Normalized defining polynomial
\( x^{16} - 170 x^{14} + 12959 x^{12} - 595620 x^{10} + 19365997 x^{8} - 240726036 x^{6} + \cdots + 20632736881 \)
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: | \(155448717458114694507131268341456449334209\) \(\medspace = 47^{8}\cdot 97^{14}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(375.38\) | 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: | $47^{1/2}97^{7/8}\approx 375.3826504989875$ | ||
Ramified primes: | \(47\), \(97\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}-\frac{1}{4}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{1}{4}a^{5}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{12}a^{10}+\frac{1}{12}a^{8}-\frac{1}{12}a^{6}+\frac{1}{12}a^{4}+\frac{5}{12}a^{2}-\frac{1}{2}a-\frac{1}{3}$, $\frac{1}{12}a^{11}+\frac{1}{12}a^{9}-\frac{1}{12}a^{7}+\frac{1}{12}a^{5}+\frac{5}{12}a^{3}-\frac{1}{2}a^{2}-\frac{1}{3}a$, $\frac{1}{72}a^{12}-\frac{1}{24}a^{11}+\frac{1}{12}a^{9}+\frac{1}{72}a^{8}-\frac{1}{12}a^{7}+\frac{17}{72}a^{6}+\frac{1}{12}a^{5}-\frac{11}{72}a^{4}-\frac{1}{12}a^{3}+\frac{1}{3}a^{2}+\frac{7}{24}a+\frac{13}{72}$, $\frac{1}{27288}a^{13}-\frac{61}{3032}a^{11}+\frac{73}{27288}a^{9}-\frac{1}{8}a^{8}+\frac{4337}{27288}a^{7}-\frac{1}{8}a^{6}-\frac{3935}{27288}a^{5}-\frac{1}{8}a^{4}-\frac{206}{1137}a^{3}-\frac{1}{8}a^{2}+\frac{4979}{13644}a-\frac{1}{8}$, $\frac{1}{18\!\cdots\!52}a^{14}+\frac{21\!\cdots\!07}{46\!\cdots\!38}a^{12}-\frac{1}{24}a^{11}+\frac{16\!\cdots\!37}{18\!\cdots\!52}a^{10}-\frac{1}{24}a^{9}-\frac{13\!\cdots\!65}{46\!\cdots\!38}a^{8}-\frac{5}{24}a^{7}-\frac{48\!\cdots\!50}{23\!\cdots\!19}a^{6}-\frac{1}{24}a^{5}-\frac{52\!\cdots\!17}{18\!\cdots\!52}a^{4}+\frac{7}{24}a^{3}-\frac{57\!\cdots\!91}{11\!\cdots\!54}a^{2}+\frac{1}{6}a-\frac{29\!\cdots\!37}{12\!\cdots\!72}$, $\frac{1}{70\!\cdots\!08}a^{15}+\frac{21\!\cdots\!07}{17\!\cdots\!02}a^{13}-\frac{34\!\cdots\!52}{87\!\cdots\!01}a^{11}-\frac{1}{24}a^{10}-\frac{50\!\cdots\!75}{35\!\cdots\!04}a^{9}+\frac{1}{12}a^{8}+\frac{68\!\cdots\!51}{35\!\cdots\!04}a^{7}-\frac{1}{12}a^{6}-\frac{21\!\cdots\!19}{70\!\cdots\!08}a^{5}+\frac{1}{12}a^{4}-\frac{23\!\cdots\!53}{88\!\cdots\!32}a^{3}-\frac{1}{12}a^{2}-\frac{31\!\cdots\!29}{24\!\cdots\!44}a-\frac{5}{24}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{4}\times C_{8}\times C_{32}$, which has order $1024$ (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: | $\frac{16\!\cdots\!83}{30\!\cdots\!92}a^{14}-\frac{43\!\cdots\!07}{51\!\cdots\!82}a^{12}+\frac{19\!\cdots\!01}{30\!\cdots\!92}a^{10}-\frac{20\!\cdots\!78}{77\!\cdots\!73}a^{8}+\frac{62\!\cdots\!52}{77\!\cdots\!73}a^{6}-\frac{64\!\cdots\!05}{10\!\cdots\!64}a^{4}+\frac{13\!\cdots\!67}{38\!\cdots\!18}a^{2}-\frac{82\!\cdots\!27}{71\!\cdots\!04}$, $\frac{42\!\cdots\!69}{15\!\cdots\!84}a^{15}+\frac{13\!\cdots\!31}{14\!\cdots\!08}a^{14}-\frac{74\!\cdots\!17}{15\!\cdots\!84}a^{13}-\frac{38\!\cdots\!53}{44\!\cdots\!24}a^{12}+\frac{29\!\cdots\!45}{76\!\cdots\!42}a^{11}+\frac{54\!\cdots\!33}{14\!\cdots\!08}a^{10}-\frac{57\!\cdots\!71}{30\!\cdots\!68}a^{9}+\frac{14\!\cdots\!25}{44\!\cdots\!24}a^{8}+\frac{19\!\cdots\!61}{30\!\cdots\!68}a^{7}-\frac{93\!\cdots\!95}{44\!\cdots\!24}a^{6}-\frac{29\!\cdots\!97}{30\!\cdots\!68}a^{5}+\frac{55\!\cdots\!19}{55\!\cdots\!53}a^{4}+\frac{49\!\cdots\!67}{77\!\cdots\!44}a^{3}-\frac{20\!\cdots\!01}{18\!\cdots\!32}a^{2}-\frac{35\!\cdots\!83}{21\!\cdots\!48}a+\frac{61\!\cdots\!23}{15\!\cdots\!32}$, $\frac{42\!\cdots\!69}{15\!\cdots\!84}a^{15}-\frac{13\!\cdots\!31}{14\!\cdots\!08}a^{14}-\frac{74\!\cdots\!17}{15\!\cdots\!84}a^{13}+\frac{38\!\cdots\!53}{44\!\cdots\!24}a^{12}+\frac{29\!\cdots\!45}{76\!\cdots\!42}a^{11}-\frac{54\!\cdots\!33}{14\!\cdots\!08}a^{10}-\frac{57\!\cdots\!71}{30\!\cdots\!68}a^{9}-\frac{14\!\cdots\!25}{44\!\cdots\!24}a^{8}+\frac{19\!\cdots\!61}{30\!\cdots\!68}a^{7}+\frac{93\!\cdots\!95}{44\!\cdots\!24}a^{6}-\frac{29\!\cdots\!97}{30\!\cdots\!68}a^{5}-\frac{55\!\cdots\!19}{55\!\cdots\!53}a^{4}+\frac{49\!\cdots\!67}{77\!\cdots\!44}a^{3}+\frac{20\!\cdots\!01}{18\!\cdots\!32}a^{2}-\frac{35\!\cdots\!83}{21\!\cdots\!48}a-\frac{61\!\cdots\!23}{15\!\cdots\!32}$, $\frac{13\!\cdots\!15}{21\!\cdots\!32}a^{15}+\frac{1173609569595}{85\!\cdots\!74}a^{14}-\frac{11\!\cdots\!65}{10\!\cdots\!66}a^{13}-\frac{198970066647075}{85\!\cdots\!74}a^{12}+\frac{85\!\cdots\!75}{10\!\cdots\!66}a^{11}+\frac{30\!\cdots\!93}{17\!\cdots\!48}a^{10}-\frac{38\!\cdots\!55}{10\!\cdots\!66}a^{9}-\frac{68\!\cdots\!57}{85\!\cdots\!74}a^{8}+\frac{12\!\cdots\!55}{10\!\cdots\!66}a^{7}+\frac{21\!\cdots\!29}{85\!\cdots\!74}a^{6}-\frac{26\!\cdots\!05}{21\!\cdots\!32}a^{5}-\frac{12\!\cdots\!53}{42\!\cdots\!37}a^{4}+\frac{84\!\cdots\!60}{13\!\cdots\!39}a^{3}+\frac{14\!\cdots\!15}{10\!\cdots\!21}a^{2}-\frac{10\!\cdots\!95}{37\!\cdots\!63}a-\frac{10\!\cdots\!03}{17\!\cdots\!48}$, $\frac{66\!\cdots\!43}{38\!\cdots\!56}a^{15}+\frac{51\!\cdots\!33}{18\!\cdots\!52}a^{14}-\frac{12\!\cdots\!51}{23\!\cdots\!36}a^{13}-\frac{35\!\cdots\!47}{18\!\cdots\!52}a^{12}+\frac{27\!\cdots\!33}{77\!\cdots\!12}a^{11}+\frac{25\!\cdots\!35}{18\!\cdots\!52}a^{10}-\frac{10\!\cdots\!47}{58\!\cdots\!34}a^{9}-\frac{20\!\cdots\!65}{46\!\cdots\!38}a^{8}+\frac{14\!\cdots\!07}{29\!\cdots\!67}a^{7}+\frac{54\!\cdots\!57}{46\!\cdots\!38}a^{6}-\frac{55\!\cdots\!95}{11\!\cdots\!68}a^{5}-\frac{21\!\cdots\!69}{18\!\cdots\!52}a^{4}+\frac{47\!\cdots\!49}{19\!\cdots\!96}a^{3}+\frac{26\!\cdots\!51}{46\!\cdots\!16}a^{2}-\frac{16\!\cdots\!11}{16\!\cdots\!96}a-\frac{31\!\cdots\!23}{12\!\cdots\!72}$, $\frac{26\!\cdots\!73}{58\!\cdots\!34}a^{15}+\frac{36\!\cdots\!05}{56\!\cdots\!08}a^{14}-\frac{22\!\cdots\!87}{29\!\cdots\!67}a^{13}-\frac{56\!\cdots\!69}{50\!\cdots\!72}a^{12}+\frac{66\!\cdots\!95}{11\!\cdots\!68}a^{11}+\frac{12\!\cdots\!26}{14\!\cdots\!77}a^{10}-\frac{50\!\cdots\!95}{19\!\cdots\!78}a^{9}-\frac{20\!\cdots\!35}{50\!\cdots\!72}a^{8}+\frac{53\!\cdots\!95}{64\!\cdots\!26}a^{7}+\frac{65\!\cdots\!51}{50\!\cdots\!72}a^{6}-\frac{54\!\cdots\!15}{58\!\cdots\!34}a^{5}-\frac{22\!\cdots\!37}{12\!\cdots\!93}a^{4}+\frac{64\!\cdots\!47}{14\!\cdots\!22}a^{3}+\frac{16\!\cdots\!61}{21\!\cdots\!46}a^{2}-\frac{18\!\cdots\!39}{81\!\cdots\!48}a-\frac{16\!\cdots\!01}{35\!\cdots\!92}$, $\frac{27\!\cdots\!63}{12\!\cdots\!52}a^{15}-\frac{10\!\cdots\!55}{16\!\cdots\!24}a^{14}-\frac{18\!\cdots\!29}{58\!\cdots\!34}a^{13}+\frac{47\!\cdots\!75}{50\!\cdots\!72}a^{12}+\frac{82\!\cdots\!77}{38\!\cdots\!56}a^{11}-\frac{26\!\cdots\!33}{42\!\cdots\!31}a^{10}-\frac{96\!\cdots\!99}{11\!\cdots\!68}a^{9}+\frac{12\!\cdots\!65}{50\!\cdots\!72}a^{8}+\frac{26\!\cdots\!99}{11\!\cdots\!68}a^{7}-\frac{32\!\cdots\!51}{50\!\cdots\!72}a^{6}+\frac{15\!\cdots\!83}{58\!\cdots\!34}a^{5}-\frac{21\!\cdots\!39}{12\!\cdots\!93}a^{4}+\frac{10\!\cdots\!59}{98\!\cdots\!48}a^{3}-\frac{13\!\cdots\!19}{21\!\cdots\!46}a^{2}+\frac{13\!\cdots\!27}{20\!\cdots\!87}a-\frac{25\!\cdots\!01}{35\!\cdots\!92}$ (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: | \( 117099354561 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 117099354561 \cdot 1024}{2\cdot\sqrt{155448717458114694507131268341456449334209}}\cr\approx \mathstrut & 0.369376989623616 \end{aligned}\] (assuming GRH)
Galois group
$\OD_{16}:C_2$ (as 16T41):
A solvable group of order 32 |
The 11 conjugacy class representatives for $\OD_{16}:C_2$ |
Character table for $\OD_{16}:C_2$ |
Intermediate fields
\(\Q(\sqrt{97}) \), 4.0.2016094657.2, 4.2.442223.2, 4.2.42895631.1, 8.0.394269853600442921953.3 x2, 8.0.4064637665983947649.4 |
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 | ${\href{/padicField/2.4.0.1}{4} }^{4}$ | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{6}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(47\) | 47.4.2.1 | $x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
47.4.2.1 | $x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
47.4.2.1 | $x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
47.4.2.1 | $x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(97\) | 97.8.7.1 | $x^{8} + 97$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |
97.8.7.1 | $x^{8} + 97$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |