Normalized defining polynomial
\( x^{16} - 2 x^{15} - 3 x^{14} - 47 x^{13} - 91 x^{12} - 117 x^{11} + 1950 x^{10} + 7956 x^{9} + 17628 x^{8} + 27001 x^{7} + 44668 x^{6} - 97565 x^{5} - 272701 x^{4} - 340644 x^{3} + 815305 x^{2} - 396615 x + 415819 \)
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: | \(882950323258772752350175913=13^{15}\cdot 29^{7}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $48.32$ | 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: | $13, 29$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{17} a^{14} - \frac{5}{17} a^{12} - \frac{6}{17} a^{11} - \frac{8}{17} a^{10} - \frac{2}{17} a^{9} + \frac{7}{17} a^{8} + \frac{1}{17} a^{7} + \frac{4}{17} a^{6} - \frac{6}{17} a^{5} + \frac{6}{17} a^{4} + \frac{5}{17} a^{3} - \frac{6}{17} a^{2} - \frac{3}{17} a + \frac{8}{17}$, $\frac{1}{55068498770464984822526128911187495492429945581} a^{15} + \frac{1573187484950068901415238930515821590621190015}{55068498770464984822526128911187495492429945581} a^{14} - \frac{17380944968475814808915182231236737791251714682}{55068498770464984822526128911187495492429945581} a^{13} - \frac{4642490648767474191772211619696825619804658361}{18356166256821661607508709637062498497476648527} a^{12} - \frac{21406012384846223093179688227218023947904699920}{55068498770464984822526128911187495492429945581} a^{11} + \frac{4660087353539811707798475366465864012528620528}{55068498770464984822526128911187495492429945581} a^{10} - \frac{14669214024957118538646078221205903583406192776}{55068498770464984822526128911187495492429945581} a^{9} + \frac{7197724499405411401497743011491233301495102197}{55068498770464984822526128911187495492429945581} a^{8} + \frac{4031000900999417871706091059458814828816854482}{55068498770464984822526128911187495492429945581} a^{7} + \frac{36677383687107923453058284426667214960417942}{346342759562672860519032257303066009386351859} a^{6} + \frac{22564310477487445842704650577997163619331324736}{55068498770464984822526128911187495492429945581} a^{5} - \frac{11077719126997394085951653107866067067345968162}{55068498770464984822526128911187495492429945581} a^{4} + \frac{20831376442755190053770392789061854873186780741}{55068498770464984822526128911187495492429945581} a^{3} + \frac{17019562397673940490580130755590273026650585328}{55068498770464984822526128911187495492429945581} a^{2} + \frac{13886825868511398791313025648680935465505156917}{55068498770464984822526128911187495492429945581} a - \frac{2988762100274263592348157877968749682005209411}{55068498770464984822526128911187495492429945581}$
Class group and class number
$C_{2}\times C_{2}$, which has order $4$ (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: | \( 1536470.17414 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 1024 |
| The 88 conjugacy class representatives for t16n1192 are not computed |
| Character table for t16n1192 is not computed |
Intermediate fields
| \(\Q(\sqrt{13}) \), 4.0.2197.1, 8.0.1819706993.1 |
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 | ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ | $16$ | $16$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{8}$ | $16$ |
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 | Data not computed | ||||||
| $29$ | 29.2.1.1 | $x^{2} - 29$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 29.2.1.1 | $x^{2} - 29$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.4.3.1 | $x^{4} - 29$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 29.4.2.2 | $x^{4} - 29 x^{2} + 2523$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 29.4.0.1 | $x^{4} - x + 19$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |