Normalized defining polynomial
\( x^{16} - 4 x^{15} - 14 x^{14} + 32 x^{13} + 383 x^{12} - 880 x^{11} - 1206 x^{10} + 632 x^{9} + 14222 x^{8} - 22724 x^{7} + 16428 x^{6} - 26060 x^{5} + 32489 x^{4} - 17792 x^{3} + 11176 x^{2} - 10148 x + 3481 \)
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: | \(9217039520504217600000000=2^{24}\cdot 3^{8}\cdot 5^{8}\cdot 11^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $36.33$ | 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$$\chi_{1320}(1,·)$, $\chi_{1320}(1099,·)$, $\chi_{1320}(571,·)$, $\chi_{1320}(769,·)$, $\chi_{1320}(329,·)$, $\chi_{1320}(331,·)$, $\chi_{1320}(529,·)$, $\chi_{1320}(131,·)$, $\chi_{1320}(89,·)$, $\chi_{1320}(859,·)$, $\chi_{1320}(1121,·)$, $\chi_{1320}(419,·)$, $\chi_{1320}(881,·)$, $\chi_{1320}(241,·)$, $\chi_{1320}(659,·)$, $\chi_{1320}(1211,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5}$, $\frac{1}{36} a^{12} + \frac{1}{6} a^{11} - \frac{1}{9} a^{10} - \frac{1}{6} a^{9} + \frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{18} a^{6} - \frac{1}{6} a^{5} - \frac{1}{3} a^{4} + \frac{1}{6} a^{3} - \frac{5}{18} a^{2} - \frac{1}{2} a + \frac{1}{36}$, $\frac{1}{36} a^{13} - \frac{1}{9} a^{11} + \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{18} a^{7} + \frac{1}{6} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{5}{18} a^{3} + \frac{1}{6} a^{2} + \frac{1}{36} a - \frac{1}{6}$, $\frac{1}{1194480} a^{14} - \frac{1009}{597240} a^{13} - \frac{13841}{1194480} a^{12} - \frac{16843}{298620} a^{11} + \frac{1681}{17064} a^{10} + \frac{4241}{24885} a^{9} + \frac{1847}{21330} a^{8} + \frac{7912}{74655} a^{7} + \frac{73783}{597240} a^{6} + \frac{1576}{8295} a^{5} - \frac{252491}{597240} a^{4} + \frac{48949}{149310} a^{3} + \frac{426539}{1194480} a^{2} + \frac{30701}{85320} a + \frac{68039}{1194480}$, $\frac{1}{47468175645843180804240} a^{15} + \frac{312750953089579}{2063833723732312208880} a^{14} + \frac{178885487778358960829}{47468175645843180804240} a^{13} + \frac{66621971513508106397}{5274241738427020089360} a^{12} - \frac{342816880332890205269}{1582272521528106026808} a^{11} + \frac{4538093959832940883709}{23734087822921590402120} a^{10} + \frac{2749942425210549007}{2966760977865198800265} a^{9} - \frac{261763363424063343883}{2966760977865198800265} a^{8} - \frac{1081485533668312920319}{7911362607640530134040} a^{7} - \frac{1095287910307891252823}{23734087822921590402120} a^{6} + \frac{1645356061646953602889}{23734087822921590402120} a^{5} - \frac{2269268836804947265369}{23734087822921590402120} a^{4} + \frac{21083776428408705587179}{47468175645843180804240} a^{3} + \frac{2879871980637051753473}{15822725215281060268080} a^{2} + \frac{4828924776851855530783}{15822725215281060268080} a - \frac{1611806055640779169}{160909069985909087472}$
Class group and class number
$C_{4}\times C_{4}$, which has order $16$ (assuming GRH)
Unit group
| Rank: | $7$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( \frac{1230726535186378519}{66762553650974937840} a^{15} - \frac{41741350245851519}{725679930988858020} a^{14} - \frac{20594687006840131651}{66762553650974937840} a^{13} + \frac{3538327536004466983}{11127092275162489640} a^{12} + \frac{11661855650419439661}{1589584610737498520} a^{11} - \frac{163014224502219661217}{16690638412743734460} a^{10} - \frac{3665002685859881095}{119218845805312389} a^{9} - \frac{129005516573570575343}{8345319206371867230} a^{8} + \frac{2761920042334765433347}{11127092275162489640} a^{7} - \frac{670723546173990555905}{3338127682548746892} a^{6} + \frac{4291731527410703555779}{33381276825487468920} a^{5} - \frac{1229486976486575721421}{3338127682548746892} a^{4} + \frac{18454861571333380896229}{66762553650974937840} a^{3} - \frac{17577240972226830333}{198698076342187315} a^{2} + \frac{2899518249900871072919}{22254184550324979280} a - \frac{5930341390079199469}{80826336139194840} \) (order $6$) | 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: | \( 381719.802089 \) (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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.4.6.2 | $x^{4} - 2 x^{2} + 4$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ |
| 2.4.6.2 | $x^{4} - 2 x^{2} + 4$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
| 2.4.6.2 | $x^{4} - 2 x^{2} + 4$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
| 2.4.6.2 | $x^{4} - 2 x^{2} + 4$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
| $3$ | 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 | $x^{4} + 143 x^{2} + 5929$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |