Normalized defining polynomial
\( x^{16} - 2 x^{15} - 45 x^{14} + 78 x^{13} + 759 x^{12} - 1218 x^{11} - 5980 x^{10} + 10232 x^{9} + 22641 x^{8} - 47290 x^{7} - 31370 x^{6} + 109240 x^{5} - 26870 x^{4} - 88500 x^{3} + 80550 x^{2} - 24300 x + 2025 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[16, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(19456426971136000000000000=2^{24}\cdot 5^{12}\cdot 41^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $38.07$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{11} - \frac{2}{5} a^{9} - \frac{1}{5} a^{8} + \frac{2}{5} a^{7} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4}$, $\frac{1}{15} a^{13} + \frac{1}{15} a^{12} - \frac{2}{5} a^{11} + \frac{1}{5} a^{10} + \frac{1}{5} a^{9} - \frac{2}{5} a^{8} - \frac{4}{15} a^{7} + \frac{2}{15} a^{6} - \frac{1}{5} a^{5} - \frac{7}{15} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{315} a^{14} + \frac{4}{315} a^{13} - \frac{4}{105} a^{12} - \frac{7}{15} a^{11} + \frac{7}{15} a^{10} - \frac{4}{15} a^{9} - \frac{58}{315} a^{8} - \frac{19}{45} a^{7} + \frac{17}{35} a^{6} - \frac{4}{315} a^{5} - \frac{31}{63} a^{4} + \frac{8}{63} a^{3} + \frac{2}{63} a^{2} + \frac{8}{21} a + \frac{2}{7}$, $\frac{1}{3535403039013195} a^{15} - \frac{121518898120}{101011515400377} a^{14} - \frac{486260726783}{168352525667295} a^{13} + \frac{6025582773199}{69321628215945} a^{12} - \frac{57351186317951}{168352525667295} a^{11} - \frac{4521528491572}{168352525667295} a^{10} - \frac{474166734547963}{3535403039013195} a^{9} + \frac{1141808088825041}{3535403039013195} a^{8} + \frac{9698011202561}{69321628215945} a^{7} + \frac{129025710744572}{505057577001885} a^{6} - \frac{1455599470564133}{3535403039013195} a^{5} - \frac{681866693183807}{3535403039013195} a^{4} + \frac{197281599924092}{707080607802639} a^{3} + \frac{12324132270136}{78564511978071} a^{2} - \frac{4083217487383}{26188170659357} a - \frac{5209294432543}{26188170659357}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $15$ | 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: | \( 26821769.1882 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^2:D_4$ (as 16T34):
| 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
| \(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), 4.4.2624.1, 4.4.65600.2, 4.4.5125.1, 4.4.328000.1, \(\Q(\sqrt{2}, \sqrt{5})\), 8.8.107584000000.4, 8.8.4303360000.1, 8.8.107584000000.1 |
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 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 | R | ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.12.1 | $x^{8} + 6 x^{6} + 8 x^{5} + 16$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ |
| 2.8.12.1 | $x^{8} + 6 x^{6} + 8 x^{5} + 16$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ | |
| $5$ | 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| $41$ | 41.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 41.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 41.2.1.2 | $x^{2} + 246$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 41.2.1.2 | $x^{2} + 246$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 41.4.2.1 | $x^{4} + 943 x^{2} + 242064$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 41.4.2.1 | $x^{4} + 943 x^{2} + 242064$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |