Normalized defining polynomial
\( x^{16} - 2 x^{15} + 12 x^{14} - 22 x^{13} + 865 x^{12} + 68 x^{11} + 5879 x^{10} + 75224 x^{9} - 163742 x^{8} + 335966 x^{7} + 4327649 x^{6} - 4700058 x^{5} + \cdots + 66232832 \)
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: | \(3922880935919264967742950184849\) \(\medspace = 13^{12}\cdot 17^{14}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(81.68\) | 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: | $13^{3/4}17^{7/8}\approx 81.67710206860912$ | ||
Ramified primes: | \(13\), \(17\) | 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 a CM field. | |||
Reflex fields: | unavailable$^{128}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{9}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{4}+\frac{1}{8}a^{3}-\frac{1}{2}a$, $\frac{1}{8}a^{10}-\frac{1}{4}a^{7}-\frac{1}{4}a^{5}+\frac{1}{8}a^{4}+\frac{1}{4}a^{2}$, $\frac{1}{32}a^{11}+\frac{1}{32}a^{10}-\frac{1}{16}a^{9}+\frac{1}{16}a^{7}+\frac{1}{16}a^{6}+\frac{7}{32}a^{5}+\frac{5}{32}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}+\frac{1}{4}a$, $\frac{1}{64}a^{12}-\frac{1}{64}a^{11}-\frac{1}{16}a^{9}-\frac{3}{32}a^{8}-\frac{5}{32}a^{7}-\frac{13}{64}a^{6}+\frac{15}{64}a^{5}+\frac{1}{32}a^{4}-\frac{1}{8}a^{3}+\frac{3}{8}a^{2}+\frac{1}{4}a$, $\frac{1}{64}a^{13}-\frac{1}{64}a^{11}-\frac{1}{16}a^{10}-\frac{1}{32}a^{9}-\frac{7}{64}a^{7}-\frac{7}{32}a^{6}-\frac{15}{64}a^{5}+\frac{5}{32}a^{4}+\frac{3}{8}a^{3}-\frac{1}{8}a^{2}-\frac{1}{4}a$, $\frac{1}{512}a^{14}+\frac{3}{512}a^{12}-\frac{1}{64}a^{11}+\frac{3}{256}a^{10}+\frac{1}{32}a^{9}-\frac{47}{512}a^{8}-\frac{35}{256}a^{7}-\frac{99}{512}a^{6}+\frac{43}{256}a^{5}+\frac{5}{64}a^{4}+\frac{3}{64}a^{3}-\frac{11}{32}a^{2}+\frac{3}{8}a$, $\frac{1}{25\!\cdots\!52}a^{15}-\frac{84\!\cdots\!09}{25\!\cdots\!52}a^{14}+\frac{14\!\cdots\!99}{25\!\cdots\!52}a^{13}+\frac{17\!\cdots\!57}{25\!\cdots\!52}a^{12}+\frac{19\!\cdots\!67}{12\!\cdots\!76}a^{11}-\frac{24\!\cdots\!59}{12\!\cdots\!76}a^{10}-\frac{14\!\cdots\!23}{25\!\cdots\!52}a^{9}-\frac{28\!\cdots\!59}{25\!\cdots\!52}a^{8}+\frac{23\!\cdots\!27}{25\!\cdots\!52}a^{7}-\frac{44\!\cdots\!03}{25\!\cdots\!52}a^{6}-\frac{76\!\cdots\!79}{12\!\cdots\!76}a^{5}+\frac{26\!\cdots\!49}{16\!\cdots\!72}a^{4}+\frac{10\!\cdots\!11}{32\!\cdots\!44}a^{3}-\frac{31\!\cdots\!61}{16\!\cdots\!72}a^{2}-\frac{49\!\cdots\!31}{40\!\cdots\!68}a-\frac{10\!\cdots\!17}{50\!\cdots\!96}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{2}\times C_{12}\times C_{408}$, which has order $9792$ (assuming GRH)
Unit group
Rank: | $7$ | 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: | $\frac{12\!\cdots\!77}{30\!\cdots\!32}a^{15}-\frac{92\!\cdots\!47}{30\!\cdots\!32}a^{14}+\frac{41\!\cdots\!39}{30\!\cdots\!32}a^{13}-\frac{14\!\cdots\!73}{30\!\cdots\!32}a^{12}+\frac{72\!\cdots\!51}{15\!\cdots\!16}a^{11}-\frac{30\!\cdots\!89}{15\!\cdots\!16}a^{10}+\frac{18\!\cdots\!49}{30\!\cdots\!32}a^{9}+\frac{47\!\cdots\!91}{30\!\cdots\!32}a^{8}-\frac{63\!\cdots\!09}{30\!\cdots\!32}a^{7}+\frac{24\!\cdots\!79}{30\!\cdots\!32}a^{6}+\frac{55\!\cdots\!95}{15\!\cdots\!16}a^{5}-\frac{35\!\cdots\!55}{37\!\cdots\!04}a^{4}+\frac{49\!\cdots\!77}{37\!\cdots\!04}a^{3}+\frac{41\!\cdots\!31}{18\!\cdots\!52}a^{2}-\frac{11\!\cdots\!67}{47\!\cdots\!63}a-\frac{15\!\cdots\!47}{47\!\cdots\!63}$, $\frac{30\!\cdots\!27}{10\!\cdots\!32}a^{15}-\frac{87\!\cdots\!83}{10\!\cdots\!32}a^{14}+\frac{40\!\cdots\!21}{10\!\cdots\!32}a^{13}-\frac{87\!\cdots\!33}{10\!\cdots\!32}a^{12}+\frac{13\!\cdots\!73}{54\!\cdots\!16}a^{11}-\frac{98\!\cdots\!85}{54\!\cdots\!16}a^{10}+\frac{16\!\cdots\!19}{10\!\cdots\!32}a^{9}+\frac{21\!\cdots\!83}{10\!\cdots\!32}a^{8}-\frac{71\!\cdots\!47}{10\!\cdots\!32}a^{7}+\frac{14\!\cdots\!11}{10\!\cdots\!32}a^{6}+\frac{64\!\cdots\!67}{54\!\cdots\!16}a^{5}-\frac{18\!\cdots\!69}{67\!\cdots\!52}a^{4}-\frac{10\!\cdots\!27}{13\!\cdots\!04}a^{3}+\frac{34\!\cdots\!53}{67\!\cdots\!52}a^{2}+\frac{32\!\cdots\!55}{16\!\cdots\!88}a+\frac{22\!\cdots\!53}{21\!\cdots\!36}$, $\frac{81\!\cdots\!75}{64\!\cdots\!88}a^{15}+\frac{84\!\cdots\!45}{64\!\cdots\!88}a^{14}+\frac{95\!\cdots\!85}{64\!\cdots\!88}a^{13}+\frac{16\!\cdots\!51}{64\!\cdots\!88}a^{12}+\frac{35\!\cdots\!33}{32\!\cdots\!44}a^{11}+\frac{11\!\cdots\!87}{32\!\cdots\!44}a^{10}+\frac{88\!\cdots\!51}{64\!\cdots\!88}a^{9}+\frac{90\!\cdots\!71}{64\!\cdots\!88}a^{8}+\frac{97\!\cdots\!25}{64\!\cdots\!88}a^{7}+\frac{39\!\cdots\!99}{64\!\cdots\!88}a^{6}+\frac{24\!\cdots\!03}{32\!\cdots\!44}a^{5}+\frac{55\!\cdots\!83}{40\!\cdots\!68}a^{4}-\frac{13\!\cdots\!67}{80\!\cdots\!36}a^{3}-\frac{35\!\cdots\!43}{40\!\cdots\!68}a^{2}-\frac{10\!\cdots\!53}{10\!\cdots\!92}a-\frac{59\!\cdots\!15}{12\!\cdots\!24}$, $\frac{14\!\cdots\!57}{12\!\cdots\!76}a^{15}-\frac{33\!\cdots\!05}{12\!\cdots\!76}a^{14}+\frac{23\!\cdots\!83}{12\!\cdots\!76}a^{13}-\frac{72\!\cdots\!07}{12\!\cdots\!76}a^{12}+\frac{64\!\cdots\!71}{64\!\cdots\!88}a^{11}-\frac{55\!\cdots\!51}{64\!\cdots\!88}a^{10}+\frac{14\!\cdots\!29}{12\!\cdots\!76}a^{9}+\frac{84\!\cdots\!93}{12\!\cdots\!76}a^{8}-\frac{25\!\cdots\!93}{12\!\cdots\!76}a^{7}+\frac{44\!\cdots\!85}{12\!\cdots\!76}a^{6}+\frac{16\!\cdots\!97}{64\!\cdots\!88}a^{5}-\frac{61\!\cdots\!59}{80\!\cdots\!36}a^{4}-\frac{31\!\cdots\!93}{16\!\cdots\!72}a^{3}+\frac{97\!\cdots\!71}{80\!\cdots\!36}a^{2}+\frac{12\!\cdots\!81}{20\!\cdots\!84}a+\frac{99\!\cdots\!15}{25\!\cdots\!48}$, $\frac{21\!\cdots\!39}{64\!\cdots\!88}a^{15}-\frac{65\!\cdots\!43}{64\!\cdots\!88}a^{14}+\frac{32\!\cdots\!29}{64\!\cdots\!88}a^{13}-\frac{78\!\cdots\!85}{64\!\cdots\!88}a^{12}+\frac{97\!\cdots\!17}{32\!\cdots\!44}a^{11}-\frac{88\!\cdots\!33}{32\!\cdots\!44}a^{10}+\frac{14\!\cdots\!23}{64\!\cdots\!88}a^{9}+\frac{15\!\cdots\!99}{64\!\cdots\!88}a^{8}-\frac{50\!\cdots\!27}{64\!\cdots\!88}a^{7}+\frac{12\!\cdots\!75}{64\!\cdots\!88}a^{6}+\frac{41\!\cdots\!39}{32\!\cdots\!44}a^{5}-\frac{11\!\cdots\!01}{40\!\cdots\!68}a^{4}-\frac{70\!\cdots\!59}{80\!\cdots\!36}a^{3}+\frac{56\!\cdots\!29}{40\!\cdots\!68}a^{2}+\frac{26\!\cdots\!31}{10\!\cdots\!92}a+\frac{25\!\cdots\!65}{12\!\cdots\!24}$, $\frac{92\!\cdots\!25}{84\!\cdots\!88}a^{15}-\frac{27\!\cdots\!37}{84\!\cdots\!88}a^{14}+\frac{22\!\cdots\!23}{84\!\cdots\!88}a^{13}-\frac{77\!\cdots\!59}{84\!\cdots\!88}a^{12}+\frac{32\!\cdots\!15}{42\!\cdots\!44}a^{11}-\frac{14\!\cdots\!99}{42\!\cdots\!44}a^{10}+\frac{55\!\cdots\!29}{84\!\cdots\!88}a^{9}+\frac{30\!\cdots\!85}{84\!\cdots\!88}a^{8}-\frac{31\!\cdots\!25}{84\!\cdots\!88}a^{7}+\frac{70\!\cdots\!45}{84\!\cdots\!88}a^{6}+\frac{29\!\cdots\!57}{42\!\cdots\!44}a^{5}-\frac{55\!\cdots\!71}{52\!\cdots\!68}a^{4}+\frac{20\!\cdots\!75}{10\!\cdots\!36}a^{3}+\frac{12\!\cdots\!47}{52\!\cdots\!68}a^{2}-\frac{62\!\cdots\!91}{13\!\cdots\!92}a-\frac{54\!\cdots\!85}{16\!\cdots\!24}$, $\frac{19\!\cdots\!25}{84\!\cdots\!88}a^{15}-\frac{17\!\cdots\!01}{84\!\cdots\!88}a^{14}+\frac{73\!\cdots\!43}{84\!\cdots\!88}a^{13}-\frac{28\!\cdots\!87}{84\!\cdots\!88}a^{12}+\frac{11\!\cdots\!07}{42\!\cdots\!44}a^{11}-\frac{61\!\cdots\!23}{42\!\cdots\!44}a^{10}+\frac{32\!\cdots\!13}{84\!\cdots\!88}a^{9}+\frac{42\!\cdots\!65}{84\!\cdots\!88}a^{8}-\frac{11\!\cdots\!73}{84\!\cdots\!88}a^{7}+\frac{43\!\cdots\!57}{84\!\cdots\!88}a^{6}-\frac{10\!\cdots\!11}{42\!\cdots\!44}a^{5}-\frac{31\!\cdots\!59}{52\!\cdots\!68}a^{4}+\frac{95\!\cdots\!59}{10\!\cdots\!36}a^{3}+\frac{10\!\cdots\!71}{52\!\cdots\!68}a^{2}-\frac{62\!\cdots\!71}{13\!\cdots\!92}a-\frac{10\!\cdots\!45}{16\!\cdots\!24}$ (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: | \( 32676067.5694 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 32676067.5694 \cdot 9792}{2\cdot\sqrt{3922880935919264967742950184849}}\cr\approx \mathstrut & 196.203873904 \end{aligned}\] (assuming GRH)
Galois group
$C_2^2:C_8$ (as 16T24):
A solvable group of order 32 |
The 20 conjugacy class representatives for $C_2^2 : C_8$ |
Character table for $C_2^2 : C_8$ |
Intermediate fields
\(\Q(\sqrt{17}) \), 4.0.3757.1, 4.4.830297.1, 4.0.63869.1, 8.0.1980626399884457.1, 8.8.1980626399884457.1, 8.0.689393108209.2 |
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 sibling: | 16.0.3922880935919264967742950184849.8 |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.2.0.1}{2} }^{4}{,}\,{\href{/padicField/2.1.0.1}{1} }^{8}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | R | R | ${\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{8}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
\(13\) | 13.8.6.1 | $x^{8} + 48 x^{7} + 872 x^{6} + 7200 x^{5} + 24242 x^{4} + 15024 x^{3} + 14408 x^{2} + 86496 x + 254881$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
13.8.6.1 | $x^{8} + 48 x^{7} + 872 x^{6} + 7200 x^{5} + 24242 x^{4} + 15024 x^{3} + 14408 x^{2} + 86496 x + 254881$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
\(17\) | 17.8.7.1 | $x^{8} + 68$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |
17.8.7.1 | $x^{8} + 68$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |