Normalized defining polynomial
\( x^{16} - x^{14} - 284 x^{12} + 3843 x^{10} - 61087 x^{8} + 514808 x^{6} - 882864 x^{4} + 717824 x^{2} + 65536 \)
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: | \(5614886866882301027209678756129=23^{6}\cdot 41^{14}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $83.53$ | 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: | $23, 41$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{8} a^{3} - \frac{3}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{8} a^{4} - \frac{1}{2} a^{3} + \frac{1}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{32} a^{11} - \frac{1}{32} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} + \frac{3}{32} a^{5} - \frac{15}{32} a^{3} - \frac{3}{8} a^{2} - \frac{3}{8} a$, $\frac{1}{1472} a^{12} + \frac{19}{1472} a^{10} - \frac{3}{46} a^{8} - \frac{11}{64} a^{6} + \frac{125}{1472} a^{4} + \frac{159}{368} a^{2} - \frac{1}{2} a - \frac{10}{23}$, $\frac{1}{5888} a^{13} - \frac{1}{2944} a^{12} + \frac{19}{5888} a^{11} - \frac{19}{2944} a^{10} - \frac{3}{184} a^{9} + \frac{3}{92} a^{8} - \frac{11}{256} a^{7} + \frac{11}{128} a^{6} - \frac{611}{5888} a^{5} + \frac{611}{2944} a^{4} - \frac{577}{1472} a^{3} - \frac{159}{736} a^{2} - \frac{5}{46} a + \frac{5}{23}$, $\frac{1}{328286881159174144} a^{14} - \frac{61983342040657}{328286881159174144} a^{12} - \frac{1}{64} a^{11} + \frac{1564059843934461}{82071720289793536} a^{10} - \frac{3}{64} a^{9} - \frac{9348312876417}{102685918410752} a^{8} - \frac{1}{4} a^{7} + \frac{20174298539573873}{328286881159174144} a^{6} - \frac{3}{64} a^{5} + \frac{6362801881205981}{41035860144896768} a^{4} + \frac{19}{64} a^{3} - \frac{3661095801210471}{20517930072448384} a^{2} + \frac{5}{16} a - \frac{2520314123125}{13938811190522}$, $\frac{1}{2626295049273393152} a^{15} + \frac{49527147483519}{2626295049273393152} a^{13} - \frac{1}{2944} a^{12} + \frac{2093734669174297}{656573762318348288} a^{11} - \frac{19}{2944} a^{10} - \frac{292025634965655}{18894208987578368} a^{9} + \frac{3}{92} a^{8} + \frac{156105585269544417}{2626295049273393152} a^{7} + \frac{11}{128} a^{6} - \frac{63707601973548113}{328286881159174144} a^{5} + \frac{611}{2944} a^{4} - \frac{60782344059969627}{164143440579587072} a^{3} + \frac{209}{736} a^{2} - \frac{232285731059875}{641185314764012} a - \frac{13}{46}$
Class group and class number
$C_{4}\times C_{4}$, which has order $16$ (assuming GRH)
Unit group
| Rank: | $7$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | 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: | \( 37533039.8671 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^5.C_2.C_2$ (as 16T257):
| A solvable group of order 128 |
| The 26 conjugacy class representatives for $C_2^5.C_2.C_2$ |
| Character table for $C_2^5.C_2.C_2$ is not computed |
Intermediate fields
| \(\Q(\sqrt{41}) \), 4.4.68921.1, 8.2.109252397543.1, 8.4.103025010883049.3, 8.2.2369575250310127.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ | R | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
| $23$ | 23.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 23.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.4.2.1 | $x^{4} + 299 x^{2} + 25921$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.4.2.1 | $x^{4} + 299 x^{2} + 25921$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.4.2.1 | $x^{4} + 299 x^{2} + 25921$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $41$ | 41.8.7.3 | $x^{8} - 53136$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |
| 41.8.7.3 | $x^{8} - 53136$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |