Normalized defining polynomial
\( x^{16} - 4x^{12} + 15x^{8} - 4x^{4} + 1 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(1846757322198614016\) \(\medspace = 2^{48}\cdot 3^{8}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(13.86\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
Galois root discriminant: | $2^{3}3^{1/2}\approx 13.856406460551018$ | ||
Ramified primes: | \(2\), \(3\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $16$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}+\frac{1}{3}a^{4}+\frac{1}{3}$, $\frac{1}{3}a^{9}+\frac{1}{3}a^{5}+\frac{1}{3}a$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{6}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}+\frac{1}{3}a^{7}+\frac{1}{3}a^{3}$, $\frac{1}{15}a^{12}-\frac{4}{15}$, $\frac{1}{15}a^{13}-\frac{4}{15}a$, $\frac{1}{15}a^{14}-\frac{4}{15}a^{2}$, $\frac{1}{15}a^{15}-\frac{4}{15}a^{3}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -\frac{3}{5} a^{14} + \frac{7}{3} a^{10} - \frac{26}{3} a^{6} + \frac{11}{15} a^{2} \) (order $24$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
Fundamental units: | $\frac{2}{15}a^{15}-\frac{2}{3}a^{11}+\frac{7}{3}a^{7}-\frac{11}{5}a^{3}$, $\frac{1}{15}a^{15}+\frac{1}{5}a^{13}-\frac{2}{3}a^{9}+\frac{7}{3}a^{5}+\frac{56}{15}a^{3}+\frac{23}{15}a$, $\frac{1}{5}a^{15}-\frac{4}{15}a^{13}-\frac{2}{3}a^{11}+a^{9}+\frac{7}{3}a^{7}-4a^{5}+\frac{23}{15}a^{3}+\frac{16}{15}a$, $\frac{2}{15}a^{14}+\frac{4}{15}a^{13}+\frac{1}{5}a^{12}-\frac{2}{3}a^{10}-a^{9}-\frac{2}{3}a^{8}+\frac{7}{3}a^{6}+4a^{5}+\frac{7}{3}a^{4}-\frac{11}{5}a^{2}-\frac{1}{15}a+\frac{8}{15}$, $\frac{2}{5}a^{14}-\frac{3}{5}a^{13}+\frac{4}{15}a^{12}-\frac{5}{3}a^{10}+\frac{7}{3}a^{9}-a^{8}+\frac{19}{3}a^{6}-\frac{26}{3}a^{5}+4a^{4}-\frac{34}{15}a^{2}+\frac{26}{15}a-\frac{1}{15}$, $\frac{1}{5}a^{14}+\frac{2}{15}a^{13}-\frac{1}{5}a^{12}-\frac{2}{3}a^{10}-\frac{2}{3}a^{9}+\frac{2}{3}a^{8}+\frac{7}{3}a^{6}+\frac{7}{3}a^{5}-\frac{7}{3}a^{4}+\frac{23}{15}a^{2}-\frac{6}{5}a-\frac{8}{15}$, $\frac{16}{15}a^{15}-\frac{2}{5}a^{14}-\frac{4}{15}a^{12}-4a^{11}+\frac{5}{3}a^{10}+a^{8}+15a^{7}-\frac{19}{3}a^{6}-4a^{4}-\frac{4}{15}a^{3}+\frac{34}{15}a^{2}+\frac{1}{15}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 4822.93617404 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 4822.93617404 \cdot 1}{24\cdot\sqrt{1846757322198614016}}\cr\approx \mathstrut & 0.359198326738 \end{aligned}\]
Galois group
$C_2\times D_4$ (as 16T9):
A solvable group of order 16 |
The 10 conjugacy class representatives for $D_4\times C_2$ |
Character table for $D_4\times C_2$ |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 8 siblings: | 8.4.1358954496.3, 8.0.1358954496.6, 8.0.1358954496.8, 8.0.150994944.2 |
Minimal sibling: | 8.0.150994944.2 |
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 | ${\href{/padicField/5.2.0.1}{2} }^{8}$ | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.16.48.4 | $x^{16} - 8 x^{14} + 8 x^{13} + 84 x^{12} + 64 x^{11} + 96 x^{10} + 192 x^{9} + 240 x^{8} + 96 x^{7} + 80 x^{6} + 400 x^{5} + 712 x^{4} + 128 x^{3} + 544 x^{2} + 480 x + 900$ | $8$ | $2$ | $48$ | $D_4\times C_2$ | $[2, 3, 4]^{2}$ |
\(3\) | 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |