Normalized defining polynomial
\( x^{32} - 7 x^{28} - 704 x^{24} - 5047 x^{20} + 565969 x^{16} - 80752 x^{12} - 180224 x^{8} - 28672 x^{4} + 65536 \)
Invariants
| Degree: | $32$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 16]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(4026692887688564776141139207792885760000000000000000=2^{64}\cdot 3^{16}\cdot 5^{16}\cdot 7^{16}\) | 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$$\chi_{840}(1,·)$, $\chi_{840}(391,·)$, $\chi_{840}(139,·)$, $\chi_{840}(659,·)$, $\chi_{840}(281,·)$, $\chi_{840}(29,·)$, $\chi_{840}(671,·)$, $\chi_{840}(419,·)$, $\chi_{840}(421,·)$, $\chi_{840}(41,·)$, $\chi_{840}(811,·)$, $\chi_{840}(559,·)$, $\chi_{840}(181,·)$, $\chi_{840}(799,·)$, $\chi_{840}(769,·)$, $\chi_{840}(701,·)$, $\chi_{840}(449,·)$, $\chi_{840}(839,·)$, $\chi_{840}(631,·)$, $\chi_{840}(589,·)$, $\chi_{840}(461,·)$, $\chi_{840}(209,·)$, $\chi_{840}(211,·)$, $\chi_{840}(601,·)$, $\chi_{840}(71,·)$, $\chi_{840}(349,·)$, $\chi_{840}(379,·)$, $\chi_{840}(491,·)$, $\chi_{840}(239,·)$, $\chi_{840}(629,·)$, $\chi_{840}(169,·)$, $\chi_{840}(251,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{4} + \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{5} + \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{6} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{7} + \frac{1}{3} a^{3}$, $\frac{1}{9} a^{12} + \frac{1}{9}$, $\frac{1}{18} a^{13} + \frac{1}{18} a$, $\frac{1}{36} a^{14} + \frac{1}{36} a^{2}$, $\frac{1}{72} a^{15} + \frac{1}{72} a^{3}$, $\frac{1}{432} a^{16} + \frac{1}{27} a^{12} + \frac{1}{9} a^{8} + \frac{169}{432} a^{4} + \frac{4}{27}$, $\frac{1}{432} a^{17} - \frac{1}{54} a^{13} + \frac{1}{9} a^{9} + \frac{169}{432} a^{5} + \frac{5}{54} a$, $\frac{1}{432} a^{18} + \frac{1}{108} a^{14} + \frac{1}{9} a^{10} + \frac{169}{432} a^{6} + \frac{13}{108} a^{2}$, $\frac{1}{432} a^{19} - \frac{1}{216} a^{15} + \frac{1}{9} a^{11} + \frac{169}{432} a^{7} + \frac{23}{216} a^{3}$, $\frac{1}{432} a^{20} - \frac{1}{27} a^{12} - \frac{23}{432} a^{8} - \frac{4}{9} a^{4} + \frac{11}{27}$, $\frac{1}{864} a^{21} - \frac{1}{864} a^{17} + \frac{1}{54} a^{13} + \frac{73}{864} a^{9} + \frac{359}{864} a^{5} - \frac{4}{27} a$, $\frac{1}{1728} a^{22} + \frac{1}{1728} a^{18} - \frac{119}{1728} a^{10} - \frac{743}{1728} a^{6} + \frac{11}{36} a^{2}$, $\frac{1}{3456} a^{23} + \frac{1}{3456} a^{19} + \frac{457}{3456} a^{11} + \frac{409}{3456} a^{7} + \frac{23}{72} a^{3}$, $\frac{1}{9020160} a^{24} + \frac{7}{601344} a^{20} - \frac{41}{37584} a^{16} - \frac{419767}{9020160} a^{12} - \frac{79313}{601344} a^{8} - \frac{4555}{18792} a^{4} + \frac{6541}{35235}$, $\frac{1}{18040320} a^{25} + \frac{7}{1202688} a^{21} - \frac{41}{75168} a^{17} - \frac{419767}{18040320} a^{13} - \frac{79313}{1202688} a^{9} + \frac{14237}{37584} a^{5} + \frac{6541}{70470} a$, $\frac{1}{36080640} a^{26} + \frac{7}{2405376} a^{22} + \frac{133}{150336} a^{18} + \frac{248393}{36080640} a^{14} + \frac{54319}{2405376} a^{10} + \frac{7235}{18792} a^{6} + \frac{16981}{140940} a^{2}$, $\frac{1}{72161280} a^{27} + \frac{7}{4810752} a^{23} + \frac{133}{300672} a^{19} + \frac{248393}{72161280} a^{15} - \frac{747473}{4810752} a^{11} - \frac{5293}{37584} a^{7} - \frac{29999}{281880} a^{3}$, $\frac{1}{417484755763200} a^{28} - \frac{9168791}{417484755763200} a^{24} + \frac{1806059933}{1739519815680} a^{20} - \frac{399491321527}{417484755763200} a^{16} + \frac{4033053919937}{417484755763200} a^{12} - \frac{8020821793}{108719988480} a^{8} - \frac{18239175839}{101924989200} a^{4} + \frac{4977957769}{101924989200}$, $\frac{1}{834969511526400} a^{29} - \frac{9168791}{834969511526400} a^{25} + \frac{1806059933}{3479039631360} a^{21} - \frac{399491321527}{834969511526400} a^{17} + \frac{4033053919937}{834969511526400} a^{13} + \frac{28219174367}{217439976960} a^{9} + \frac{49710816961}{203849978400} a^{5} - \frac{62972035031}{203849978400} a$, $\frac{1}{1669939023052800} a^{30} - \frac{9168791}{1669939023052800} a^{26} + \frac{1806059933}{6958079262720} a^{22} + \frac{1533308473673}{1669939023052800} a^{18} - \frac{11429344441663}{1669939023052800} a^{14} + \frac{52379171807}{434879953920} a^{10} + \frac{129457683511}{407699956800} a^{6} - \frac{44097037031}{407699956800} a^{2}$, $\frac{1}{3339878046105600} a^{31} - \frac{9168791}{3339878046105600} a^{27} + \frac{1806059933}{13916158525440} a^{23} - \frac{2332291116727}{3339878046105600} a^{19} + \frac{19495452281537}{3339878046105600} a^{15} - \frac{140900807713}{869759907840} a^{11} - \frac{301836020789}{815399913600} a^{7} - \frac{217747018631}{815399913600} a^{3}$
Class group and class number
Not computed
Unit group
| Rank: | $15$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( \frac{10002394073}{3339878046105600} a^{31} - \frac{47178166783}{3339878046105600} a^{27} - \frac{30022469771}{13916158525440} a^{23} - \frac{66534600394271}{3339878046105600} a^{19} + \frac{5548482660848281}{3339878046105600} a^{15} + \frac{3160937174461}{869759907840} a^{11} - \frac{708699417311}{407699956800} a^{7} - \frac{1076519987503}{815399913600} a^{3} \) (order $24$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Not computed | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | Not computed | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| An abelian group of order 32 |
| The 32 conjugacy class representatives for $C_2^5$ |
| Character table for $C_2^5$ is not computed |
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 | ${\href{/LocalNumberField/11.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{16}$ |
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 | $x^{8} + 4 x^{6} + 8 x^{2} + 4$ | $4$ | $2$ | $16$ | $C_2^3$ | $[2, 3]^{2}$ | |
| 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 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 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 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 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 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 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 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |