Normalized defining polynomial
\( x^{16} - 4 x^{15} + 49 x^{14} - 154 x^{13} + 848 x^{12} - 1754 x^{11} + 6979 x^{10} + 35819 x^{9} + 284435 x^{8} - 556257 x^{7} + 7200689 x^{6} - 31438382 x^{5} + \cdots + 31088575 \)
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: | \(670188096064724500217499812793861721\) \(\medspace = 29^{14}\cdot 83^{8}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(173.44\) | 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: | $29^{7/8}83^{1/2}\approx 173.4352195192488$ | ||
Ramified primes: | \(29\), \(83\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7}a^{7}-\frac{1}{7}a$, $\frac{1}{7}a^{8}-\frac{1}{7}a^{2}$, $\frac{1}{7}a^{9}-\frac{1}{7}a^{3}$, $\frac{1}{7}a^{10}-\frac{1}{7}a^{4}$, $\frac{1}{7}a^{11}-\frac{1}{7}a^{5}$, $\frac{1}{49}a^{12}+\frac{3}{49}a^{11}+\frac{2}{49}a^{10}-\frac{1}{49}a^{9}-\frac{3}{49}a^{8}-\frac{2}{49}a^{7}+\frac{6}{49}a^{6}-\frac{24}{49}a^{5}-\frac{16}{49}a^{4}+\frac{8}{49}a^{3}+\frac{24}{49}a^{2}+\frac{16}{49}a-\frac{2}{7}$, $\frac{1}{245}a^{13}+\frac{2}{245}a^{12}-\frac{1}{245}a^{11}+\frac{11}{245}a^{10}-\frac{9}{245}a^{9}+\frac{3}{49}a^{8}-\frac{13}{245}a^{7}-\frac{79}{245}a^{6}+\frac{57}{245}a^{5}+\frac{108}{245}a^{4}+\frac{23}{245}a^{3}-\frac{24}{49}a^{2}-\frac{58}{245}a-\frac{1}{7}$, $\frac{1}{121755902708755}a^{14}-\frac{119615023182}{121755902708755}a^{13}+\frac{1119061807081}{121755902708755}a^{12}-\frac{1493886611362}{24351180541751}a^{11}+\frac{13493425973}{239206095695}a^{10}+\frac{3284043991066}{121755902708755}a^{9}-\frac{4577724990648}{121755902708755}a^{8}-\frac{5165636889477}{121755902708755}a^{7}-\frac{60843182233072}{121755902708755}a^{6}+\frac{4720472585841}{24351180541751}a^{5}+\frac{43520087800236}{121755902708755}a^{4}-\frac{51642933771197}{121755902708755}a^{3}-\frac{41037158909078}{121755902708755}a^{2}-\frac{38839989511178}{121755902708755}a-\frac{21073538876}{58961696227}$, $\frac{1}{91\!\cdots\!15}a^{15}+\frac{28\!\cdots\!16}{91\!\cdots\!15}a^{14}+\frac{13\!\cdots\!27}{91\!\cdots\!15}a^{13}+\frac{23\!\cdots\!22}{91\!\cdots\!15}a^{12}-\frac{26\!\cdots\!05}{18\!\cdots\!03}a^{11}+\frac{45\!\cdots\!14}{91\!\cdots\!15}a^{10}-\frac{10\!\cdots\!38}{91\!\cdots\!15}a^{9}+\frac{12\!\cdots\!64}{91\!\cdots\!15}a^{8}+\frac{16\!\cdots\!73}{13\!\cdots\!45}a^{7}-\frac{98\!\cdots\!94}{91\!\cdots\!15}a^{6}+\frac{31\!\cdots\!12}{79\!\cdots\!61}a^{5}+\frac{10\!\cdots\!97}{91\!\cdots\!15}a^{4}-\frac{37\!\cdots\!68}{91\!\cdots\!15}a^{3}+\frac{36\!\cdots\!38}{91\!\cdots\!15}a^{2}+\frac{21\!\cdots\!16}{18\!\cdots\!03}a+\frac{25\!\cdots\!08}{44\!\cdots\!31}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $7$ |
Class group and class number
$C_{30}$, which has order $30$ (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: | $\frac{18\!\cdots\!76}{12\!\cdots\!55}a^{15}-\frac{93\!\cdots\!76}{12\!\cdots\!55}a^{14}+\frac{98\!\cdots\!13}{12\!\cdots\!55}a^{13}-\frac{37\!\cdots\!06}{12\!\cdots\!55}a^{12}+\frac{18\!\cdots\!83}{12\!\cdots\!55}a^{11}-\frac{48\!\cdots\!18}{12\!\cdots\!55}a^{10}+\frac{16\!\cdots\!42}{12\!\cdots\!55}a^{9}+\frac{10\!\cdots\!52}{24\!\cdots\!71}a^{8}+\frac{45\!\cdots\!34}{12\!\cdots\!55}a^{7}-\frac{26\!\cdots\!07}{20\!\cdots\!45}a^{6}+\frac{14\!\cdots\!39}{12\!\cdots\!55}a^{5}-\frac{12\!\cdots\!41}{20\!\cdots\!45}a^{4}+\frac{43\!\cdots\!07}{12\!\cdots\!55}a^{3}+\frac{26\!\cdots\!41}{20\!\cdots\!45}a^{2}-\frac{26\!\cdots\!07}{34\!\cdots\!53}a+\frac{95\!\cdots\!49}{83\!\cdots\!81}$, $\frac{20\!\cdots\!84}{18\!\cdots\!03}a^{15}-\frac{44\!\cdots\!01}{91\!\cdots\!15}a^{14}+\frac{17\!\cdots\!69}{91\!\cdots\!15}a^{13}-\frac{14\!\cdots\!62}{91\!\cdots\!15}a^{12}+\frac{40\!\cdots\!93}{91\!\cdots\!15}a^{11}+\frac{17\!\cdots\!31}{18\!\cdots\!03}a^{10}+\frac{42\!\cdots\!21}{91\!\cdots\!15}a^{9}-\frac{40\!\cdots\!17}{91\!\cdots\!15}a^{8}-\frac{92\!\cdots\!31}{18\!\cdots\!35}a^{7}-\frac{15\!\cdots\!11}{91\!\cdots\!15}a^{6}+\frac{17\!\cdots\!38}{39\!\cdots\!05}a^{5}+\frac{68\!\cdots\!03}{18\!\cdots\!03}a^{4}-\frac{78\!\cdots\!72}{91\!\cdots\!15}a^{3}-\frac{14\!\cdots\!07}{91\!\cdots\!15}a^{2}+\frac{85\!\cdots\!67}{91\!\cdots\!15}a-\frac{18\!\cdots\!98}{44\!\cdots\!31}$, $\frac{19\!\cdots\!06}{18\!\cdots\!03}a^{15}-\frac{33\!\cdots\!99}{91\!\cdots\!15}a^{14}+\frac{22\!\cdots\!46}{91\!\cdots\!15}a^{13}-\frac{54\!\cdots\!63}{91\!\cdots\!15}a^{12}+\frac{24\!\cdots\!82}{91\!\cdots\!15}a^{11}-\frac{42\!\cdots\!65}{18\!\cdots\!03}a^{10}+\frac{47\!\cdots\!64}{91\!\cdots\!15}a^{9}+\frac{41\!\cdots\!62}{91\!\cdots\!15}a^{8}+\frac{41\!\cdots\!07}{13\!\cdots\!45}a^{7}-\frac{13\!\cdots\!34}{91\!\cdots\!15}a^{6}-\frac{31\!\cdots\!48}{39\!\cdots\!05}a^{5}+\frac{64\!\cdots\!45}{18\!\cdots\!03}a^{4}+\frac{45\!\cdots\!72}{91\!\cdots\!15}a^{3}-\frac{21\!\cdots\!83}{91\!\cdots\!15}a^{2}+\frac{97\!\cdots\!23}{91\!\cdots\!15}a+\frac{67\!\cdots\!26}{44\!\cdots\!31}$, $\frac{14\!\cdots\!59}{83\!\cdots\!65}a^{15}-\frac{14\!\cdots\!29}{83\!\cdots\!65}a^{14}+\frac{96\!\cdots\!01}{83\!\cdots\!65}a^{13}-\frac{61\!\cdots\!36}{83\!\cdots\!65}a^{12}+\frac{21\!\cdots\!08}{83\!\cdots\!65}a^{11}-\frac{88\!\cdots\!63}{83\!\cdots\!65}a^{10}+\frac{19\!\cdots\!87}{83\!\cdots\!65}a^{9}-\frac{22\!\cdots\!96}{16\!\cdots\!73}a^{8}+\frac{68\!\cdots\!24}{83\!\cdots\!65}a^{7}-\frac{59\!\cdots\!32}{14\!\cdots\!35}a^{6}+\frac{12\!\cdots\!49}{83\!\cdots\!65}a^{5}-\frac{17\!\cdots\!96}{14\!\cdots\!35}a^{4}+\frac{40\!\cdots\!53}{16\!\cdots\!73}a^{3}+\frac{71\!\cdots\!64}{14\!\cdots\!35}a^{2}-\frac{26\!\cdots\!11}{11\!\cdots\!95}a-\frac{20\!\cdots\!25}{82\!\cdots\!29}$, $\frac{65\!\cdots\!57}{91\!\cdots\!15}a^{15}-\frac{36\!\cdots\!00}{18\!\cdots\!03}a^{14}+\frac{28\!\cdots\!22}{91\!\cdots\!15}a^{13}-\frac{12\!\cdots\!51}{18\!\cdots\!03}a^{12}+\frac{43\!\cdots\!21}{91\!\cdots\!15}a^{11}-\frac{53\!\cdots\!37}{91\!\cdots\!15}a^{10}+\frac{32\!\cdots\!91}{91\!\cdots\!15}a^{9}+\frac{28\!\cdots\!74}{91\!\cdots\!15}a^{8}+\frac{21\!\cdots\!54}{91\!\cdots\!15}a^{7}-\frac{23\!\cdots\!44}{18\!\cdots\!03}a^{6}+\frac{18\!\cdots\!96}{39\!\cdots\!05}a^{5}-\frac{14\!\cdots\!76}{91\!\cdots\!15}a^{4}-\frac{61\!\cdots\!20}{18\!\cdots\!03}a^{3}+\frac{31\!\cdots\!62}{91\!\cdots\!15}a^{2}+\frac{12\!\cdots\!69}{91\!\cdots\!15}a+\frac{33\!\cdots\!32}{44\!\cdots\!31}$, $\frac{72\!\cdots\!39}{59\!\cdots\!55}a^{15}-\frac{59\!\cdots\!73}{83\!\cdots\!37}a^{14}+\frac{20\!\cdots\!96}{41\!\cdots\!85}a^{13}-\frac{19\!\cdots\!89}{41\!\cdots\!85}a^{12}+\frac{11\!\cdots\!06}{41\!\cdots\!85}a^{11}-\frac{10\!\cdots\!73}{83\!\cdots\!37}a^{10}+\frac{26\!\cdots\!47}{41\!\cdots\!85}a^{9}-\frac{43\!\cdots\!84}{41\!\cdots\!85}a^{8}+\frac{13\!\cdots\!42}{41\!\cdots\!85}a^{7}+\frac{18\!\cdots\!78}{41\!\cdots\!85}a^{6}-\frac{24\!\cdots\!12}{41\!\cdots\!85}a^{5}-\frac{61\!\cdots\!10}{83\!\cdots\!37}a^{4}+\frac{68\!\cdots\!99}{41\!\cdots\!85}a^{3}+\frac{63\!\cdots\!08}{41\!\cdots\!85}a^{2}-\frac{40\!\cdots\!63}{41\!\cdots\!85}a+\frac{10\!\cdots\!13}{20\!\cdots\!49}$, $\frac{10\!\cdots\!54}{13\!\cdots\!45}a^{15}-\frac{11\!\cdots\!39}{91\!\cdots\!15}a^{14}-\frac{80\!\cdots\!04}{91\!\cdots\!15}a^{13}+\frac{15\!\cdots\!46}{91\!\cdots\!15}a^{12}-\frac{12\!\cdots\!39}{15\!\cdots\!85}a^{11}+\frac{12\!\cdots\!39}{91\!\cdots\!15}a^{10}+\frac{26\!\cdots\!44}{18\!\cdots\!03}a^{9}-\frac{26\!\cdots\!22}{91\!\cdots\!15}a^{8}+\frac{11\!\cdots\!69}{91\!\cdots\!15}a^{7}+\frac{12\!\cdots\!53}{91\!\cdots\!15}a^{6}-\frac{25\!\cdots\!96}{39\!\cdots\!05}a^{5}-\frac{24\!\cdots\!93}{91\!\cdots\!15}a^{4}+\frac{13\!\cdots\!98}{91\!\cdots\!15}a^{3}+\frac{35\!\cdots\!40}{18\!\cdots\!03}a^{2}-\frac{10\!\cdots\!47}{91\!\cdots\!15}a+\frac{25\!\cdots\!19}{44\!\cdots\!31}$ (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: | \( 89005085005.7 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 89005085005.7 \cdot 30}{2\cdot\sqrt{670188096064724500217499812793861721}}\cr\approx \mathstrut & 3.96138151253 \end{aligned}\] (assuming GRH)
Galois group
$\OD_{16}:C_2$ (as 16T36):
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
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 | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.2.0.1}{2} }^{8}$ | ${\href{/padicField/7.1.0.1}{1} }^{16}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{8}$ | R | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.1.0.1}{1} }^{16}$ |
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 |
---|---|---|---|---|---|---|---|
\(29\) | 29.8.7.2 | $x^{8} + 29$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ |
29.8.7.2 | $x^{8} + 29$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ | |
\(83\) | 83.2.1.2 | $x^{2} + 83$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
83.2.1.2 | $x^{2} + 83$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
83.2.1.2 | $x^{2} + 83$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
83.2.1.2 | $x^{2} + 83$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
83.4.2.1 | $x^{4} + 164 x^{3} + 6894 x^{2} + 13940 x + 564653$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
83.4.2.1 | $x^{4} + 164 x^{3} + 6894 x^{2} + 13940 x + 564653$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |