Normalized defining polynomial
\( x^{16} - 4 x^{15} - 36 x^{14} + 274 x^{13} - 16 x^{12} - 6092 x^{11} + 67960 x^{10} - 468046 x^{9} + 2156282 x^{8} - 6421886 x^{7} + 10535912 x^{6} - 5431518 x^{5} + \cdots + 15280961 \)
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: | \(43690605516556003276152578767301401\) \(\medspace = 29^{14}\cdot 59^{8}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(146.23\) | 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: | $29^{7/8}59^{1/2}\approx 146.22588347393037$ | ||
Ramified primes: | \(29\), \(59\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{116}a^{8}-\frac{1}{58}a^{7}-\frac{5}{29}a^{6}+\frac{5}{58}a^{5}-\frac{7}{58}a^{4}+\frac{13}{58}a^{3}+\frac{33}{116}a^{2}+\frac{2}{29}a+\frac{7}{116}$, $\frac{1}{116}a^{9}-\frac{6}{29}a^{7}+\frac{7}{29}a^{6}+\frac{3}{58}a^{5}-\frac{1}{58}a^{4}+\frac{27}{116}a^{3}+\frac{4}{29}a^{2}+\frac{23}{116}a+\frac{7}{58}$, $\frac{1}{116}a^{10}-\frac{5}{29}a^{7}-\frac{5}{58}a^{6}+\frac{3}{58}a^{5}-\frac{19}{116}a^{4}-\frac{14}{29}a^{3}+\frac{3}{116}a^{2}+\frac{8}{29}a-\frac{3}{58}$, $\frac{1}{116}a^{11}+\frac{2}{29}a^{7}+\frac{3}{29}a^{6}+\frac{7}{116}a^{5}+\frac{3}{29}a^{4}-\frac{57}{116}a^{3}-\frac{1}{29}a^{2}+\frac{19}{58}a-\frac{17}{58}$, $\frac{1}{1624}a^{12}-\frac{1}{406}a^{11}-\frac{3}{812}a^{10}-\frac{1}{232}a^{9}-\frac{1}{1624}a^{8}-\frac{59}{406}a^{7}+\frac{351}{1624}a^{6}-\frac{179}{812}a^{5}+\frac{33}{1624}a^{4}+\frac{659}{1624}a^{3}+\frac{613}{1624}a^{2}-\frac{495}{1624}a+\frac{185}{1624}$, $\frac{1}{1624}a^{13}+\frac{3}{812}a^{11}-\frac{3}{1624}a^{10}-\frac{1}{1624}a^{9}-\frac{1}{812}a^{8}+\frac{359}{1624}a^{7}+\frac{187}{812}a^{6}-\frac{111}{1624}a^{5}-\frac{51}{232}a^{4}-\frac{279}{1624}a^{3}+\frac{487}{1624}a^{2}-\frac{535}{1624}a+\frac{27}{812}$, $\frac{1}{1830248}a^{14}-\frac{177}{915124}a^{13}-\frac{5}{261464}a^{12}+\frac{2041}{1830248}a^{11}-\frac{6841}{1830248}a^{10}+\frac{1927}{1830248}a^{9}-\frac{3}{39788}a^{8}-\frac{20299}{457562}a^{7}+\frac{193139}{915124}a^{6}+\frac{377995}{1830248}a^{5}+\frac{51873}{915124}a^{4}+\frac{119263}{915124}a^{3}+\frac{79372}{228781}a^{2}+\frac{322187}{1830248}a+\frac{440213}{1830248}$, $\frac{1}{27\!\cdots\!96}a^{15}-\frac{12\!\cdots\!81}{13\!\cdots\!48}a^{14}-\frac{45\!\cdots\!33}{30\!\cdots\!38}a^{13}+\frac{64\!\cdots\!13}{27\!\cdots\!96}a^{12}+\frac{79\!\cdots\!85}{27\!\cdots\!96}a^{11}-\frac{12\!\cdots\!08}{49\!\cdots\!41}a^{10}-\frac{62\!\cdots\!29}{27\!\cdots\!96}a^{9}+\frac{59\!\cdots\!13}{69\!\cdots\!74}a^{8}-\frac{28\!\cdots\!97}{27\!\cdots\!96}a^{7}+\frac{31\!\cdots\!41}{27\!\cdots\!96}a^{6}+\frac{40\!\cdots\!85}{27\!\cdots\!96}a^{5}-\frac{10\!\cdots\!61}{39\!\cdots\!28}a^{4}-\frac{19\!\cdots\!11}{39\!\cdots\!28}a^{3}-\frac{11\!\cdots\!03}{34\!\cdots\!87}a^{2}+\frac{19\!\cdots\!95}{69\!\cdots\!74}a+\frac{19\!\cdots\!13}{69\!\cdots\!74}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $7$ |
Class group and class number
$C_{42}$, which has order $42$ (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: | $\frac{17\!\cdots\!75}{80\!\cdots\!16}a^{15}-\frac{47\!\cdots\!83}{40\!\cdots\!08}a^{14}-\frac{52\!\cdots\!51}{80\!\cdots\!16}a^{13}+\frac{56\!\cdots\!35}{80\!\cdots\!16}a^{12}-\frac{75\!\cdots\!17}{80\!\cdots\!16}a^{11}-\frac{10\!\cdots\!49}{80\!\cdots\!16}a^{10}+\frac{17\!\cdots\!30}{10\!\cdots\!77}a^{9}-\frac{25\!\cdots\!95}{20\!\cdots\!54}a^{8}+\frac{64\!\cdots\!23}{10\!\cdots\!77}a^{7}-\frac{17\!\cdots\!79}{80\!\cdots\!16}a^{6}+\frac{19\!\cdots\!07}{40\!\cdots\!08}a^{5}-\frac{11\!\cdots\!37}{20\!\cdots\!54}a^{4}+\frac{17\!\cdots\!51}{10\!\cdots\!77}a^{3}+\frac{37\!\cdots\!87}{80\!\cdots\!16}a^{2}-\frac{26\!\cdots\!75}{80\!\cdots\!16}a-\frac{23\!\cdots\!74}{10\!\cdots\!77}$, $\frac{13\!\cdots\!17}{39\!\cdots\!28}a^{15}-\frac{13\!\cdots\!23}{57\!\cdots\!04}a^{14}-\frac{68\!\cdots\!51}{39\!\cdots\!28}a^{13}+\frac{16\!\cdots\!75}{39\!\cdots\!28}a^{12}+\frac{23\!\cdots\!85}{71\!\cdots\!63}a^{11}-\frac{14\!\cdots\!53}{99\!\cdots\!82}a^{10}+\frac{29\!\cdots\!33}{19\!\cdots\!64}a^{9}-\frac{37\!\cdots\!13}{39\!\cdots\!28}a^{8}+\frac{12\!\cdots\!82}{49\!\cdots\!41}a^{7}-\frac{11\!\cdots\!85}{49\!\cdots\!41}a^{6}-\frac{14\!\cdots\!41}{57\!\cdots\!04}a^{5}+\frac{11\!\cdots\!93}{39\!\cdots\!28}a^{4}+\frac{46\!\cdots\!85}{39\!\cdots\!28}a^{3}-\frac{11\!\cdots\!83}{19\!\cdots\!64}a^{2}+\frac{20\!\cdots\!37}{39\!\cdots\!28}a+\frac{33\!\cdots\!99}{99\!\cdots\!82}$, $\frac{15\!\cdots\!87}{13\!\cdots\!48}a^{15}+\frac{89\!\cdots\!27}{12\!\cdots\!52}a^{14}-\frac{39\!\cdots\!83}{69\!\cdots\!74}a^{13}-\frac{31\!\cdots\!01}{13\!\cdots\!48}a^{12}+\frac{52\!\cdots\!51}{27\!\cdots\!96}a^{11}+\frac{24\!\cdots\!03}{27\!\cdots\!96}a^{10}+\frac{23\!\cdots\!51}{13\!\cdots\!48}a^{9}+\frac{28\!\cdots\!89}{39\!\cdots\!28}a^{8}-\frac{87\!\cdots\!83}{69\!\cdots\!74}a^{7}+\frac{15\!\cdots\!49}{27\!\cdots\!96}a^{6}-\frac{35\!\cdots\!91}{27\!\cdots\!96}a^{5}-\frac{46\!\cdots\!31}{27\!\cdots\!96}a^{4}+\frac{75\!\cdots\!19}{27\!\cdots\!96}a^{3}+\frac{63\!\cdots\!57}{17\!\cdots\!36}a^{2}-\frac{74\!\cdots\!84}{34\!\cdots\!87}a+\frac{12\!\cdots\!63}{69\!\cdots\!74}$, $\frac{28\!\cdots\!45}{27\!\cdots\!96}a^{15}+\frac{30\!\cdots\!19}{27\!\cdots\!96}a^{14}-\frac{92\!\cdots\!27}{13\!\cdots\!48}a^{13}-\frac{12\!\cdots\!29}{27\!\cdots\!96}a^{12}+\frac{18\!\cdots\!91}{69\!\cdots\!74}a^{11}+\frac{13\!\cdots\!01}{27\!\cdots\!96}a^{10}-\frac{30\!\cdots\!09}{27\!\cdots\!96}a^{9}+\frac{21\!\cdots\!71}{81\!\cdots\!72}a^{8}-\frac{80\!\cdots\!61}{27\!\cdots\!96}a^{7}+\frac{15\!\cdots\!77}{13\!\cdots\!48}a^{6}-\frac{14\!\cdots\!05}{69\!\cdots\!74}a^{5}+\frac{78\!\cdots\!24}{34\!\cdots\!87}a^{4}-\frac{41\!\cdots\!93}{13\!\cdots\!48}a^{3}+\frac{29\!\cdots\!15}{39\!\cdots\!28}a^{2}-\frac{26\!\cdots\!27}{69\!\cdots\!74}a+\frac{28\!\cdots\!63}{13\!\cdots\!48}$, $\frac{87\!\cdots\!01}{69\!\cdots\!74}a^{15}+\frac{18\!\cdots\!31}{12\!\cdots\!52}a^{14}-\frac{15\!\cdots\!81}{27\!\cdots\!96}a^{13}+\frac{10\!\cdots\!71}{34\!\cdots\!87}a^{12}+\frac{30\!\cdots\!79}{27\!\cdots\!96}a^{11}-\frac{35\!\cdots\!01}{13\!\cdots\!48}a^{10}+\frac{20\!\cdots\!35}{39\!\cdots\!28}a^{9}-\frac{76\!\cdots\!97}{27\!\cdots\!96}a^{8}+\frac{21\!\cdots\!45}{39\!\cdots\!28}a^{7}-\frac{25\!\cdots\!35}{27\!\cdots\!96}a^{6}+\frac{32\!\cdots\!89}{13\!\cdots\!48}a^{5}-\frac{16\!\cdots\!25}{34\!\cdots\!87}a^{4}+\frac{16\!\cdots\!45}{34\!\cdots\!87}a^{3}+\frac{29\!\cdots\!07}{30\!\cdots\!38}a^{2}+\frac{71\!\cdots\!63}{27\!\cdots\!96}a-\frac{57\!\cdots\!63}{69\!\cdots\!74}$, $\frac{25\!\cdots\!93}{34\!\cdots\!87}a^{15}-\frac{40\!\cdots\!07}{69\!\cdots\!74}a^{14}-\frac{78\!\cdots\!63}{27\!\cdots\!96}a^{13}+\frac{37\!\cdots\!28}{34\!\cdots\!87}a^{12}+\frac{23\!\cdots\!17}{69\!\cdots\!74}a^{11}-\frac{92\!\cdots\!07}{27\!\cdots\!96}a^{10}+\frac{10\!\cdots\!55}{27\!\cdots\!96}a^{9}-\frac{74\!\cdots\!19}{34\!\cdots\!87}a^{8}+\frac{10\!\cdots\!69}{12\!\cdots\!52}a^{7}-\frac{36\!\cdots\!49}{19\!\cdots\!64}a^{6}+\frac{42\!\cdots\!15}{27\!\cdots\!96}a^{5}+\frac{31\!\cdots\!67}{27\!\cdots\!96}a^{4}-\frac{58\!\cdots\!41}{27\!\cdots\!96}a^{3}-\frac{23\!\cdots\!39}{27\!\cdots\!96}a^{2}+\frac{14\!\cdots\!81}{27\!\cdots\!96}a-\frac{22\!\cdots\!51}{69\!\cdots\!74}$, $\frac{74\!\cdots\!39}{13\!\cdots\!48}a^{15}-\frac{48\!\cdots\!08}{49\!\cdots\!41}a^{14}-\frac{29\!\cdots\!95}{13\!\cdots\!48}a^{13}+\frac{14\!\cdots\!75}{13\!\cdots\!48}a^{12}+\frac{38\!\cdots\!21}{19\!\cdots\!64}a^{11}-\frac{39\!\cdots\!23}{13\!\cdots\!48}a^{10}+\frac{21\!\cdots\!27}{69\!\cdots\!74}a^{9}-\frac{12\!\cdots\!87}{69\!\cdots\!74}a^{8}+\frac{52\!\cdots\!37}{69\!\cdots\!74}a^{7}-\frac{25\!\cdots\!03}{13\!\cdots\!48}a^{6}+\frac{80\!\cdots\!91}{49\!\cdots\!41}a^{5}+\frac{24\!\cdots\!61}{34\!\cdots\!87}a^{4}-\frac{22\!\cdots\!85}{69\!\cdots\!74}a^{3}-\frac{12\!\cdots\!51}{13\!\cdots\!48}a^{2}+\frac{70\!\cdots\!25}{13\!\cdots\!48}a-\frac{12\!\cdots\!37}{34\!\cdots\!87}$ (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: | \( 2123693498.09 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 2123693498.09 \cdot 42}{2\cdot\sqrt{43690605516556003276152578767301401}}\cr\approx \mathstrut & 0.518269972159 \end{aligned}\] (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 | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | ${\href{/padicField/7.2.0.1}{2} }^{4}{,}\,{\href{/padicField/7.1.0.1}{1} }^{8}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | R |
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 | $x^{8} + 29$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ | |
\(59\) | 59.2.1.2 | $x^{2} + 59$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
59.2.1.2 | $x^{2} + 59$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
59.2.1.2 | $x^{2} + 59$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
59.2.1.2 | $x^{2} + 59$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
59.4.2.1 | $x^{4} + 116 x^{3} + 3486 x^{2} + 7076 x + 201725$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
59.4.2.1 | $x^{4} + 116 x^{3} + 3486 x^{2} + 7076 x + 201725$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |