Normalized defining polynomial
\( x^{16} - 44 x^{14} + 788 x^{12} - 7446 x^{10} + 40268 x^{8} - 124812 x^{6} + 226937 x^{4} - 166712 x^{2} + 59536 \)
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: | \(2852586422067225600000000=2^{16}\cdot 3^{8}\cdot 5^{8}\cdot 19^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $33.76$ | 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: | $2, 3, 5, 19$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1140=2^{2}\cdot 3\cdot 5\cdot 19\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1140}(1,·)$, $\chi_{1140}(569,·)$, $\chi_{1140}(911,·)$, $\chi_{1140}(721,·)$, $\chi_{1140}(341,·)$, $\chi_{1140}(151,·)$, $\chi_{1140}(989,·)$, $\chi_{1140}(799,·)$, $\chi_{1140}(419,·)$, $\chi_{1140}(229,·)$, $\chi_{1140}(379,·)$, $\chi_{1140}(1139,·)$, $\chi_{1140}(949,·)$, $\chi_{1140}(761,·)$, $\chi_{1140}(571,·)$, $\chi_{1140}(191,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} + \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} + \frac{1}{4} a^{4}$, $\frac{1}{4} a^{11} + \frac{1}{4} a^{5}$, $\frac{1}{24} a^{12} - \frac{1}{24} a^{10} - \frac{1}{8} a^{8} + \frac{1}{24} a^{6} + \frac{1}{8} a^{4} - \frac{7}{24} a^{2} + \frac{1}{6}$, $\frac{1}{1464} a^{13} - \frac{109}{1464} a^{11} - \frac{19}{488} a^{9} - \frac{203}{1464} a^{7} - \frac{235}{488} a^{5} - \frac{541}{1464} a^{3} - \frac{40}{183} a$, $\frac{1}{4028892713196288} a^{14} + \frac{715014301747}{35972256367824} a^{12} + \frac{120916502968381}{1007223178299072} a^{10} - \frac{198939194219971}{2014446356598144} a^{8} + \frac{238533770829857}{1007223178299072} a^{6} + \frac{108449175461537}{1007223178299072} a^{4} - \frac{1956954698086327}{4028892713196288} a^{2} - \frac{5575825681723}{16511855381952}$, $\frac{1}{8057785426392576} a^{15} + \frac{2449114133}{71944512735648} a^{13} - \frac{74130830060457}{671482118866048} a^{11} - \frac{442489061103763}{4028892713196288} a^{9} - \frac{81250001705997}{671482118866048} a^{7} - \frac{430250106374647}{2014446356598144} a^{5} + \frac{3951537552755497}{8057785426392576} a^{3} + \frac{168243744597147}{671482118866048} a$
Class group and class number
$C_{4}\times C_{8}$, which has order $32$ (assuming GRH)
Unit group
| Rank: | $7$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( \frac{1394549221}{83069952849408} a^{15} - \frac{178554861}{247232002528} a^{13} + \frac{258926389241}{20767488212352} a^{11} - \frac{4587298247935}{41534976424704} a^{9} + \frac{10986581283613}{20767488212352} a^{7} - \frac{26435153049331}{20767488212352} a^{5} + \frac{38136086563471}{27689984283136} a^{3} + \frac{12629680759477}{20767488212352} a \) (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: | \( 274735.259425 \) (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 | ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ | R | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $19$ | 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |