Normalized defining polynomial
\( x^{16} - 8 x^{15} + 28 x^{14} - 56 x^{13} + 140 x^{12} - 476 x^{11} + 2262 x^{10} - 7328 x^{9} + 23357 x^{8} - 54124 x^{7} + 123426 x^{6} - 206968 x^{5} + 355430 x^{4} - 416728 x^{3} + 541996 x^{2} - 360952 x + 344644 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 8]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(2359562117249079705600000000=2^{32}\cdot 3^{8}\cdot 5^{8}\cdot 11^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $51.38$ | 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: | $2, 3, 5, 11$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1320=2^{3}\cdot 3\cdot 5\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1320}(1,·)$, $\chi_{1320}(131,·)$, $\chi_{1320}(901,·)$, $\chi_{1320}(769,·)$, $\chi_{1320}(529,·)$, $\chi_{1320}(659,·)$, $\chi_{1320}(661,·)$, $\chi_{1320}(791,·)$, $\chi_{1320}(419,·)$, $\chi_{1320}(1189,·)$, $\chi_{1320}(1319,·)$, $\chi_{1320}(551,·)$, $\chi_{1320}(109,·)$, $\chi_{1320}(241,·)$, $\chi_{1320}(1079,·)$, $\chi_{1320}(1211,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6}$, $\frac{1}{14} a^{11} - \frac{1}{7} a^{10} - \frac{1}{14} a^{9} + \frac{1}{7} a^{8} - \frac{5}{14} a^{7} + \frac{3}{14} a^{5} + \frac{3}{7} a^{4} - \frac{1}{7} a^{3} + \frac{1}{7} a^{2} - \frac{1}{7} a + \frac{3}{7}$, $\frac{1}{210} a^{12} - \frac{1}{35} a^{11} - \frac{7}{30} a^{10} - \frac{1}{14} a^{9} - \frac{8}{35} a^{8} + \frac{1}{35} a^{7} - \frac{19}{42} a^{6} + \frac{1}{14} a^{5} + \frac{31}{70} a^{4} - \frac{17}{35} a^{3} + \frac{16}{105} a^{2} - \frac{1}{5} a + \frac{23}{105}$, $\frac{1}{210} a^{13} + \frac{1}{42} a^{11} + \frac{6}{35} a^{10} - \frac{3}{35} a^{9} + \frac{1}{70} a^{8} - \frac{89}{210} a^{7} - \frac{1}{7} a^{6} + \frac{11}{70} a^{5} + \frac{17}{70} a^{4} + \frac{8}{21} a^{3} - \frac{3}{7} a^{2} + \frac{17}{105} a - \frac{4}{35}$, $\frac{1}{17460757518330} a^{14} - \frac{1}{2494393931190} a^{13} - \frac{3704165704}{8730378759165} a^{12} + \frac{44449988539}{17460757518330} a^{11} - \frac{1102255041754}{8730378759165} a^{10} + \frac{314118904989}{2910126253055} a^{9} + \frac{13842545089}{105185286255} a^{8} - \frac{510004625231}{3492151503666} a^{7} - \frac{5343820548817}{17460757518330} a^{6} + \frac{1239395565}{2726113586} a^{5} + \frac{3886606846657}{8730378759165} a^{4} + \frac{182477667389}{1247196965595} a^{3} - \frac{95058105293}{1247196965595} a^{2} + \frac{3202038774247}{8730378759165} a + \frac{104041991726}{1746075751833}$, $\frac{1}{328663838767525590} a^{15} + \frac{4702}{164331919383762795} a^{14} + \frac{23515356940616}{54777306461254265} a^{13} - \frac{1438658328958}{2314534075827645} a^{12} + \frac{1967959943114693}{328663838767525590} a^{11} - \frac{236860899864497}{65732767753505118} a^{10} + \frac{2213238804693005}{9390395393357874} a^{9} - \frac{47465592648659947}{328663838767525590} a^{8} - \frac{76340623008688534}{164331919383762795} a^{7} - \frac{153301641946791827}{328663838767525590} a^{6} - \frac{2200493842423283}{4629068151655290} a^{5} - \frac{132176360268731}{925813630331058} a^{4} - \frac{3891974436304401}{7825329494464895} a^{3} + \frac{27103661212517909}{164331919383762795} a^{2} + \frac{66035305476120476}{164331919383762795} a + \frac{54630086171234372}{164331919383762795}$
Class group and class number
$C_{2}\times C_{2}\times C_{8}\times C_{48}$, which has order $1536$ (assuming GRH)
Unit group
| Rank: | $7$ | 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: | \( 15197.42445606848 \) (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 | ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ | R | ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $11$ | 11.4.2.1 | $x^{4} + 143 x^{2} + 5929$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 11.4.2.1 | $x^{4} + 143 x^{2} + 5929$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 11.4.2.1 | $x^{4} + 143 x^{2} + 5929$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 11.4.2.1 | $x^{4} + 143 x^{2} + 5929$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |