Normalized defining polynomial
\( x^{16} + 28 x^{14} - 12 x^{13} + 344 x^{12} - 96 x^{11} + 2102 x^{10} + 468 x^{9} + 5861 x^{8} + 7896 x^{7} + 8258 x^{6} + 25200 x^{5} + 21946 x^{4} + 32952 x^{3} + 43108 x^{2} + 36264 x + 29124 \)
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: | \(1368569050405273600000000=2^{32}\cdot 5^{8}\cdot 13^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $32.25$ | 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: | $2, 5, 13$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(520=2^{3}\cdot 5\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{520}(1,·)$, $\chi_{520}(259,·)$, $\chi_{520}(261,·)$, $\chi_{520}(129,·)$, $\chi_{520}(79,·)$, $\chi_{520}(209,·)$, $\chi_{520}(339,·)$, $\chi_{520}(469,·)$, $\chi_{520}(389,·)$, $\chi_{520}(391,·)$, $\chi_{520}(519,·)$, $\chi_{520}(51,·)$, $\chi_{520}(131,·)$, $\chi_{520}(311,·)$, $\chi_{520}(441,·)$, $\chi_{520}(181,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{5} + \frac{1}{3} a$, $\frac{1}{6} a^{10} - \frac{1}{2} a^{6} + \frac{1}{3} a^{2}$, $\frac{1}{6} a^{11} - \frac{1}{2} a^{7} + \frac{1}{3} a^{3}$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{4}$, $\frac{1}{18} a^{13} - \frac{1}{18} a^{12} + \frac{1}{18} a^{11} - \frac{1}{18} a^{10} + \frac{1}{6} a^{7} - \frac{1}{2} a^{6} - \frac{1}{18} a^{5} + \frac{1}{18} a^{4} - \frac{2}{9} a^{3} - \frac{4}{9} a^{2}$, $\frac{1}{11427570} a^{14} + \frac{81394}{5713785} a^{13} + \frac{8113}{1632510} a^{12} - \frac{302677}{11427570} a^{11} + \frac{29}{6790} a^{10} - \frac{46198}{1904595} a^{9} + \frac{16351}{173145} a^{8} - \frac{177159}{1269730} a^{7} + \frac{1318081}{5713785} a^{6} - \frac{48233}{336105} a^{5} - \frac{1077191}{2285514} a^{4} + \frac{1132046}{5713785} a^{3} + \frac{4699}{57715} a^{2} - \frac{675659}{1904595} a - \frac{35822}{634865}$, $\frac{1}{24485764507129101364650} a^{15} - \frac{785890201399921}{24485764507129101364650} a^{14} + \frac{422438319723550886909}{24485764507129101364650} a^{13} - \frac{468440208187173182531}{24485764507129101364650} a^{12} - \frac{36628324736213549714}{2448576450712910136465} a^{11} + \frac{1125917638953689384639}{24485764507129101364650} a^{10} - \frac{20993514583636153049}{388662928684588910550} a^{9} - \frac{2743249327514872934}{18137603338614149159} a^{8} + \frac{281518472612702555201}{24485764507129101364650} a^{7} + \frac{173308275570751549381}{2448576450712910136465} a^{6} + \frac{3440786500189794335264}{12242882253564550682325} a^{5} - \frac{5774978777132783892643}{24485764507129101364650} a^{4} + \frac{110878282064043409441}{1748983179080650097475} a^{3} - \frac{4540507855549664585836}{12242882253564550682325} a^{2} + \frac{60268847426180399024}{163238430047527342431} a + \frac{92038319214794884828}{1360320250396061186925}$
Class group and class number
$C_{2}\times C_{24}$, which has order $48$ (assuming GRH)
Unit group
| Rank: | $7$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{5601851706033731}{126215280964582996725} a^{15} + \frac{5393198377799101}{36061508847023713350} a^{14} - \frac{251566633083475643}{252430561929165993450} a^{13} + \frac{561146159444189701}{126215280964582996725} a^{12} - \frac{566636135377062911}{50486112385833198690} a^{11} + \frac{11274612429068231167}{252430561929165993450} a^{10} - \frac{639113086984164322}{14023920107175888525} a^{9} + \frac{511000774219000267}{2804784021435177705} a^{8} + \frac{22676278787065879183}{252430561929165993450} a^{7} + \frac{2668908851314878899}{25243056192916599345} a^{6} + \frac{234965097930513743729}{252430561929165993450} a^{5} + \frac{174568650662256349}{7424428292034293925} a^{4} + \frac{165756968597374182536}{126215280964582996725} a^{3} + \frac{30176740352863900366}{18030754423511856675} a^{2} + \frac{1132044330354554671}{764941096755048465} a + \frac{24582193446877811279}{14023920107175888525} \) (order $8$) | 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: | \( 57386.4139318 \) (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 | ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ | R | ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ | R | ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 | $x^{8} + 4 x^{6} + 8 x^{2} + 4$ | $4$ | $2$ | $16$ | $C_2^3$ | $[2, 3]^{2}$ | |
| $5$ | 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 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $13$ | 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |