Normalized defining polynomial
\( x^{36} - x^{33} - 70 x^{30} + 1289 x^{27} + 2848 x^{24} - 33978 x^{21} + 415031 x^{18} + \cdots + 594823321 \)
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: | \(4999758568289868528789868885747458073284974388119052886962890625\) \(\medspace = 3^{54}\cdot 5^{18}\cdot 7^{30}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(58.81\) | 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: | $3^{3/2}5^{1/2}7^{5/6}\approx 58.8051349456126$ | ||
Ramified primes: | \(3\), \(5\), \(7\) | 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: | \(315=3^{2}\cdot 5\cdot 7\) | ||
Dirichlet character group: | $\lbrace$$\chi_{315}(256,·)$, $\chi_{315}(1,·)$, $\chi_{315}(139,·)$, $\chi_{315}(269,·)$, $\chi_{315}(16,·)$, $\chi_{315}(19,·)$, $\chi_{315}(151,·)$, $\chi_{315}(281,·)$, $\chi_{315}(304,·)$, $\chi_{315}(34,·)$, $\chi_{315}(164,·)$, $\chi_{315}(296,·)$, $\chi_{315}(299,·)$, $\chi_{315}(46,·)$, $\chi_{315}(176,·)$, $\chi_{315}(116,·)$, $\chi_{315}(314,·)$, $\chi_{315}(59,·)$, $\chi_{315}(191,·)$, $\chi_{315}(194,·)$, $\chi_{315}(11,·)$, $\chi_{315}(199,·)$, $\chi_{315}(209,·)$, $\chi_{315}(211,·)$, $\chi_{315}(86,·)$, $\chi_{315}(89,·)$, $\chi_{315}(71,·)$, $\chi_{315}(221,·)$, $\chi_{315}(94,·)$, $\chi_{315}(226,·)$, $\chi_{315}(229,·)$, $\chi_{315}(104,·)$, $\chi_{315}(106,·)$, $\chi_{315}(244,·)$, $\chi_{315}(121,·)$, $\chi_{315}(124,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{131072}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{13}a^{18}+\frac{5}{13}a^{15}+\frac{5}{13}a^{9}+\frac{5}{13}a^{3}-\frac{1}{13}$, $\frac{1}{13}a^{19}+\frac{5}{13}a^{16}+\frac{5}{13}a^{10}+\frac{5}{13}a^{4}-\frac{1}{13}a$, $\frac{1}{13}a^{20}+\frac{5}{13}a^{17}+\frac{5}{13}a^{11}+\frac{5}{13}a^{5}-\frac{1}{13}a^{2}$, $\frac{1}{13}a^{21}+\frac{1}{13}a^{15}+\frac{5}{13}a^{12}+\frac{1}{13}a^{9}+\frac{5}{13}a^{6}+\frac{5}{13}$, $\frac{1}{13}a^{22}+\frac{1}{13}a^{16}+\frac{5}{13}a^{13}+\frac{1}{13}a^{10}+\frac{5}{13}a^{7}+\frac{5}{13}a$, $\frac{1}{13}a^{23}+\frac{1}{13}a^{17}+\frac{5}{13}a^{14}+\frac{1}{13}a^{11}+\frac{5}{13}a^{8}+\frac{5}{13}a^{2}$, $\frac{1}{13}a^{24}+\frac{1}{13}a^{12}+\frac{1}{13}$, $\frac{1}{13}a^{25}+\frac{1}{13}a^{13}+\frac{1}{13}a$, $\frac{1}{13}a^{26}+\frac{1}{13}a^{14}+\frac{1}{13}a^{2}$, $\frac{1}{26}a^{27}-\frac{6}{13}a^{15}-\frac{1}{2}a^{9}-\frac{6}{13}a^{3}-\frac{1}{2}$, $\frac{1}{26}a^{28}-\frac{6}{13}a^{16}-\frac{1}{2}a^{10}-\frac{6}{13}a^{4}-\frac{1}{2}a$, $\frac{1}{26}a^{29}-\frac{6}{13}a^{17}-\frac{1}{2}a^{11}-\frac{6}{13}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{4112401046}a^{30}-\frac{25393591}{4112401046}a^{27}+\frac{12696760}{2056200523}a^{24}+\frac{14910783}{2056200523}a^{21}-\frac{31936432}{2056200523}a^{18}+\frac{454998186}{2056200523}a^{15}-\frac{1362198399}{4112401046}a^{12}+\frac{1864147897}{4112401046}a^{9}-\frac{722929869}{2056200523}a^{6}-\frac{889789065}{4112401046}a^{3}+\frac{484278341}{4112401046}$, $\frac{1}{119259630334}a^{31}-\frac{408119973}{59629815167}a^{28}-\frac{1885334492}{59629815167}a^{25}+\frac{173080054}{59629815167}a^{22}+\frac{442571381}{59629815167}a^{19}+\frac{1562183083}{59629815167}a^{16}+\frac{16985437037}{119259630334}a^{13}+\frac{22996687253}{59629815167}a^{10}-\frac{16381687698}{59629815167}a^{7}-\frac{44228169319}{119259630334}a^{4}-\frac{12648656416}{59629815167}a$, $\frac{1}{3458529279686}a^{32}+\frac{3770668913}{3458529279686}a^{29}-\frac{1885334492}{1729264639843}a^{26}+\frac{9346897772}{1729264639843}a^{23}-\frac{54600334927}{1729264639843}a^{20}-\frac{649778874895}{1729264639843}a^{17}+\frac{1062800656889}{3458529279686}a^{14}-\frac{426458237971}{3458529279686}a^{11}+\frac{89117216059}{1729264639843}a^{8}+\frac{1496973207305}{3458529279686}a^{5}+\frac{951714274135}{3458529279686}a^{2}$, $\frac{1}{69\!\cdots\!06}a^{33}+\frac{51\!\cdots\!52}{34\!\cdots\!03}a^{30}-\frac{86\!\cdots\!25}{69\!\cdots\!06}a^{27}+\frac{66\!\cdots\!22}{34\!\cdots\!03}a^{24}+\frac{74\!\cdots\!44}{34\!\cdots\!03}a^{21}-\frac{41\!\cdots\!05}{34\!\cdots\!03}a^{18}-\frac{13\!\cdots\!87}{69\!\cdots\!06}a^{15}-\frac{80\!\cdots\!43}{34\!\cdots\!03}a^{12}-\frac{34\!\cdots\!11}{69\!\cdots\!06}a^{9}-\frac{84\!\cdots\!97}{69\!\cdots\!06}a^{6}+\frac{11\!\cdots\!49}{34\!\cdots\!03}a^{3}-\frac{12\!\cdots\!53}{28\!\cdots\!54}$, $\frac{1}{20\!\cdots\!74}a^{34}+\frac{51\!\cdots\!52}{10\!\cdots\!87}a^{31}-\frac{98\!\cdots\!71}{10\!\cdots\!87}a^{28}+\frac{81\!\cdots\!15}{10\!\cdots\!87}a^{25}+\frac{22\!\cdots\!92}{10\!\cdots\!87}a^{22}+\frac{23\!\cdots\!74}{10\!\cdots\!87}a^{19}+\frac{28\!\cdots\!85}{20\!\cdots\!74}a^{16}-\frac{20\!\cdots\!37}{10\!\cdots\!87}a^{13}-\frac{48\!\cdots\!15}{10\!\cdots\!87}a^{10}+\frac{27\!\cdots\!89}{20\!\cdots\!74}a^{7}-\frac{13\!\cdots\!87}{10\!\cdots\!87}a^{4}+\frac{23\!\cdots\!46}{41\!\cdots\!83}a$, $\frac{1}{58\!\cdots\!46}a^{35}+\frac{51\!\cdots\!52}{29\!\cdots\!23}a^{32}+\frac{83\!\cdots\!47}{58\!\cdots\!46}a^{29}+\frac{24\!\cdots\!12}{29\!\cdots\!23}a^{26}+\frac{56\!\cdots\!85}{29\!\cdots\!23}a^{23}+\frac{11\!\cdots\!60}{29\!\cdots\!23}a^{20}-\frac{18\!\cdots\!77}{58\!\cdots\!46}a^{17}-\frac{12\!\cdots\!23}{29\!\cdots\!23}a^{14}-\frac{14\!\cdots\!59}{58\!\cdots\!46}a^{11}-\frac{14\!\cdots\!21}{58\!\cdots\!46}a^{8}+\frac{13\!\cdots\!40}{29\!\cdots\!23}a^{5}-\frac{54\!\cdots\!51}{24\!\cdots\!14}a^{2}$
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: | \( \frac{6921108753981151158125696013}{8292087143854494049769968630493817166} a^{34} - \frac{941567415194979960768417862}{142967019721629207754654631560238227} a^{31} - \frac{150652800368259247394568166143}{4146043571927247024884984315246908583} a^{28} + \frac{6095073959946102864921771660252}{4146043571927247024884984315246908583} a^{25} - \frac{26254952032201136127001100887891}{4146043571927247024884984315246908583} a^{22} - \frac{105340678329434777534244437457400}{4146043571927247024884984315246908583} a^{19} + \frac{5039082461939296081592750135788093}{8292087143854494049769968630493817166} a^{16} - \frac{21832931462959167005202935547890502}{4146043571927247024884984315246908583} a^{13} + \frac{111571895622143247251603156194403821}{4146043571927247024884984315246908583} a^{10} - \frac{374570198742050087237942648772652019}{8292087143854494049769968630493817166} a^{7} - \frac{18185607450678345871428102267198351}{142967019721629207754654631560238227} a^{4} + \frac{2347059412181283559934910054110578416}{4146043571927247024884984315246908583} a \) (order $18$) | 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 | ${\href{/padicField/2.6.0.1}{6} }^{6}$ | R | R | R | ${\href{/padicField/11.6.0.1}{6} }^{6}$ | ${\href{/padicField/13.3.0.1}{3} }^{12}$ | ${\href{/padicField/17.6.0.1}{6} }^{6}$ | ${\href{/padicField/19.6.0.1}{6} }^{6}$ | ${\href{/padicField/23.6.0.1}{6} }^{6}$ | ${\href{/padicField/29.6.0.1}{6} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }^{6}$ | ${\href{/padicField/37.6.0.1}{6} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{6}$ | ${\href{/padicField/43.6.0.1}{6} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{6}$ | ${\href{/padicField/53.6.0.1}{6} }^{6}$ | ${\href{/padicField/59.6.0.1}{6} }^{6}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | Deg $18$ | $6$ | $3$ | $27$ | |||
Deg $18$ | $6$ | $3$ | $27$ | ||||
\(5\) | 5.12.6.1 | $x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
5.12.6.1 | $x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
5.12.6.1 | $x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
\(7\) | 7.18.15.1 | $x^{18} + 315 x^{12} + 9555 x^{6} + 76489$ | $6$ | $3$ | $15$ | $C_6 \times C_3$ | $[\ ]_{6}^{3}$ |
7.18.15.1 | $x^{18} + 315 x^{12} + 9555 x^{6} + 76489$ | $6$ | $3$ | $15$ | $C_6 \times C_3$ | $[\ ]_{6}^{3}$ |