Normalized defining polynomial
\( x^{16} - x^{15} + 8 x^{14} + 4 x^{13} + 8 x^{12} + 25 x^{11} + 39 x^{10} + 88 x^{8} + 39 x^{6} + \cdots + 1 \)
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: | \(1040864112987605361\) \(\medspace = 3^{12}\cdot 7^{4}\cdot 13^{8}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(13.37\) | 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: | $3^{3/4}7^{1/2}13^{1/2}\approx 21.745111415357325$ | ||
Ramified primes: | \(3\), \(7\), \(13\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}+\frac{1}{4}a^{2}+\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{8}a^{10}-\frac{1}{8}$, $\frac{1}{8}a^{11}-\frac{1}{8}a$, $\frac{1}{24}a^{12}+\frac{1}{24}a^{11}+\frac{1}{24}a^{10}+\frac{1}{12}a^{9}-\frac{1}{12}a^{8}-\frac{1}{12}a^{7}+\frac{1}{12}a^{6}-\frac{1}{12}a^{5}-\frac{1}{12}a^{4}+\frac{1}{12}a^{3}+\frac{1}{24}a^{2}-\frac{11}{24}a+\frac{1}{24}$, $\frac{1}{24}a^{13}+\frac{1}{24}a^{10}+\frac{1}{12}a^{9}-\frac{1}{4}a^{8}-\frac{1}{12}a^{7}+\frac{1}{12}a^{6}-\frac{1}{4}a^{5}-\frac{1}{12}a^{4}+\frac{5}{24}a^{3}-\frac{1}{4}a^{2}+\frac{1}{4}a+\frac{5}{24}$, $\frac{1}{96}a^{14}+\frac{1}{96}a^{13}+\frac{1}{96}a^{12}+\frac{5}{96}a^{11}+\frac{1}{96}a^{10}+\frac{5}{48}a^{9}-\frac{5}{48}a^{8}-\frac{7}{48}a^{7}+\frac{11}{48}a^{6}-\frac{11}{48}a^{5}-\frac{11}{96}a^{4}-\frac{23}{96}a^{3}-\frac{35}{96}a^{2}-\frac{9}{32}a+\frac{7}{32}$, $\frac{1}{1152}a^{15}+\frac{1}{288}a^{14}-\frac{5}{288}a^{13}-\frac{5}{144}a^{11}+\frac{17}{1152}a^{10}+\frac{19}{288}a^{9}-\frac{25}{288}a^{8}+\frac{17}{288}a^{7}-\frac{35}{288}a^{6}+\frac{11}{1152}a^{5}+\frac{35}{72}a^{4}-\frac{5}{48}a^{3}+\frac{91}{288}a^{2}+\frac{13}{288}a+\frac{547}{1152}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
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: | \( -\frac{31}{48} a^{15} + \frac{31}{48} a^{14} - \frac{241}{48} a^{13} - \frac{133}{48} a^{12} - \frac{193}{48} a^{11} - \frac{379}{24} a^{10} - \frac{197}{8} a^{9} + \frac{79}{24} a^{8} - \frac{1243}{24} a^{7} - \frac{19}{8} a^{6} - \frac{655}{48} a^{5} - \frac{785}{48} a^{4} - \frac{77}{48} a^{3} + \frac{43}{48} a^{2} - \frac{157}{48} a + \frac{13}{12} \) (order $6$) | 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: | $\frac{103}{1152}a^{15}-\frac{11}{288}a^{14}+\frac{187}{288}a^{13}+\frac{41}{48}a^{12}+\frac{25}{36}a^{11}+\frac{3551}{1152}a^{10}+\frac{1369}{288}a^{9}+\frac{365}{288}a^{8}+\frac{2363}{288}a^{7}+\frac{1783}{288}a^{6}-\frac{1219}{1152}a^{5}+\frac{937}{144}a^{4}+\frac{7}{24}a^{3}-\frac{317}{288}a^{2}+\frac{385}{288}a+\frac{445}{1152}$, $\frac{239}{576}a^{15}-\frac{173}{288}a^{14}+\frac{919}{288}a^{13}+\frac{13}{32}a^{12}+\frac{77}{288}a^{11}+\frac{4009}{576}a^{10}+\frac{685}{72}a^{9}-\frac{521}{36}a^{8}+\frac{419}{18}a^{7}-\frac{1205}{72}a^{6}-\frac{2759}{576}a^{5}-\frac{391}{288}a^{4}-\frac{223}{32}a^{3}-\frac{1421}{288}a^{2}+\frac{295}{288}a-\frac{913}{576}$, $\frac{5}{128}a^{15}-\frac{1}{48}a^{14}+\frac{11}{24}a^{13}+\frac{23}{96}a^{12}+\frac{47}{32}a^{11}+\frac{329}{128}a^{10}+\frac{275}{96}a^{9}+\frac{129}{32}a^{8}+\frac{351}{32}a^{7}+\frac{221}{96}a^{6}+\frac{1087}{128}a^{5}+\frac{197}{32}a^{4}-\frac{19}{96}a^{3}+\frac{5}{6}a^{2}+\frac{59}{48}a-\frac{77}{128}$, $\frac{851}{1152}a^{15}-\frac{139}{288}a^{14}+\frac{1643}{288}a^{13}+\frac{235}{48}a^{12}+\frac{529}{72}a^{11}+\frac{23323}{1152}a^{10}+\frac{10085}{288}a^{9}+\frac{2905}{288}a^{8}+\frac{18607}{288}a^{7}+\frac{5387}{288}a^{6}+\frac{34369}{1152}a^{5}+\frac{2915}{144}a^{4}+\frac{119}{12}a^{3}+\frac{83}{288}a^{2}+\frac{881}{288}a-\frac{367}{1152}$, $\frac{449}{576}a^{15}-\frac{53}{72}a^{14}+\frac{415}{72}a^{13}+\frac{173}{48}a^{12}+\frac{487}{144}a^{11}+\frac{9181}{576}a^{10}+\frac{3965}{144}a^{9}-\frac{1307}{144}a^{8}+\frac{6715}{144}a^{7}-\frac{589}{144}a^{6}+\frac{1267}{576}a^{5}+\frac{733}{144}a^{4}-\frac{55}{16}a^{3}-\frac{299}{72}a^{2}+\frac{139}{72}a-\frac{529}{576}$, $\frac{49}{384}a^{15}-\frac{11}{48}a^{14}+\frac{7}{8}a^{13}-\frac{1}{32}a^{12}-\frac{39}{32}a^{11}+\frac{183}{128}a^{10}+\frac{173}{96}a^{9}-\frac{289}{32}a^{8}+\frac{97}{32}a^{7}-\frac{397}{96}a^{6}-\frac{1279}{128}a^{5}+\frac{3}{32}a^{4}+\frac{19}{96}a^{3}-\frac{89}{24}a^{2}+\frac{11}{16}a-\frac{25}{384}$, $\frac{319}{288}a^{15}-\frac{263}{288}a^{14}+\frac{2473}{288}a^{13}+\frac{571}{96}a^{12}+\frac{2597}{288}a^{11}+\frac{253}{9}a^{10}+\frac{6659}{144}a^{9}+\frac{625}{144}a^{8}+\frac{13195}{144}a^{7}+\frac{1565}{144}a^{6}+\frac{9623}{288}a^{5}+\frac{8225}{288}a^{4}+\frac{695}{96}a^{3}+\frac{217}{288}a^{2}+\frac{2209}{288}a-\frac{91}{144}$ | 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: | \( 616.818979676 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 616.818979676 \cdot 1}{6\cdot\sqrt{1040864112987605361}}\cr\approx \mathstrut & 0.244764487825 \end{aligned}\]
Galois group
$C_2^2\wr C_2$ (as 16T39):
A solvable group of order 32 |
The 14 conjugacy class representatives for $C_2^2\wr C_2$ |
Character table for $C_2^2\wr C_2$ |
Intermediate fields
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 | ${\href{/padicField/2.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.8.6.2 | $x^{8} + 6 x^{5} + 6 x^{4} + 18 x^{2} + 18 x + 9$ | $4$ | $2$ | $6$ | $D_4$ | $[\ ]_{4}^{2}$ |
3.8.6.2 | $x^{8} + 6 x^{5} + 6 x^{4} + 18 x^{2} + 18 x + 9$ | $4$ | $2$ | $6$ | $D_4$ | $[\ ]_{4}^{2}$ | |
\(7\) | 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(13\) | 13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |