Normalized defining polynomial
\( x^{16} - 44 x^{14} - 36 x^{13} + 644 x^{12} + 816 x^{11} - 3894 x^{10} - 5844 x^{9} + 10061 x^{8} + 15576 x^{7} - 13122 x^{6} - 17952 x^{5} + 9374 x^{4} + 8568 x^{3} - 3164 x^{2} - 1176 x + 196 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[16, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(63456228123711897600000000=2^{32}\cdot 3^{8}\cdot 5^{8}\cdot 7^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $40.99$ | 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: | $2, 3, 5, 7$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(840=2^{3}\cdot 3\cdot 5\cdot 7\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{840}(1,·)$, $\chi_{840}(491,·)$, $\chi_{840}(391,·)$, $\chi_{840}(139,·)$, $\chi_{840}(461,·)$, $\chi_{840}(589,·)$, $\chi_{840}(209,·)$, $\chi_{840}(659,·)$, $\chi_{840}(71,·)$, $\chi_{840}(239,·)$, $\chi_{840}(421,·)$, $\chi_{840}(169,·)$, $\chi_{840}(811,·)$, $\chi_{840}(559,·)$, $\chi_{840}(629,·)$, $\chi_{840}(41,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{10} a^{10} + \frac{1}{5} a^{9} + \frac{1}{5} a^{8} - \frac{1}{5} a^{7} + \frac{1}{10} a^{6} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2} + \frac{2}{5}$, $\frac{1}{10} a^{11} - \frac{1}{5} a^{9} - \frac{1}{10} a^{8} - \frac{1}{2} a^{7} - \frac{1}{5} a^{6} - \frac{3}{10} a^{4} + \frac{1}{5} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{330} a^{12} - \frac{7}{330} a^{10} + \frac{13}{110} a^{9} + \frac{3}{22} a^{8} + \frac{3}{55} a^{7} - \frac{25}{66} a^{6} + \frac{39}{110} a^{5} - \frac{13}{55} a^{3} + \frac{52}{165} a^{2} + \frac{17}{55} a - \frac{2}{33}$, $\frac{1}{330} a^{13} - \frac{7}{330} a^{11} + \frac{1}{55} a^{10} - \frac{7}{110} a^{9} - \frac{8}{55} a^{8} - \frac{59}{330} a^{7} + \frac{14}{55} a^{6} - \frac{13}{55} a^{4} + \frac{19}{165} a^{3} - \frac{1}{11} a^{2} - \frac{2}{33} a - \frac{2}{5}$, $\frac{1}{2310} a^{14} - \frac{1}{1155} a^{12} + \frac{1}{385} a^{11} - \frac{4}{165} a^{10} + \frac{52}{385} a^{9} + \frac{83}{1155} a^{8} + \frac{139}{385} a^{7} + \frac{139}{462} a^{6} + \frac{167}{385} a^{5} + \frac{19}{1155} a^{4} + \frac{19}{77} a^{3} - \frac{16}{231} a^{2} + \frac{9}{55} a + \frac{8}{33}$, $\frac{1}{19921391829570} a^{15} + \frac{64417735}{569182623702} a^{14} + \frac{1730497176}{3320231971595} a^{13} - \frac{26918110871}{19921391829570} a^{12} - \frac{20414052569}{1422956559255} a^{11} + \frac{212043282919}{6640463943190} a^{10} + \frac{2455181374757}{9960695914785} a^{9} + \frac{187500556766}{1992139182957} a^{8} + \frac{486482488181}{1811035620870} a^{7} - \frac{323339990445}{664046394319} a^{6} - \frac{195713694214}{585923289105} a^{5} + \frac{8923186049}{1171846578210} a^{4} + \frac{1003026139051}{3320231971595} a^{3} - \frac{618842151674}{1422956559255} a^{2} - \frac{89506918805}{284591311851} a + \frac{29275458202}{67759836155}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $15$ | 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: | \( 100622555.559 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| An abelian group of order 16 |
| The 16 conjugacy class representatives for $C_2^4$ |
| Character table for $C_2^4$ |
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 | R | R | ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.16.6 | $x^{8} + 4 x^{6} + 8 x^{2} + 4$ | $4$ | $2$ | $16$ | $C_2^3$ | $[2, 3]^{2}$ |
| 2.8.16.6 | $x^{8} + 4 x^{6} + 8 x^{2} + 4$ | $4$ | $2$ | $16$ | $C_2^3$ | $[2, 3]^{2}$ | |
| $3$ | 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $5$ | 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $7$ | 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |