Normalized defining polynomial
\( x^{16} - 2 x^{14} - 7 x^{12} + 4 x^{10} + 34 x^{8} + 50 x^{6} + 32 x^{4} + 8 x^{2} + 1 \)
Invariants
| Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
| |
| Signature: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
| |
| Discriminant: | \(5572562780160000\)\(\medspace = 2^{24}\cdot 3^{12}\cdot 5^{4}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
| Root discriminant: | $9.64$ | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
| |
| Ramified primes: | $2, 3, 5$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
| |
| $|\Aut(K/\Q)|$: | $8$ | ||
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{572} a^{14} + \frac{43}{572} a^{12} - \frac{1}{4} a^{11} + \frac{69}{572} a^{10} + \frac{53}{286} a^{8} - \frac{1}{4} a^{7} + \frac{57}{143} a^{6} + \frac{7}{286} a^{4} - \frac{1}{4} a^{3} + \frac{233}{572} a^{2} - \frac{1}{4} a + \frac{197}{572}$, $\frac{1}{572} a^{15} + \frac{43}{572} a^{13} - \frac{37}{286} a^{11} - \frac{1}{4} a^{10} - \frac{37}{572} a^{9} - \frac{29}{286} a^{7} - \frac{1}{4} a^{6} - \frac{68}{143} a^{5} - \frac{49}{143} a^{3} - \frac{1}{4} a^{2} + \frac{197}{572} a - \frac{1}{4}$
Class group and class number
Trivial group, which has order $1$
Unit group
| Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
| |
| Torsion generator: | \( \frac{19}{143} a^{14} - \frac{41}{143} a^{12} - \frac{119}{143} a^{10} + \frac{167}{286} a^{8} + \frac{1085}{286} a^{6} + \frac{981}{143} a^{4} + \frac{709}{143} a^{2} + \frac{479}{286} \) (order $12$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
| |
| 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 | sage: UK.fundamental_units()
gp: K.fu
magma: [K!f(g): g in Generators(UK)];
| |
| Regulator: | \( 62.6647388247 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
|
Class number formula
Galois group
$C_2^2:D_4$ (as 16T43):
| A solvable group of order 32 |
| The 14 conjugacy class representatives for $C_2^2:D_4$ |
| Character table for $C_2^2:D_4$ |
Intermediate fields
| \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), 4.0.320.1, 4.0.2880.1, 4.0.432.1 x2, 4.2.1728.1 x2, \(\Q(\zeta_{12})\), 8.0.2985984.1, 8.0.4665600.1, 8.0.8294400.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | Deg 32 |
| Degree 16 siblings: | 16.0.3482851737600000000.1, 16.0.217678233600000000.2, 16.4.3482851737600000000.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 | Data not computed | ||||||
| 3 | Data not computed | ||||||
| $5$ | 5.4.0.1 | $x^{4} + x^{2} - 2 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 5.4.0.1 | $x^{4} + x^{2} - 2 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |