Normalized defining polynomial
\( x^{16} - 8 x^{15} + 23 x^{14} - 5 x^{13} - 135 x^{12} + 284 x^{11} + 123 x^{10} - 1204 x^{9} + 1099 x^{8} + 1747 x^{7} - 3431 x^{6} - 398 x^{5} + 4310 x^{4} - 1715 x^{3} - 650 x^{2} + 1061 x - 383 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[2, 7]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-3031849240553934703791403=-\,29^{8}\cdot 67^{7}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $33.89$ | 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: | $29, 67$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{11} + \frac{1}{5} a^{10} + \frac{1}{5} a^{9} + \frac{2}{5} a^{8} - \frac{2}{5} a^{7} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2} - \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{10} - \frac{2}{5} a^{9} - \frac{1}{5} a^{6} + \frac{1}{5} a^{4} - \frac{1}{5} a^{3} + \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{145} a^{14} - \frac{12}{145} a^{13} + \frac{9}{145} a^{12} - \frac{22}{145} a^{11} + \frac{33}{145} a^{10} + \frac{8}{145} a^{9} - \frac{12}{145} a^{8} + \frac{1}{145} a^{7} + \frac{14}{29} a^{6} - \frac{16}{145} a^{5} - \frac{51}{145} a^{4} + \frac{44}{145} a^{3} - \frac{14}{29} a^{2} + \frac{71}{145} a - \frac{16}{145}$, $\frac{1}{29044158205045716005} a^{15} + \frac{55401985618505989}{29044158205045716005} a^{14} + \frac{1651696634378856489}{29044158205045716005} a^{13} + \frac{2014685266045967469}{29044158205045716005} a^{12} - \frac{532383485927126187}{2234166015772747385} a^{11} + \frac{2946043057241913172}{29044158205045716005} a^{10} - \frac{152741153854983171}{319166573681821055} a^{9} + \frac{397690887922800599}{4149165457863673715} a^{8} + \frac{1166848105291253351}{4149165457863673715} a^{7} + \frac{12610390058029332601}{29044158205045716005} a^{6} - \frac{168807226652077599}{4149165457863673715} a^{5} - \frac{10929168126163953539}{29044158205045716005} a^{4} - \frac{1818282557414473401}{4149165457863673715} a^{3} - \frac{253494203119723901}{4149165457863673715} a^{2} + \frac{1416972688647099958}{5808831641009143201} a - \frac{12478669744522448608}{29044158205045716005}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $8$ | 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: | \( 1464412.76758 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 32 |
| The 11 conjugacy class representatives for $D_{16}$ |
| Character table for $D_{16}$ |
Intermediate fields
| \(\Q(\sqrt{29}) \), 4.2.56347.1, 8.2.212723955403.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | data not computed |
| Degree 16 sibling: | 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 | $16$ | $16$ | ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/7.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ | $16$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | R | $16$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ | $16$ | $16$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $29$ | 29.2.1.1 | $x^{2} - 29$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 29.2.1.1 | $x^{2} - 29$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.2.1.1 | $x^{2} - 29$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.2.1.1 | $x^{2} - 29$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.2.1.1 | $x^{2} - 29$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.2.1.1 | $x^{2} - 29$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.2.1.1 | $x^{2} - 29$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.2.1.1 | $x^{2} - 29$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| $67$ | $\Q_{67}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{67}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 67.2.1.2 | $x^{2} + 268$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 67.2.1.2 | $x^{2} + 268$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 67.2.1.2 | $x^{2} + 268$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 67.2.1.2 | $x^{2} + 268$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 67.2.1.2 | $x^{2} + 268$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 67.2.1.2 | $x^{2} + 268$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 67.2.1.2 | $x^{2} + 268$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |