Normalized defining polynomial
\( x^{16} - 3 x^{15} + 162 x^{14} - 2159 x^{13} + 43399 x^{12} - 251492 x^{11} + 2281047 x^{10} + \cdots + 1323645815808 \)
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: | \(410614370925299326221754150224933859206386689\) \(\medspace = 29^{14}\cdot 53^{14}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(614.24\) | 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: | $29^{7/8}53^{7/8}\approx 614.2421770815839$ | ||
Ramified primes: | \(29\), \(53\) | 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 a CM field. | |||
Reflex fields: | unavailable$^{128}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{6}a^{6}+\frac{1}{6}a^{5}+\frac{1}{6}a^{4}-\frac{1}{6}a^{3}-\frac{1}{2}a^{2}+\frac{1}{6}a$, $\frac{1}{6}a^{7}+\frac{1}{6}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{12}a^{8}-\frac{1}{6}a^{5}-\frac{1}{6}a^{4}+\frac{1}{3}a^{3}+\frac{5}{12}a^{2}-\frac{1}{2}a$, $\frac{1}{36}a^{9}+\frac{1}{36}a^{8}+\frac{1}{18}a^{7}+\frac{1}{9}a^{5}+\frac{1}{12}a^{3}-\frac{17}{36}a^{2}-\frac{1}{2}a$, $\frac{1}{1368}a^{10}+\frac{17}{1368}a^{9}+\frac{5}{228}a^{8}+\frac{5}{342}a^{7}+\frac{13}{342}a^{6}-\frac{41}{342}a^{5}+\frac{77}{456}a^{4}-\frac{89}{1368}a^{3}+\frac{275}{684}a^{2}+\frac{16}{57}a+\frac{8}{19}$, $\frac{1}{1368}a^{11}+\frac{7}{1368}a^{9}+\frac{1}{342}a^{8}+\frac{2}{171}a^{7}+\frac{23}{342}a^{6}-\frac{7}{456}a^{5}-\frac{35}{342}a^{4}+\frac{353}{1368}a^{3}+\frac{35}{114}a^{2}+\frac{17}{114}a-\frac{3}{19}$, $\frac{1}{8208}a^{12}+\frac{1}{4104}a^{10}-\frac{5}{8208}a^{9}+\frac{85}{4104}a^{8}+\frac{31}{684}a^{7}-\frac{53}{8208}a^{6}+\frac{1}{114}a^{5}+\frac{967}{4104}a^{4}-\frac{1415}{8208}a^{3}+\frac{25}{216}a^{2}+\frac{335}{684}a-\frac{20}{57}$, $\frac{1}{49248}a^{13}+\frac{1}{6156}a^{11}+\frac{13}{49248}a^{10}+\frac{259}{24624}a^{9}+\frac{5}{152}a^{8}+\frac{1771}{49248}a^{7}+\frac{2}{513}a^{6}-\frac{499}{3078}a^{5}-\frac{3569}{49248}a^{4}-\frac{6791}{24624}a^{3}+\frac{1427}{4104}a^{2}-\frac{34}{171}a-\frac{6}{19}$, $\frac{1}{935712}a^{14}+\frac{7}{935712}a^{13}-\frac{5}{116964}a^{12}+\frac{83}{311904}a^{11}+\frac{25}{103968}a^{10}+\frac{5029}{467856}a^{9}-\frac{2969}{935712}a^{8}-\frac{73811}{935712}a^{7}-\frac{157}{58482}a^{6}-\frac{63943}{311904}a^{5}-\frac{22391}{311904}a^{4}+\frac{19051}{467856}a^{3}-\frac{12593}{25992}a^{2}-\frac{1361}{3249}a-\frac{317}{1083}$, $\frac{1}{39\!\cdots\!96}a^{15}+\frac{19\!\cdots\!97}{39\!\cdots\!96}a^{14}-\frac{19\!\cdots\!43}{19\!\cdots\!48}a^{13}+\frac{53\!\cdots\!49}{39\!\cdots\!96}a^{12}-\frac{13\!\cdots\!61}{39\!\cdots\!96}a^{11}-\frac{94\!\cdots\!27}{98\!\cdots\!24}a^{10}-\frac{15\!\cdots\!99}{13\!\cdots\!32}a^{9}+\frac{61\!\cdots\!87}{39\!\cdots\!96}a^{8}-\frac{58\!\cdots\!57}{19\!\cdots\!48}a^{7}-\frac{21\!\cdots\!49}{39\!\cdots\!96}a^{6}-\frac{14\!\cdots\!91}{39\!\cdots\!96}a^{5}+\frac{94\!\cdots\!69}{98\!\cdots\!24}a^{4}+\frac{10\!\cdots\!37}{32\!\cdots\!08}a^{3}-\frac{44\!\cdots\!41}{20\!\cdots\!38}a^{2}-\frac{47\!\cdots\!17}{27\!\cdots\!84}a+\frac{32\!\cdots\!17}{11\!\cdots\!41}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$, $3$ |
Class group and class number
$C_{2}\times C_{3118}\times C_{6236}$, which has order $38887696$ (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: | $\frac{15\!\cdots\!57}{34\!\cdots\!12}a^{15}+\frac{12\!\cdots\!29}{34\!\cdots\!12}a^{14}+\frac{12\!\cdots\!29}{17\!\cdots\!56}a^{13}-\frac{27\!\cdots\!79}{42\!\cdots\!52}a^{12}+\frac{58\!\cdots\!19}{34\!\cdots\!12}a^{11}-\frac{37\!\cdots\!13}{86\!\cdots\!28}a^{10}+\frac{10\!\cdots\!25}{11\!\cdots\!04}a^{9}-\frac{46\!\cdots\!01}{34\!\cdots\!12}a^{8}+\frac{81\!\cdots\!55}{17\!\cdots\!56}a^{7}-\frac{35\!\cdots\!91}{11\!\cdots\!04}a^{6}+\frac{15\!\cdots\!81}{34\!\cdots\!12}a^{5}-\frac{16\!\cdots\!69}{86\!\cdots\!28}a^{4}+\frac{93\!\cdots\!81}{86\!\cdots\!28}a^{3}-\frac{28\!\cdots\!05}{71\!\cdots\!44}a^{2}+\frac{33\!\cdots\!59}{79\!\cdots\!16}a+\frac{92\!\cdots\!54}{99\!\cdots\!27}$, $\frac{66\!\cdots\!77}{57\!\cdots\!52}a^{15}-\frac{38\!\cdots\!75}{17\!\cdots\!56}a^{14}+\frac{51\!\cdots\!91}{31\!\cdots\!64}a^{13}-\frac{38\!\cdots\!13}{17\!\cdots\!56}a^{12}+\frac{76\!\cdots\!27}{17\!\cdots\!56}a^{11}-\frac{83\!\cdots\!47}{43\!\cdots\!64}a^{10}+\frac{29\!\cdots\!01}{19\!\cdots\!84}a^{9}-\frac{67\!\cdots\!85}{17\!\cdots\!56}a^{8}+\frac{49\!\cdots\!45}{28\!\cdots\!76}a^{7}-\frac{56\!\cdots\!67}{17\!\cdots\!56}a^{6}+\frac{19\!\cdots\!25}{17\!\cdots\!56}a^{5}-\frac{29\!\cdots\!19}{43\!\cdots\!64}a^{4}+\frac{29\!\cdots\!71}{14\!\cdots\!88}a^{3}-\frac{12\!\cdots\!07}{35\!\cdots\!72}a^{2}+\frac{34\!\cdots\!77}{11\!\cdots\!24}a-\frac{54\!\cdots\!37}{99\!\cdots\!27}$, $\frac{41\!\cdots\!25}{13\!\cdots\!32}a^{15}-\frac{16\!\cdots\!71}{13\!\cdots\!32}a^{14}+\frac{32\!\cdots\!55}{65\!\cdots\!16}a^{13}-\frac{72\!\cdots\!03}{13\!\cdots\!32}a^{12}+\frac{15\!\cdots\!83}{13\!\cdots\!32}a^{11}-\frac{12\!\cdots\!41}{27\!\cdots\!84}a^{10}+\frac{86\!\cdots\!35}{15\!\cdots\!48}a^{9}-\frac{16\!\cdots\!81}{14\!\cdots\!48}a^{8}+\frac{25\!\cdots\!85}{65\!\cdots\!16}a^{7}-\frac{30\!\cdots\!29}{13\!\cdots\!32}a^{6}+\frac{50\!\cdots\!69}{13\!\cdots\!32}a^{5}-\frac{78\!\cdots\!75}{54\!\cdots\!68}a^{4}+\frac{30\!\cdots\!97}{32\!\cdots\!08}a^{3}-\frac{30\!\cdots\!39}{82\!\cdots\!52}a^{2}+\frac{10\!\cdots\!81}{27\!\cdots\!84}a-\frac{14\!\cdots\!41}{11\!\cdots\!41}$, $\frac{38\!\cdots\!73}{65\!\cdots\!16}a^{15}-\frac{30\!\cdots\!33}{19\!\cdots\!48}a^{14}+\frac{38\!\cdots\!01}{41\!\cdots\!76}a^{13}-\frac{20\!\cdots\!57}{19\!\cdots\!48}a^{12}+\frac{44\!\cdots\!69}{19\!\cdots\!48}a^{11}-\frac{82\!\cdots\!01}{98\!\cdots\!24}a^{10}+\frac{78\!\cdots\!63}{75\!\cdots\!24}a^{9}-\frac{40\!\cdots\!83}{19\!\cdots\!48}a^{8}+\frac{19\!\cdots\!41}{27\!\cdots\!84}a^{7}-\frac{82\!\cdots\!47}{19\!\cdots\!48}a^{6}+\frac{14\!\cdots\!31}{19\!\cdots\!48}a^{5}-\frac{26\!\cdots\!51}{98\!\cdots\!24}a^{4}+\frac{96\!\cdots\!51}{54\!\cdots\!68}a^{3}-\frac{72\!\cdots\!87}{10\!\cdots\!69}a^{2}+\frac{26\!\cdots\!96}{34\!\cdots\!23}a-\frac{27\!\cdots\!81}{11\!\cdots\!41}$, $\frac{27\!\cdots\!81}{43\!\cdots\!44}a^{15}+\frac{13\!\cdots\!09}{39\!\cdots\!96}a^{14}+\frac{77\!\cdots\!17}{73\!\cdots\!24}a^{13}-\frac{42\!\cdots\!19}{39\!\cdots\!96}a^{12}+\frac{98\!\cdots\!95}{39\!\cdots\!96}a^{11}-\frac{41\!\cdots\!73}{49\!\cdots\!12}a^{10}+\frac{62\!\cdots\!67}{45\!\cdots\!44}a^{9}-\frac{93\!\cdots\!41}{39\!\cdots\!96}a^{8}+\frac{51\!\cdots\!23}{65\!\cdots\!16}a^{7}-\frac{19\!\cdots\!37}{39\!\cdots\!96}a^{6}+\frac{31\!\cdots\!89}{39\!\cdots\!96}a^{5}-\frac{79\!\cdots\!21}{24\!\cdots\!56}a^{4}+\frac{63\!\cdots\!21}{32\!\cdots\!08}a^{3}-\frac{30\!\cdots\!69}{41\!\cdots\!76}a^{2}+\frac{21\!\cdots\!05}{27\!\cdots\!84}a+\frac{20\!\cdots\!72}{11\!\cdots\!41}$, $\frac{10\!\cdots\!35}{39\!\cdots\!96}a^{15}-\frac{22\!\cdots\!81}{39\!\cdots\!96}a^{14}+\frac{68\!\cdots\!35}{19\!\cdots\!48}a^{13}-\frac{20\!\cdots\!89}{39\!\cdots\!96}a^{12}+\frac{39\!\cdots\!53}{39\!\cdots\!96}a^{11}-\frac{14\!\cdots\!89}{32\!\cdots\!08}a^{10}+\frac{40\!\cdots\!87}{13\!\cdots\!32}a^{9}-\frac{34\!\cdots\!15}{39\!\cdots\!96}a^{8}+\frac{80\!\cdots\!85}{19\!\cdots\!48}a^{7}-\frac{11\!\cdots\!31}{39\!\cdots\!96}a^{6}+\frac{96\!\cdots\!55}{39\!\cdots\!96}a^{5}-\frac{55\!\cdots\!57}{32\!\cdots\!08}a^{4}+\frac{36\!\cdots\!79}{98\!\cdots\!24}a^{3}-\frac{74\!\cdots\!27}{82\!\cdots\!52}a^{2}-\frac{58\!\cdots\!69}{27\!\cdots\!84}a+\frac{40\!\cdots\!32}{38\!\cdots\!47}$, $\frac{14\!\cdots\!17}{79\!\cdots\!28}a^{15}-\frac{55\!\cdots\!71}{79\!\cdots\!28}a^{14}+\frac{33\!\cdots\!67}{11\!\cdots\!92}a^{13}-\frac{24\!\cdots\!35}{79\!\cdots\!28}a^{12}+\frac{16\!\cdots\!61}{23\!\cdots\!84}a^{11}-\frac{15\!\cdots\!55}{59\!\cdots\!96}a^{10}+\frac{26\!\cdots\!51}{81\!\cdots\!28}a^{9}-\frac{49\!\cdots\!09}{79\!\cdots\!28}a^{8}+\frac{25\!\cdots\!17}{11\!\cdots\!92}a^{7}-\frac{38\!\cdots\!43}{29\!\cdots\!64}a^{6}+\frac{51\!\cdots\!07}{23\!\cdots\!84}a^{5}-\frac{48\!\cdots\!59}{59\!\cdots\!96}a^{4}+\frac{30\!\cdots\!85}{59\!\cdots\!96}a^{3}-\frac{17\!\cdots\!07}{82\!\cdots\!68}a^{2}+\frac{36\!\cdots\!81}{16\!\cdots\!36}a-\frac{48\!\cdots\!63}{68\!\cdots\!89}$ (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: | \( 1383941352.52 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 1383941352.52 \cdot 38887696}{2\cdot\sqrt{410614370925299326221754150224933859206386689}}\cr\approx \mathstrut & 3.22568368845 \end{aligned}\] (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
\(\Q(\sqrt{1537}) \), 4.4.3630961153.1, 4.4.125205557.1, 4.4.68508701.1, 8.0.20263621860992652421633.1 x2, 8.8.13183878894595089409.3 |
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 | ${\href{/padicField/2.2.0.1}{2} }^{8}$ | ${\href{/padicField/3.2.0.1}{2} }^{8}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(29\) | 29.16.14.1 | $x^{16} - 11832 x^{8} - 3693672$ | $8$ | $2$ | $14$ | $C_8: C_2$ | $[\ ]_{8}^{2}$ |
\(53\) | 53.16.14.3 | $x^{16} + 392 x^{15} + 67244 x^{14} + 6593832 x^{13} + 404342918 x^{12} + 15884530312 x^{11} + 390788364316 x^{10} + 5520959909976 x^{9} + 34792850359499 x^{8} + 11041919840728 x^{7} + 1563157014412 x^{6} + 127424552136 x^{5} + 27785666398 x^{4} + 835108627976 x^{3} + 20437943180972 x^{2} + 285892602154488 x + 1749632335115022$ | $8$ | $2$ | $14$ | $C_8: C_2$ | $[\ ]_{8}^{2}$ |