Normalized defining polynomial
\( x^{16} - 6 x^{15} + 11 x^{14} - 55 x^{13} + 140 x^{12} - 125 x^{11} + 295 x^{10} + 424 x^{9} - 1932 x^{8} + 3263 x^{7} - 8562 x^{6} + 5689 x^{5} + 10442 x^{4} - 27935 x^{3} + 31002 x^{2} - 5990 x - 3229 \)
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: | \(17521464151279710991512529=17^{14}\cdot 101^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $37.82$ | 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: | $17, 101$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{137} a^{14} - \frac{36}{137} a^{13} - \frac{12}{137} a^{12} + \frac{9}{137} a^{11} - \frac{46}{137} a^{10} - \frac{41}{137} a^{9} + \frac{66}{137} a^{8} - \frac{36}{137} a^{7} + \frac{56}{137} a^{6} + \frac{54}{137} a^{5} - \frac{25}{137} a^{4} + \frac{33}{137} a^{3} + \frac{37}{137} a^{2} + \frac{42}{137} a + \frac{28}{137}$, $\frac{1}{174122210410167826547529580219327} a^{15} + \frac{561899033355403423732004740913}{174122210410167826547529580219327} a^{14} - \frac{3237414789491484843377702979936}{174122210410167826547529580219327} a^{13} + \frac{23971714258999904578349447789638}{174122210410167826547529580219327} a^{12} - \frac{87037864745922413225034896190741}{174122210410167826547529580219327} a^{11} + \frac{20900668486621333583262886309161}{174122210410167826547529580219327} a^{10} - \frac{19004288106709606651105437202211}{174122210410167826547529580219327} a^{9} - \frac{3442954713730870443255435849447}{174122210410167826547529580219327} a^{8} + \frac{7608240858750740984860021110119}{174122210410167826547529580219327} a^{7} - \frac{58721079363493327170894252409517}{174122210410167826547529580219327} a^{6} + \frac{41423589646950911325502152155357}{174122210410167826547529580219327} a^{5} + \frac{17164682048566409442688033935730}{174122210410167826547529580219327} a^{4} - \frac{65391244661465344974568908302537}{174122210410167826547529580219327} a^{3} + \frac{49775222344414281525953042419663}{174122210410167826547529580219327} a^{2} - \frac{57699655023917640787140173157477}{174122210410167826547529580219327} a + \frac{72517042088101841390134098009581}{174122210410167826547529580219327}$
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: | \( 3421001.30832 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 1024 |
| The 40 conjugacy class representatives for t16n1194 |
| Character table for t16n1194 is not computed |
Intermediate fields
| \(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\) |
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 | ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $17$ | 17.8.7.3 | $x^{8} - 17$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |
| 17.8.7.3 | $x^{8} - 17$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ | |
| $101$ | $\Q_{101}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{101}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{101}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{101}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 101.2.1.2 | $x^{2} + 202$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 101.2.1.2 | $x^{2} + 202$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 101.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 101.2.1.2 | $x^{2} + 202$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 101.2.1.2 | $x^{2} + 202$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 101.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |