacy class representatives for $C_6^2$ |
Character table for $C_6^2$ 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 | R | ${\href{/padicField/5.6.0.1}{6} }^{6}$ | R | ${\href{/padicField/11.3.0.1}{3} }^{12}$ | ${\href{/padicField/13.6.0.1}{6} }^{6}$ | ${\href{/padicField/17.6.0.1}{6} }^{6}$ | ${\href{/padicField/19.6.0.1}{6} }^{6}$ | ${\href{/padicField/23.6.0.1}{6} }^{6}$ | ${\href{/padicField/29.3.0.1}{3} }^{12}$ | ${\href{/padicField/31.6.0.1}{6} }^{6}$ | ${\href{/padicField/37.6.0.1}{6} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{6}$ | ${\href{/padicField/43.6.0.1}{6} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{6}$ | ${\href{/padicField/53.3.0.1}{3} }^{12}$ | ${\href{/padicField/59.6.0.1}{6} }^{6}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.6.9.3 | $x^{6} + 12 x^{5} + 86 x^{4} + 352 x^{3} + 892 x^{2} + 1552 x + 1384$ | $2$ | $3$ | $9$ | $C_6$ | $[3]^{3}$ |
2.6.9.3 | $x^{6} + 12 x^{5} + 86 x^{4} + 352 x^{3} + 892 x^{2} + 1552 x + 1384$ | $2$ | $3$ | $9$ | $C_6$ | $[3]^{3}$ | |
2.6.9.3 | $x^{6} + 12 x^{5} + 86 x^{4} + 352 x^{3} + 892 x^{2} + 1552 x + 1384$ | $2$ | $3$ | $9$ | $C_6$ | $[3]^{3}$ | |
2.6.9.3 | $x^{6} + 12 x^{5} + 86 x^{4} + 352 x^{3} + 892 x^{2} + 1552 x + 1384$ | $2$ | $3$ | $9$ | $C_6$ | $[3]^{3}$ | |
2.6.9.3 | $x^{6} + 12 x^{5} + 86 x^{4} + 352 x^{3} + 892 x^{2} + 1552 x + 1384$ | $2$ | $3$ | $9$ | $C_6$ | $[3]^{3}$ | |
2.6.9.3 | $x^{6} + 12 x^{5} + 86 x^{4} + 352 x^{3} + 892 x^{2} + 1552 x + 1384$ | $2$ | $3$ | $9$ | $C_6$ | $[3]^{3}$ | |
\(3\) | Deg $36$ | $6$ | $6$ | $54$ | |||
\(7\) | 7.18.15.5 | $x^{18} + 36 x^{17} + 540 x^{16} + 4344 x^{15} + 20160 x^{14} + 55296 x^{13} + 98757 x^{12} + 161784 x^{11} + 246024 x^{10} + 264920 x^{9} + 530640 x^{8} + 156384 x^{7} - 1885725 x^{6} - 6133212 x^{5} - 3645540 x^{4} + 5968464 x^{3} + 5011344 x^{2} + 1820448 x + 2358791$ | $6$ | $3$ | $15$ | $C_6 \times C_3$ | $[\ ]_{6}^{3}$ |
7.18.15.5 | $x^{18} + 36 x^{17} + 540 x^{16} + 4344 x^{15} + 20160 x^{14} + 55296 x^{13} + 98757 x^{12} + 161784 x^{11} + 246024 x^{10} + 264920 x^{9} + 530640 x^{8} + 156384 x^{7} - 1885725 x^{6} - 6133212 x^{5} - 3645540 x^{4} + 5968464 x^{3} + 5011344 x^{2} + 1820448 x + 2358791$ | $6$ | $3$ | $15$ | $C_6 \times C_3$ | $[\ ]_{6}^{3}$ |