Normalized defining polynomial
\( x^{16} - 13 x^{14} + 147 x^{12} - 1215 x^{10} + 5461 x^{8} - 7380 x^{6} - 13612 x^{4} + 26896 x^{2} + 26896 \)
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: | \(1216026685696000000000000=2^{20}\cdot 5^{12}\cdot 41^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $32.01$ | 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: | $2, 5, 41$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{6232} a^{12} - \frac{259}{6232} a^{10} + \frac{107}{328} a^{8} + \frac{671}{6232} a^{6} - \frac{689}{6232} a^{4} - \frac{1}{76} a^{2} - \frac{13}{38}$, $\frac{1}{6232} a^{13} - \frac{259}{6232} a^{11} + \frac{107}{328} a^{9} + \frac{671}{6232} a^{7} - \frac{689}{6232} a^{5} - \frac{1}{76} a^{3} - \frac{13}{38} a$, $\frac{1}{32715641449744} a^{14} - \frac{2302381139}{32715641449744} a^{12} - \frac{7058665487119}{32715641449744} a^{10} + \frac{3096569481191}{32715641449744} a^{8} - \frac{3376360130489}{32715641449744} a^{6} + \frac{6938487682987}{16357820724872} a^{4} + \frac{1195255323}{18135056236} a^{2} + \frac{11815388021}{49871404649}$, $\frac{1}{32715641449744} a^{15} - \frac{2302381139}{32715641449744} a^{13} - \frac{7058665487119}{32715641449744} a^{11} + \frac{3096569481191}{32715641449744} a^{9} - \frac{3376360130489}{32715641449744} a^{7} + \frac{6938487682987}{16357820724872} a^{5} + \frac{1195255323}{18135056236} a^{3} + \frac{11815388021}{49871404649} a$
Class group and class number
$C_{2}$, which has order $2$ (assuming GRH)
Unit group
| Rank: | $7$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{10561847}{270377202064} a^{14} + \frac{198903713}{270377202064} a^{12} - \frac{2339366947}{270377202064} a^{10} + \frac{21687085619}{270377202064} a^{8} - \frac{6873338855}{14230379056} a^{6} + \frac{201787519457}{135188601032} a^{4} - \frac{1729196607}{1648641476} a^{2} - \frac{1010581407}{412160369} \) (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: | \( 939543.897763 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^3.C_2^4.C_2$ (as 16T467):
| A solvable group of order 256 |
| The 46 conjugacy class representatives for $C_2^3.C_2^4.C_2$ |
| Character table for $C_2^3.C_2^4.C_2$ is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.0.1025.1, 4.4.5125.1, 8.0.26265625.1 |
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 | R | ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.12.4 | $x^{8} + 2 x^{6} + 16$ | $2$ | $4$ | $12$ | $C_2^3: C_4$ | $[2, 2, 3]^{4}$ |
| 2.8.8.3 | $x^{8} + 2 x^{7} + 2 x^{6} + 16$ | $2$ | $4$ | $8$ | $C_2^3: C_4$ | $[2, 2, 2]^{4}$ | |
| $5$ | 5.4.3.2 | $x^{4} - 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 5.4.3.2 | $x^{4} - 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| $41$ | 41.4.3.3 | $x^{4} + 246$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 41.4.3.3 | $x^{4} + 246$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 41.4.0.1 | $x^{4} - x + 17$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 41.4.0.1 | $x^{4} - x + 17$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |