Normalized defining polynomial
\( x^{16} - 5 x^{15} + 10 x^{14} - 50 x^{13} + 106 x^{12} + 15 x^{11} - 5 x^{10} - 20 x^{9} - 240 x^{8} + 20 x^{7} - 5 x^{6} - 15 x^{5} + 106 x^{4} + 50 x^{3} + 10 x^{2} + 5 x + 1 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[8, 4]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(20837857125684480535711201=31^{4}\cdot 41^{12}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $38.23$ | 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: | $31, 41$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{10} - \frac{1}{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a$, $\frac{1}{820} a^{12} - \frac{1}{82} a^{11} + \frac{5}{164} a^{10} + \frac{9}{41} a^{9} - \frac{4}{41} a^{8} - \frac{1}{82} a^{7} + \frac{3}{82} a^{6} + \frac{1}{82} a^{5} - \frac{4}{41} a^{4} - \frac{9}{41} a^{3} + \frac{5}{164} a^{2} - \frac{20}{41} a + \frac{1}{820}$, $\frac{1}{820} a^{13} - \frac{15}{164} a^{11} + \frac{1}{41} a^{10} + \frac{4}{41} a^{9} + \frac{1}{82} a^{8} - \frac{7}{82} a^{7} - \frac{5}{41} a^{6} + \frac{1}{41} a^{5} - \frac{8}{41} a^{4} - \frac{27}{164} a^{3} - \frac{15}{82} a^{2} - \frac{309}{820} a - \frac{20}{41}$, $\frac{1}{820} a^{14} + \frac{9}{82} a^{11} - \frac{19}{164} a^{10} - \frac{1}{41} a^{9} + \frac{4}{41} a^{8} - \frac{3}{82} a^{7} - \frac{19}{82} a^{6} + \frac{9}{41} a^{5} + \frac{3}{164} a^{4} - \frac{6}{41} a^{3} - \frac{37}{410} a^{2} - \frac{3}{41} a + \frac{15}{164}$, $\frac{1}{67240} a^{15} + \frac{1}{16810} a^{14} - \frac{9}{16810} a^{13} + \frac{9}{16810} a^{12} + \frac{207}{6724} a^{11} - \frac{453}{13448} a^{10} + \frac{149}{3362} a^{9} - \frac{669}{3362} a^{8} + \frac{366}{1681} a^{7} - \frac{293}{1681} a^{6} - \frac{2893}{13448} a^{5} + \frac{353}{1681} a^{4} + \frac{3752}{8405} a^{3} - \frac{819}{16810} a^{2} - \frac{5143}{33620} a - \frac{29839}{67240}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $11$ | 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: | \( 8033741.83521 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\wr C_4$ (as 16T157):
| A solvable group of order 64 |
| The 13 conjugacy class representatives for $C_2\wr C_4$ |
| Character table for $C_2\wr C_4$ |
Intermediate fields
| \(\Q(\sqrt{41}) \), 4.4.68921.1, 8.6.147253231471.1 x2, 8.4.4564850175601.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 8 siblings: | data not computed |
| Degree 16 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ | ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ | R | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $31$ | 31.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 31.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.4.2.1 | $x^{4} + 713 x^{2} + 138384$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 31.4.2.1 | $x^{4} + 713 x^{2} + 138384$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $41$ | 41.4.3.1 | $x^{4} - 41$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 41.4.3.1 | $x^{4} - 41$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 41.4.3.1 | $x^{4} - 41$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 41.4.3.1 | $x^{4} - 41$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |