Normalized defining polynomial
\( x^{16} - 5 x^{15} + 23 x^{14} - 92 x^{13} + 5 x^{12} + 373 x^{11} - 2967 x^{10} + 3754 x^{9} + 14359 x^{8} + 20408 x^{7} + 75241 x^{6} + 28094 x^{5} - 81462 x^{4} - 129250 x^{3} - 391063 x^{2} - 386833 x - 77607 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[4, 6]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(313426397802392026311505288813=13^{11}\cdot 53^{10}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $69.74$ | 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: | $13, 53$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{13} a^{14} + \frac{1}{13} a^{13} - \frac{6}{13} a^{12} + \frac{6}{13} a^{11} + \frac{4}{13} a^{10} + \frac{5}{13} a^{9} + \frac{4}{13} a^{8} + \frac{2}{13} a^{7} - \frac{4}{13} a^{6} - \frac{5}{13} a^{5} + \frac{3}{13} a^{4} - \frac{1}{13} a^{3} + \frac{2}{13} a^{2} + \frac{4}{13} a - \frac{4}{13}$, $\frac{1}{102910977661495117352255838514152145785267} a^{15} - \frac{432239188993916196483188365086524100524}{34303659220498372450751946171384048595089} a^{14} + \frac{38098975365144475740870600827864686568690}{102910977661495117352255838514152145785267} a^{13} + \frac{42235899637575516498715814529676341748952}{102910977661495117352255838514152145785267} a^{12} - \frac{4375673823221222867200904661392602348087}{11434553073499457483583982057128016198363} a^{11} - \frac{18080917371459746947978542298883111670923}{102910977661495117352255838514152145785267} a^{10} - \frac{11251700551301488863054089717450001352429}{102910977661495117352255838514152145785267} a^{9} + \frac{43224243400712281313814778900735706503811}{102910977661495117352255838514152145785267} a^{8} - \frac{29653441204931562614696017622244681000649}{102910977661495117352255838514152145785267} a^{7} - \frac{9277249566876018787920449838688657148387}{34303659220498372450751946171384048595089} a^{6} + \frac{34112812060814118965970671286405791527102}{102910977661495117352255838514152145785267} a^{5} + \frac{27324165523294127251399304872913166507157}{102910977661495117352255838514152145785267} a^{4} + \frac{23847758292817528217854411895027368641467}{102910977661495117352255838514152145785267} a^{3} + \frac{2608729260019079570531119272652936225755}{11434553073499457483583982057128016198363} a^{2} + \frac{7016503328834074707566198249833149610331}{102910977661495117352255838514152145785267} a - \frac{1504784791193081921844696100386089943634}{11434553073499457483583982057128016198363}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $9$ | 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: | \( 863343322.007 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_4:D_4.D_4$ (as 16T681):
| A solvable group of order 256 |
| The 19 conjugacy class representatives for $C_4:D_4.D_4$ |
| Character table for $C_4:D_4.D_4$ |
Intermediate fields
| \(\Q(\sqrt{13}) \), 4.4.8957.1, 8.4.2929680361933.3 |
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 | $16$ | ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{4}$ | $16$ | ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ | R | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ | $16$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ | $16$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ | $16$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ | R | ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $13$ | 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 13.4.3.1 | $x^{4} - 13$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 13.8.6.1 | $x^{8} - 13 x^{4} + 2704$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| $53$ | 53.4.0.1 | $x^{4} - x + 18$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 53.4.3.3 | $x^{4} + 106$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 53.8.7.2 | $x^{8} - 212$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ |