Normalized defining polynomial
\( x^{16} - 8 x^{15} + 64 x^{14} - 308 x^{13} + 1364 x^{12} - 4544 x^{11} + 12774 x^{10} - 28604 x^{9} + 50171 x^{8} - 68176 x^{7} + 60630 x^{6} - 24820 x^{5} - 5062 x^{4} + 9920 x^{3} + 2902 x^{2} - 6304 x + 17191 \)
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$$\chi_{1320}(1,·)$, $\chi_{1320}(901,·)$, $\chi_{1320}(769,·)$, $\chi_{1320}(329,·)$, $\chi_{1320}(461,·)$, $\chi_{1320}(749,·)$, $\chi_{1320}(529,·)$, $\chi_{1320}(661,·)$, $\chi_{1320}(89,·)$, $\chi_{1320}(989,·)$, $\chi_{1320}(1121,·)$, $\chi_{1320}(1189,·)$, $\chi_{1320}(881,·)$, $\chi_{1320}(109,·)$, $\chi_{1320}(221,·)$, $\chi_{1320}(241,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{192} a^{12} - \frac{1}{32} a^{11} + \frac{17}{192} a^{10} + \frac{3}{32} a^{9} - \frac{1}{32} a^{8} + \frac{7}{32} a^{7} + \frac{1}{192} a^{6} - \frac{3}{32} a^{5} - \frac{5}{32} a^{4} - \frac{15}{32} a^{3} + \frac{89}{192} a^{2} + \frac{5}{32} a + \frac{25}{192}$, $\frac{1}{192} a^{13} - \frac{19}{192} a^{11} - \frac{1}{8} a^{10} + \frac{1}{32} a^{9} + \frac{1}{32} a^{8} - \frac{35}{192} a^{7} + \frac{3}{16} a^{6} - \frac{7}{32} a^{5} + \frac{3}{32} a^{4} - \frac{67}{192} a^{3} - \frac{5}{16} a^{2} + \frac{13}{192} a + \frac{9}{32}$, $\frac{1}{9414450252096} a^{14} - \frac{7}{9414450252096} a^{13} - \frac{893525525}{1569075042016} a^{12} + \frac{32166918991}{9414450252096} a^{11} - \frac{828727392877}{9414450252096} a^{10} - \frac{143075264319}{1569075042016} a^{9} + \frac{694488039733}{9414450252096} a^{8} - \frac{1980631660309}{9414450252096} a^{7} + \frac{837903259279}{9414450252096} a^{6} - \frac{158833212435}{1569075042016} a^{5} + \frac{1721734325993}{9414450252096} a^{4} + \frac{81560324791}{9414450252096} a^{3} - \frac{155936636923}{1569075042016} a^{2} + \frac{2193938019617}{9414450252096} a - \frac{2505273358637}{9414450252096}$, $\frac{1}{299840826079005504} a^{15} + \frac{15917}{299840826079005504} a^{14} + \frac{570500517293387}{299840826079005504} a^{13} + \frac{27637594108769}{74960206519751376} a^{12} + \frac{865496307741643}{49973471013167584} a^{11} + \frac{14343038782490759}{299840826079005504} a^{10} + \frac{10056302875508629}{299840826079005504} a^{9} + \frac{33607008879380927}{299840826079005504} a^{8} + \frac{2596673914448925}{49973471013167584} a^{7} + \frac{21465263229166459}{299840826079005504} a^{6} - \frac{35619082646412379}{299840826079005504} a^{5} - \frac{3334734377893505}{299840826079005504} a^{4} + \frac{100203348492942457}{299840826079005504} a^{3} + \frac{332387211371857}{3656595439987872} a^{2} + \frac{7955459650390341}{49973471013167584} a + \frac{51567656404152367}{299840826079005504}$
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: | \( \frac{32239}{1235167968} a^{14} - \frac{225673}{1235167968} a^{13} + \frac{3525053}{2470335936} a^{12} - \frac{3820705}{617583984} a^{11} + \frac{63894299}{2470335936} a^{10} - \frac{31689343}{411722656} a^{9} + \frac{28561709}{154395996} a^{8} - \frac{209200835}{617583984} a^{7} + \frac{856775731}{2470335936} a^{6} - \frac{48556445}{411722656} a^{5} - \frac{108219545}{154395996} a^{4} + \frac{409789387}{308791992} a^{3} - \frac{1407524915}{2470335936} a^{2} - \frac{46628033}{617583984} a + \frac{1900176307}{2470335936} \) (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: | \( 405538.035702 \) (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 | ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.4.6.1 | $x^{4} - 6 x^{2} + 4$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ |
| 2.4.6.1 | $x^{4} - 6 x^{2} + 4$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
| 2.4.6.1 | $x^{4} - 6 x^{2} + 4$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
| 2.4.6.1 | $x^{4} - 6 x^{2} + 4$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
| $3$ | 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 | $x^{4} + 143 x^{2} + 5929$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |