Normalized defining polynomial
\( x^{16} - 36x^{12} + 336x^{8} - 504x^{4} + 36 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[8, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(1378596953991976568487936\) \(\medspace = 2^{58}\cdot 3^{14}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(32.26\) | 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: | $2^{29/8}3^{7/8}\approx 32.263749133641326$ | ||
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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $8$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{24}a^{8}-\frac{1}{2}a^{6}+\frac{1}{4}a^{4}+\frac{1}{4}$, $\frac{1}{24}a^{9}-\frac{1}{2}a^{7}+\frac{1}{4}a^{5}+\frac{1}{4}a$, $\frac{1}{24}a^{10}+\frac{1}{4}a^{6}+\frac{1}{4}a^{2}$, $\frac{1}{24}a^{11}+\frac{1}{4}a^{7}+\frac{1}{4}a^{3}$, $\frac{1}{1200}a^{12}-\frac{7}{600}a^{8}-\frac{1}{2}a^{7}-\frac{87}{200}a^{4}-\frac{1}{2}a^{2}+\frac{13}{50}$, $\frac{1}{1200}a^{13}-\frac{7}{600}a^{9}-\frac{87}{200}a^{5}-\frac{1}{2}a^{3}+\frac{13}{50}a$, $\frac{1}{3600}a^{14}+\frac{1}{100}a^{10}+\frac{163}{600}a^{6}-\frac{1}{2}a^{4}+\frac{17}{100}a^{2}$, $\frac{1}{3600}a^{15}+\frac{1}{100}a^{11}+\frac{163}{600}a^{7}-\frac{1}{2}a^{5}+\frac{17}{100}a^{3}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -1 \) (order $2$) | 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: | $\frac{13}{900}a^{14}-\frac{313}{600}a^{10}+\frac{1463}{300}a^{6}-\frac{741}{100}a^{2}$, $\frac{11}{3600}a^{14}+\frac{1}{300}a^{12}-\frac{59}{600}a^{10}-\frac{13}{100}a^{8}+\frac{443}{600}a^{6}+\frac{63}{50}a^{4}-\frac{19}{50}a^{2}-\frac{23}{50}$, $\frac{11}{3600}a^{14}-\frac{1}{300}a^{12}-\frac{59}{600}a^{10}+\frac{13}{100}a^{8}+\frac{443}{600}a^{6}-\frac{63}{50}a^{4}-\frac{19}{50}a^{2}+\frac{23}{50}$, $\frac{1}{72}a^{14}-\frac{1}{2}a^{10}+\frac{55}{12}a^{6}-\frac{11}{2}a^{2}-1$, $\frac{1}{400}a^{14}-\frac{1}{300}a^{12}-\frac{23}{300}a^{10}+\frac{13}{100}a^{8}+\frac{89}{200}a^{6}-\frac{63}{50}a^{4}+\frac{153}{100}a^{2}+\frac{73}{50}$, $\frac{23}{720}a^{14}-\frac{1}{300}a^{12}-\frac{137}{120}a^{10}+\frac{13}{100}a^{8}+\frac{1259}{120}a^{6}-\frac{63}{50}a^{4}-\frac{71}{5}a^{2}+\frac{123}{50}$, $\frac{1}{720}a^{14}-\frac{1}{120}a^{12}-\frac{1}{30}a^{10}+\frac{17}{60}a^{8}-\frac{17}{120}a^{6}-\frac{43}{20}a^{4}+\frac{87}{20}a^{2}-\frac{8}{5}$, $\frac{3}{200}a^{15}+\frac{1}{400}a^{14}-\frac{1}{400}a^{13}+\frac{1}{80}a^{12}-\frac{163}{300}a^{11}-\frac{23}{300}a^{10}+\frac{23}{300}a^{9}-\frac{17}{40}a^{8}+\frac{517}{100}a^{7}+\frac{89}{200}a^{6}-\frac{89}{200}a^{5}+\frac{139}{40}a^{4}-\frac{441}{50}a^{3}+\frac{103}{100}a^{2}-\frac{153}{100}a-\frac{13}{5}$, $\frac{3}{400}a^{15}-\frac{1}{1800}a^{14}+\frac{3}{400}a^{13}-\frac{1}{240}a^{12}-\frac{163}{600}a^{11}+\frac{13}{600}a^{10}-\frac{163}{600}a^{9}+\frac{17}{120}a^{8}+\frac{517}{200}a^{7}-\frac{22}{75}a^{6}+\frac{517}{200}a^{5}-\frac{53}{40}a^{4}-\frac{104}{25}a^{3}+\frac{141}{100}a^{2}-\frac{183}{50}a+\frac{11}{5}$, $\frac{43}{1800}a^{15}+\frac{17}{1800}a^{14}+\frac{1}{75}a^{13}+\frac{1}{150}a^{12}-\frac{509}{600}a^{11}-\frac{49}{150}a^{10}-\frac{287}{600}a^{9}-\frac{131}{600}a^{8}+\frac{571}{75}a^{7}+\frac{821}{300}a^{6}+\frac{429}{100}a^{5}+\frac{177}{100}a^{4}-\frac{863}{100}a^{3}-\frac{61}{50}a^{2}-\frac{459}{100}a-\frac{67}{100}$, $\frac{19}{1800}a^{15}+\frac{19}{1800}a^{14}+\frac{1}{150}a^{13}+\frac{1}{200}a^{12}-\frac{37}{100}a^{11}-\frac{37}{100}a^{10}-\frac{131}{600}a^{9}-\frac{67}{600}a^{8}+\frac{997}{300}a^{7}+\frac{997}{300}a^{6}+\frac{177}{100}a^{5}+\frac{7}{50}a^{4}-\frac{126}{25}a^{3}-\frac{126}{25}a^{2}-\frac{267}{100}a-\frac{69}{100}$ (assuming GRH) | 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: | \( 1533253.77477 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 1533253.77477 \cdot 1}{2\cdot\sqrt{1378596953991976568487936}}\cr\approx \mathstrut & 0.260510257612 \end{aligned}\] (assuming GRH)
Galois group
$Q_{16}:C_2$ (as 16T50):
A solvable group of order 32 |
The 11 conjugacy class representatives for $Q_{16}:C_2$ |
Character table for $Q_{16}:C_2$ |
Intermediate fields
\(\Q(\sqrt{6}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}) \), 4.4.27648.1 x2, 4.4.13824.1 x2, \(\Q(\sqrt{2}, \sqrt{3})\), 8.8.3057647616.1 |
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 sibling: | 16.0.1378596953991976568487936.72 |
Minimal sibling: | 16.0.1378596953991976568487936.72 |
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 | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.16.58.68 | $x^{16} + 4 x^{14} + 8 x^{13} + 8 x^{12} + 8 x^{11} + 8 x^{10} + 10 x^{8} + 20 x^{4} + 14$ | $16$ | $1$ | $58$ | 16T50 | $[2, 3, 7/2, 9/2]^{2}$ |
\(3\) | 3.16.14.1 | $x^{16} + 16 x^{15} + 128 x^{14} + 672 x^{13} + 2576 x^{12} + 7616 x^{11} + 17920 x^{10} + 34182 x^{9} + 53410 x^{8} + 68544 x^{7} + 71344 x^{6} + 57904 x^{5} + 34832 x^{4} + 16128 x^{3} + 7241 x^{2} + 2966 x + 634$ | $8$ | $2$ | $14$ | $QD_{16}$ | $[\ ]_{8}^{2}$ |