Normalized defining polynomial
\( x^{27} - x^{26} - 42 x^{25} + 37 x^{24} + 728 x^{23} - 564 x^{22} - 6817 x^{21} + 4664 x^{20} + 37948 x^{19} - 23103 x^{18} - 130429 x^{17} + 71289 x^{16} + 279661 x^{15} - 138143 x^{14} - 372684 x^{13} + 166778 x^{12} + 305327 x^{11} - 124486 x^{10} - 150120 x^{9} + 56020 x^{8} + 42107 x^{7} - 14253 x^{6} - 6122 x^{5} + 1790 x^{4} + 395 x^{3} - 85 x^{2} - 10 x + 1 \)
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: | \(7977212169716289044333767743376433611324896529=7^{18}\cdot 19^{24}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $50.13$ | 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: | $7, 19$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(133=7\cdot 19\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{133}(64,·)$, $\chi_{133}(1,·)$, $\chi_{133}(130,·)$, $\chi_{133}(4,·)$, $\chi_{133}(100,·)$, $\chi_{133}(9,·)$, $\chi_{133}(74,·)$, $\chi_{133}(11,·)$, $\chi_{133}(16,·)$, $\chi_{133}(81,·)$, $\chi_{133}(85,·)$, $\chi_{133}(23,·)$, $\chi_{133}(25,·)$, $\chi_{133}(92,·)$, $\chi_{133}(93,·)$, $\chi_{133}(30,·)$, $\chi_{133}(99,·)$, $\chi_{133}(36,·)$, $\chi_{133}(102,·)$, $\chi_{133}(39,·)$, $\chi_{133}(106,·)$, $\chi_{133}(43,·)$, $\chi_{133}(44,·)$, $\chi_{133}(120,·)$, $\chi_{133}(121,·)$, $\chi_{133}(58,·)$, $\chi_{133}(123,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{113} a^{24} + \frac{27}{113} a^{23} + \frac{32}{113} a^{22} + \frac{6}{113} a^{21} + \frac{12}{113} a^{20} - \frac{18}{113} a^{19} + \frac{34}{113} a^{18} + \frac{41}{113} a^{17} + \frac{27}{113} a^{16} + \frac{41}{113} a^{15} - \frac{22}{113} a^{14} + \frac{32}{113} a^{13} + \frac{48}{113} a^{12} - \frac{33}{113} a^{11} + \frac{12}{113} a^{10} + \frac{18}{113} a^{9} + \frac{25}{113} a^{8} - \frac{7}{113} a^{7} + \frac{48}{113} a^{6} - \frac{34}{113} a^{5} + \frac{27}{113} a^{4} - \frac{15}{113} a^{3} - \frac{48}{113} a^{2} - \frac{24}{113} a - \frac{4}{113}$, $\frac{1}{113} a^{25} - \frac{19}{113} a^{23} + \frac{46}{113} a^{22} - \frac{37}{113} a^{21} - \frac{3}{113} a^{20} - \frac{45}{113} a^{19} + \frac{27}{113} a^{18} + \frac{50}{113} a^{17} - \frac{10}{113} a^{16} + \frac{1}{113} a^{15} - \frac{52}{113} a^{14} - \frac{25}{113} a^{13} + \frac{27}{113} a^{12} - \frac{1}{113} a^{11} + \frac{33}{113} a^{10} - \frac{9}{113} a^{9} - \frac{4}{113} a^{8} + \frac{11}{113} a^{7} + \frac{26}{113} a^{6} + \frac{41}{113} a^{5} + \frac{47}{113} a^{4} + \frac{18}{113} a^{3} + \frac{29}{113} a^{2} - \frac{34}{113} a - \frac{5}{113}$, $\frac{1}{8206544682495678472680089967879773} a^{26} + \frac{14669387966349457650803639027094}{8206544682495678472680089967879773} a^{25} - \frac{27945535847640849390740210846844}{8206544682495678472680089967879773} a^{24} + \frac{3742464948774644485192036085284255}{8206544682495678472680089967879773} a^{23} - \frac{1327541569494088178244529599129901}{8206544682495678472680089967879773} a^{22} - \frac{658886807479998098622240062336646}{8206544682495678472680089967879773} a^{21} - \frac{694961768360920198946127230809891}{8206544682495678472680089967879773} a^{20} - \frac{2309153772045912282863070608472196}{8206544682495678472680089967879773} a^{19} - \frac{1574994466038791749158137346267}{8206544682495678472680089967879773} a^{18} + \frac{3301909933939954784265101539864587}{8206544682495678472680089967879773} a^{17} + \frac{459648920194588690607000112177860}{8206544682495678472680089967879773} a^{16} + \frac{2757068035596579369764651155621985}{8206544682495678472680089967879773} a^{15} + \frac{3257553843522999448324846275435568}{8206544682495678472680089967879773} a^{14} + \frac{2342967951202671702013075889463633}{8206544682495678472680089967879773} a^{13} - \frac{3430477967030387439023911928846622}{8206544682495678472680089967879773} a^{12} + \frac{188613811539485153419623697366638}{8206544682495678472680089967879773} a^{11} + \frac{4009723612964788006229743716129247}{8206544682495678472680089967879773} a^{10} + \frac{1063182427617136563970509047153085}{8206544682495678472680089967879773} a^{9} - \frac{3914265650992212190793577659924791}{8206544682495678472680089967879773} a^{8} + \frac{3463845486354937268700293132362653}{8206544682495678472680089967879773} a^{7} - \frac{1066498817418124604979083074586013}{8206544682495678472680089967879773} a^{6} + \frac{3486686275783114067746518820906586}{8206544682495678472680089967879773} a^{5} - \frac{1775750959556524252412577928499423}{8206544682495678472680089967879773} a^{4} - \frac{573928709447212093576965780811753}{8206544682495678472680089967879773} a^{3} + \frac{958037281550647932241699355858838}{8206544682495678472680089967879773} a^{2} - \frac{1468986731164605974315582480867614}{8206544682495678472680089967879773} a - \frac{2548044515053912648408539789703558}{8206544682495678472680089967879773}$
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: | \( 173879598132270.75 \) (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
| 3.3.17689.1, 3.3.361.1, 3.3.17689.2, \(\Q(\zeta_{7})^+\), 9.9.5534900853769.1, 9.9.1998099208210609.1, 9.9.1998099208210609.2, \(\Q(\zeta_{19})^+\) |
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 | ${\href{/LocalNumberField/2.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/3.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/5.9.0.1}{9} }^{3}$ | R | ${\href{/LocalNumberField/11.3.0.1}{3} }^{9}$ | ${\href{/LocalNumberField/13.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/17.9.0.1}{9} }^{3}$ | R | ${\href{/LocalNumberField/23.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/29.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{9}$ | ${\href{/LocalNumberField/37.3.0.1}{3} }^{9}$ | ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/43.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/47.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/53.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/59.9.0.1}{9} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
| $7$ | 7.9.6.1 | $x^{9} + 42 x^{6} + 539 x^{3} + 2744$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ |
| 7.9.6.1 | $x^{9} + 42 x^{6} + 539 x^{3} + 2744$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ | |
| 7.9.6.1 | $x^{9} + 42 x^{6} + 539 x^{3} + 2744$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ | |
| 19 | Data not computed | ||||||