Normalized defining polynomial
\( x^{16} + 8 x^{14} - 2 x^{12} - 84 x^{10} + 275 x^{8} + 2471 x^{6} + 8088 x^{4} + 17303 x^{2} + 14641 \)
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: | \(5205633773443603515625=5^{14}\cdot 31^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $22.77$ | 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: | $5, 31$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{22} a^{13} - \frac{3}{22} a^{11} - \frac{1}{11} a^{9} - \frac{7}{22} a^{7} - \frac{1}{2} a^{4} - \frac{2}{11} a^{3} - \frac{1}{2} a^{2} - \frac{4}{11} a - \frac{1}{2}$, $\frac{1}{1317591106942} a^{14} + \frac{145680040037}{658795553471} a^{12} - \frac{226410669641}{1317591106942} a^{10} + \frac{144449869291}{658795553471} a^{8} - \frac{1}{2} a^{7} - \frac{27812300821}{119781009722} a^{6} - \frac{1}{2} a^{5} + \frac{104265494109}{1317591106942} a^{4} - \frac{196964896279}{1317591106942} a^{2} - \frac{1}{2} a - \frac{2185893561}{5444591351}$, $\frac{1}{14493502176362} a^{15} + \frac{145680040037}{7246751088181} a^{13} - \frac{1760194218498}{7246751088181} a^{11} + \frac{2265286398995}{14493502176362} a^{9} - \frac{283413422285}{658795553471} a^{7} - \frac{277265029681}{7246751088181} a^{5} - \frac{1}{2} a^{4} + \frac{5073399531489}{14493502176362} a^{3} + \frac{14147880492}{59890504861} a - \frac{1}{2}$
Class group and class number
$C_{6}$, which has order $6$ (assuming GRH)
Unit group
| Rank: | $7$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{6666645970}{7246751088181} a^{15} - \frac{368311}{265589822} a^{14} - \frac{26945027839}{7246751088181} a^{13} - \frac{788343}{132794911} a^{12} + \frac{124426774495}{7246751088181} a^{11} + \frac{6690323}{265589822} a^{10} + \frac{146107364267}{14493502176362} a^{9} + \frac{6589819}{265589822} a^{8} - \frac{405209935551}{1317591106942} a^{7} - \frac{125468079}{265589822} a^{6} - \frac{7285065842723}{7246751088181} a^{5} - \frac{447192047}{265589822} a^{4} - \frac{46334818117569}{14493502176362} a^{3} - \frac{665679685}{132794911} a^{2} - \frac{382550349263}{119781009722} a - \frac{1244023225}{265589822} \) (order $10$) | 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: | \( 17871.2949174 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^3.C_4$ (as 16T36):
| A solvable group of order 32 |
| The 11 conjugacy class representatives for $C_2^3.C_4$ |
| Character table for $C_2^3.C_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-155}) \), \(\Q(\sqrt{-31}) \), 4.4.120125.1, \(\Q(\zeta_{5})\), \(\Q(\sqrt{5}, \sqrt{-31})\), 8.4.72150078125.1 x2, 8.0.75078125.1 x2, 8.0.14430015625.1 |
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 8 siblings: | data not computed |
| Degree 16 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 | ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ | R | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/41.1.0.1}{1} }^{16}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $5$ | 5.8.7.2 | $x^{8} - 20$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ |
| 5.8.7.2 | $x^{8} - 20$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ | |
| $31$ | 31.2.1.2 | $x^{2} + 217$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 31.2.1.2 | $x^{2} + 217$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 31.2.1.2 | $x^{2} + 217$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 31.2.1.2 | $x^{2} + 217$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 31.4.2.1 | $x^{4} + 713 x^{2} + 138384$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 31.4.2.1 | $x^{4} + 713 x^{2} + 138384$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |