Normalized defining polynomial
\( x^{16} + 28 x^{14} + 392 x^{12} + 3498 x^{10} + 21368 x^{8} + 89784 x^{6} + 270713 x^{4} + 429916 x^{2} + 364816 \)
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: | \(1171659381002265600000000=2^{16}\cdot 3^{8}\cdot 5^{8}\cdot 17^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $31.94$ | 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: | $2, 3, 5, 17$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1020=2^{2}\cdot 3\cdot 5\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1020}(1,·)$, $\chi_{1020}(781,·)$, $\chi_{1020}(271,·)$, $\chi_{1020}(851,·)$, $\chi_{1020}(341,·)$, $\chi_{1020}(919,·)$, $\chi_{1020}(409,·)$, $\chi_{1020}(611,·)$, $\chi_{1020}(101,·)$, $\chi_{1020}(679,·)$, $\chi_{1020}(169,·)$, $\chi_{1020}(749,·)$, $\chi_{1020}(239,·)$, $\chi_{1020}(1019,·)$, $\chi_{1020}(509,·)$, $\chi_{1020}(511,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{3}{8} a^{3}$, $\frac{1}{8} a^{10} - \frac{3}{8} a^{4}$, $\frac{1}{8} a^{11} - \frac{3}{8} a^{5}$, $\frac{1}{24} a^{12} - \frac{1}{24} a^{10} - \frac{11}{24} a^{6} + \frac{1}{8} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{24} a^{13} - \frac{1}{24} a^{11} + \frac{1}{24} a^{7} + \frac{1}{8} a^{5} + \frac{1}{3} a^{3} + \frac{1}{6} a$, $\frac{1}{3991545649222824} a^{14} + \frac{53597482954721}{3991545649222824} a^{12} + \frac{23498099367609}{665257608203804} a^{10} - \frac{107928934165433}{3991545649222824} a^{8} + \frac{42610751386169}{190073602343944} a^{6} + \frac{905832883744447}{1995772824611412} a^{4} + \frac{6637868142901}{25919127592356} a^{2} - \frac{26662287607124}{166314402050951}$, $\frac{1}{602723393032646424} a^{15} + \frac{53597482954721}{602723393032646424} a^{13} - \frac{4300676353957117}{100453898838774404} a^{11} + \frac{25837117785782923}{602723393032646424} a^{9} + \frac{1753273172481665}{28701113953935544} a^{7} + \frac{55789585560558277}{301361696516323212} a^{5} - \frac{1587388478786993}{3913788266445756} a^{3} - \frac{1024548699912830}{25113474709693601} a$
Class group and class number
$C_{2}\times C_{12}$, which has order $24$ (assuming GRH)
Unit group
| Rank: | $7$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{44313676931}{4531754834831928} a^{15} - \frac{391829787207}{1510584944943976} a^{13} - \frac{3853072845403}{1132938708707982} a^{11} - \frac{15696314103655}{566469354353991} a^{9} - \frac{677842836954125}{4531754834831928} a^{7} - \frac{585125404438291}{1132938708707982} a^{5} - \frac{164563131123853}{137325904085816} a^{3} - \frac{667268528605511}{1132938708707982} a \) (order $12$) | 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: | \( 197754.791017 \) (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 | ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ | R | ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.4.4.1 | $x^{4} + 8 x^{2} + 4$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ |
| 2.4.4.1 | $x^{4} + 8 x^{2} + 4$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
| 2.4.4.1 | $x^{4} + 8 x^{2} + 4$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
| 2.4.4.1 | $x^{4} + 8 x^{2} + 4$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
| $3$ | 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 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $17$ | 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |