Normalized defining polynomial
\( x^{27} - 9 x^{26} - 27 x^{25} + 420 x^{24} + 9 x^{23} - 8244 x^{22} + 7758 x^{21} + 88713 x^{20} - 125694 x^{19} - 570907 x^{18} + 991197 x^{17} + 2239173 x^{16} - 4595436 x^{15} - 5142816 x^{14} + 13108698 x^{13} + 5797086 x^{12} - 22773501 x^{11} - 112752 x^{10} + 22776726 x^{9} - 6585183 x^{8} - 11690658 x^{7} + 5812863 x^{6} + 2561958 x^{5} - 1837044 x^{4} - 142095 x^{3} + 213804 x^{2} - 7074 x - 7019 \)
Invariants
| Degree: | $27$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[27, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(3475226802198116057554377769989579566636047832552241=3^{66}\cdot 13^{18}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $81.08$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $3, 13$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(351=3^{3}\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{351}(256,·)$, $\chi_{351}(1,·)$, $\chi_{351}(196,·)$, $\chi_{351}(133,·)$, $\chi_{351}(328,·)$, $\chi_{351}(139,·)$, $\chi_{351}(334,·)$, $\chi_{351}(79,·)$, $\chi_{351}(16,·)$, $\chi_{351}(274,·)$, $\chi_{351}(211,·)$, $\chi_{351}(22,·)$, $\chi_{351}(217,·)$, $\chi_{351}(157,·)$, $\chi_{351}(94,·)$, $\chi_{351}(289,·)$, $\chi_{351}(100,·)$, $\chi_{351}(295,·)$, $\chi_{351}(40,·)$, $\chi_{351}(235,·)$, $\chi_{351}(172,·)$, $\chi_{351}(178,·)$, $\chi_{351}(118,·)$, $\chi_{351}(55,·)$, $\chi_{351}(313,·)$, $\chi_{351}(250,·)$, $\chi_{351}(61,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{53} a^{22} + \frac{16}{53} a^{17} + \frac{5}{53} a^{16} - \frac{9}{53} a^{15} - \frac{19}{53} a^{14} + \frac{1}{53} a^{13} + \frac{20}{53} a^{12} - \frac{11}{53} a^{11} - \frac{19}{53} a^{10} + \frac{6}{53} a^{9} - \frac{6}{53} a^{8} + \frac{23}{53} a^{7} + \frac{26}{53} a^{6} + \frac{7}{53} a^{5} - \frac{9}{53} a^{4} + \frac{14}{53} a^{3} - \frac{1}{53} a^{2} - \frac{9}{53} a + \frac{17}{53}$, $\frac{1}{53} a^{23} + \frac{16}{53} a^{18} + \frac{5}{53} a^{17} - \frac{9}{53} a^{16} - \frac{19}{53} a^{15} + \frac{1}{53} a^{14} + \frac{20}{53} a^{13} - \frac{11}{53} a^{12} - \frac{19}{53} a^{11} + \frac{6}{53} a^{10} - \frac{6}{53} a^{9} + \frac{23}{53} a^{8} + \frac{26}{53} a^{7} + \frac{7}{53} a^{6} - \frac{9}{53} a^{5} + \frac{14}{53} a^{4} - \frac{1}{53} a^{3} - \frac{9}{53} a^{2} + \frac{17}{53} a$, $\frac{1}{53} a^{24} + \frac{16}{53} a^{19} + \frac{5}{53} a^{18} - \frac{9}{53} a^{17} - \frac{19}{53} a^{16} + \frac{1}{53} a^{15} + \frac{20}{53} a^{14} - \frac{11}{53} a^{13} - \frac{19}{53} a^{12} + \frac{6}{53} a^{11} - \frac{6}{53} a^{10} + \frac{23}{53} a^{9} + \frac{26}{53} a^{8} + \frac{7}{53} a^{7} - \frac{9}{53} a^{6} + \frac{14}{53} a^{5} - \frac{1}{53} a^{4} - \frac{9}{53} a^{3} + \frac{17}{53} a^{2}$, $\frac{1}{5777} a^{25} - \frac{15}{5777} a^{24} - \frac{21}{5777} a^{23} - \frac{47}{5777} a^{22} + \frac{47}{109} a^{21} + \frac{705}{5777} a^{20} + \frac{2256}{5777} a^{19} - \frac{1374}{5777} a^{18} - \frac{1536}{5777} a^{17} + \frac{2042}{5777} a^{16} + \frac{1728}{5777} a^{15} - \frac{1188}{5777} a^{14} - \frac{1116}{5777} a^{13} + \frac{1543}{5777} a^{12} + \frac{1350}{5777} a^{11} - \frac{233}{5777} a^{10} + \frac{1009}{5777} a^{9} - \frac{2333}{5777} a^{8} + \frac{1704}{5777} a^{7} - \frac{372}{5777} a^{6} + \frac{497}{5777} a^{5} + \frac{1672}{5777} a^{4} - \frac{2817}{5777} a^{3} - \frac{443}{5777} a^{2} - \frac{146}{5777} a + \frac{102}{5777}$, $\frac{1}{342547841334140361116757614627009154759727950212586227909831} a^{26} - \frac{19314627126488732583275223068870823739826868072459349512}{342547841334140361116757614627009154759727950212586227909831} a^{25} + \frac{1701396580697436797198072761469493720248574986067296663645}{342547841334140361116757614627009154759727950212586227909831} a^{24} + \frac{1924642012332203843423767354128678990366518462596888561436}{342547841334140361116757614627009154759727950212586227909831} a^{23} - \frac{2161431186373159963704290378875745091949138263529956395709}{342547841334140361116757614627009154759727950212586227909831} a^{22} + \frac{127350438971072545153166197876652119711019993818021898910943}{342547841334140361116757614627009154759727950212586227909831} a^{21} - \frac{111451448557936383200756741087619562789008010062896530289219}{342547841334140361116757614627009154759727950212586227909831} a^{20} + \frac{102037967475027652969481161583548221523226473060090718416186}{342547841334140361116757614627009154759727950212586227909831} a^{19} - \frac{142229050120045829971956569519451659931794769858464763124260}{342547841334140361116757614627009154759727950212586227909831} a^{18} - \frac{5902021400676739709340354437115843613251569740648179804879}{342547841334140361116757614627009154759727950212586227909831} a^{17} - \frac{22462743255373069525823751123753318225044098091888271281143}{342547841334140361116757614627009154759727950212586227909831} a^{16} - \frac{73528812566611398520687186404118219835976672519559500780952}{342547841334140361116757614627009154759727950212586227909831} a^{15} - \frac{119387383342788357095765390975488681156909118605800030291982}{342547841334140361116757614627009154759727950212586227909831} a^{14} - \frac{127372370454076535467173474559847580533737817611254742001222}{342547841334140361116757614627009154759727950212586227909831} a^{13} - \frac{147565499220604738731896062812643471675329548015795651459648}{342547841334140361116757614627009154759727950212586227909831} a^{12} + \frac{145234307831714483762373563783518636184092547365325160289793}{342547841334140361116757614627009154759727950212586227909831} a^{11} - \frac{68812414065066797729308453449356903235816589834675137513120}{342547841334140361116757614627009154759727950212586227909831} a^{10} + \frac{159435461068902152770593685531780159589071160439075067410271}{342547841334140361116757614627009154759727950212586227909831} a^{9} + \frac{103860331111293273810748217650949330393591948271749464768898}{342547841334140361116757614627009154759727950212586227909831} a^{8} - \frac{65430537313999265786696742853601376505436792496625208726845}{342547841334140361116757614627009154759727950212586227909831} a^{7} - \frac{63729202263093812640865007065724641780714842621007903261274}{342547841334140361116757614627009154759727950212586227909831} a^{6} + \frac{357588025499342756597326654850352804689243409541821582766}{6463166817625289832391653106169984052070338683256343922827} a^{5} + \frac{117095038980697706950636119226404998770007256447210137481096}{342547841334140361116757614627009154759727950212586227909831} a^{4} - \frac{167909817644306060839818543726892582350398379950965181638005}{342547841334140361116757614627009154759727950212586227909831} a^{3} + \frac{635275536900028296162340858721966570945813940926319956772}{3142640746184773955199611143367056465685577524886112182659} a^{2} - \frac{94373318003504921956567583065794464231724600409962986431783}{342547841334140361116757614627009154759727950212586227909831} a + \frac{112201501983927612858314805552571572054475547521922360097027}{342547841334140361116757614627009154759727950212586227909831}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $26$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 183824455555719360 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_3\times C_9$ (as 27T2):
| An abelian group of order 27 |
| The 27 conjugacy class representatives for $C_3\times C_9$ |
| Character table for $C_3\times C_9$ is not computed |
Intermediate fields
| 3.3.13689.2, \(\Q(\zeta_{9})^+\), 3.3.13689.1, 3.3.169.1, 9.9.2565164201769.1, 9.9.151470380950257681.2, 9.9.151470380950257681.1, \(\Q(\zeta_{27})^+\) |
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 | ${\href{/LocalNumberField/2.9.0.1}{9} }^{3}$ | R | ${\href{/LocalNumberField/5.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/7.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/11.9.0.1}{9} }^{3}$ | R | ${\href{/LocalNumberField/17.3.0.1}{3} }^{9}$ | ${\href{/LocalNumberField/19.3.0.1}{3} }^{9}$ | ${\href{/LocalNumberField/23.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/29.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/31.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/37.3.0.1}{3} }^{9}$ | ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/43.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/47.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/53.1.0.1}{1} }^{27}$ | ${\href{/LocalNumberField/59.9.0.1}{9} }^{3}$ |
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 | Data not computed | ||||||
| 13 | Data not computed | ||||||