Normalized defining polynomial
\( x^{16} - 2 x^{15} - 13 x^{14} + 20 x^{13} + 117 x^{12} - 202 x^{11} - 299 x^{10} + 768 x^{9} + \cdots + 16384 \)
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: | \(12180359347643040469140625\) \(\medspace = 5^{8}\cdot 89^{10}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(36.97\) | 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: | $5^{1/2}89^{3/4}\approx 64.79291322178058$ | ||
Ramified primes: | \(5\), \(89\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{9}-\frac{1}{8}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{10}-\frac{1}{8}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{16}a^{11}-\frac{1}{16}a^{10}-\frac{1}{16}a^{9}-\frac{1}{8}a^{7}+\frac{1}{8}a^{6}-\frac{3}{16}a^{5}-\frac{1}{16}a^{4}+\frac{7}{16}a^{3}-\frac{3}{8}a^{2}+\frac{1}{4}a$, $\frac{1}{32}a^{12}-\frac{1}{32}a^{11}-\frac{1}{32}a^{10}-\frac{1}{16}a^{9}+\frac{1}{16}a^{8}+\frac{1}{16}a^{7}+\frac{5}{32}a^{6}-\frac{1}{32}a^{5}+\frac{7}{32}a^{4}-\frac{3}{8}a^{3}+\frac{1}{4}a^{2}+\frac{1}{4}a$, $\frac{1}{64}a^{13}-\frac{1}{64}a^{12}-\frac{1}{64}a^{11}-\frac{1}{32}a^{10}-\frac{1}{32}a^{9}+\frac{1}{32}a^{8}-\frac{3}{64}a^{7}+\frac{7}{64}a^{6}-\frac{1}{64}a^{5}-\frac{1}{16}a^{4}+\frac{1}{16}a^{3}-\frac{1}{4}a$, $\frac{1}{64}a^{14}-\frac{1}{64}a^{11}-\frac{1}{32}a^{10}+\frac{3}{64}a^{8}-\frac{1}{8}a^{6}+\frac{13}{64}a^{5}+\frac{1}{32}a^{4}+\frac{1}{8}a^{2}-\frac{1}{4}a$, $\frac{1}{15\!\cdots\!32}a^{15}-\frac{41\!\cdots\!17}{76\!\cdots\!16}a^{14}+\frac{14\!\cdots\!51}{15\!\cdots\!32}a^{13}+\frac{36\!\cdots\!47}{38\!\cdots\!08}a^{12}-\frac{60\!\cdots\!03}{15\!\cdots\!32}a^{11}+\frac{26\!\cdots\!47}{76\!\cdots\!16}a^{10}-\frac{75\!\cdots\!07}{15\!\cdots\!32}a^{9}-\frac{51\!\cdots\!65}{47\!\cdots\!76}a^{8}+\frac{16\!\cdots\!99}{15\!\cdots\!32}a^{7}-\frac{29\!\cdots\!11}{76\!\cdots\!16}a^{6}+\frac{24\!\cdots\!41}{15\!\cdots\!32}a^{5}-\frac{15\!\cdots\!87}{19\!\cdots\!04}a^{4}+\frac{95\!\cdots\!95}{38\!\cdots\!08}a^{3}+\frac{35\!\cdots\!51}{95\!\cdots\!52}a^{2}-\frac{27\!\cdots\!13}{95\!\cdots\!52}a-\frac{45\!\cdots\!69}{29\!\cdots\!11}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{37}$, which has order $37$ (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: | $\frac{8123920365}{168745744951808}a^{15}-\frac{7989401573}{84372872475904}a^{14}-\frac{102876800689}{168745744951808}a^{13}+\frac{34331108667}{42186436237952}a^{12}+\frac{872093323561}{168745744951808}a^{11}-\frac{622346405897}{84372872475904}a^{10}-\frac{1402484333887}{168745744951808}a^{9}+\frac{95510554703}{5273304529744}a^{8}-\frac{3255775041669}{168745744951808}a^{7}-\frac{7640923597883}{84372872475904}a^{6}+\frac{43399694323697}{168745744951808}a^{5}-\frac{2442581826359}{21093218118976}a^{4}-\frac{6007125172985}{42186436237952}a^{3}+\frac{7753522083479}{10546609059488}a^{2}-\frac{1178220800837}{10546609059488}a-\frac{83603486560}{329581533109}$, $\frac{94\!\cdots\!91}{25\!\cdots\!68}a^{15}+\frac{10\!\cdots\!29}{12\!\cdots\!84}a^{14}-\frac{17\!\cdots\!83}{25\!\cdots\!68}a^{13}-\frac{11\!\cdots\!87}{64\!\cdots\!92}a^{12}+\frac{14\!\cdots\!91}{25\!\cdots\!68}a^{11}+\frac{20\!\cdots\!73}{12\!\cdots\!84}a^{10}-\frac{62\!\cdots\!89}{25\!\cdots\!68}a^{9}-\frac{26\!\cdots\!41}{40\!\cdots\!12}a^{8}+\frac{13\!\cdots\!05}{25\!\cdots\!68}a^{7}+\frac{14\!\cdots\!71}{12\!\cdots\!84}a^{6}-\frac{49\!\cdots\!21}{25\!\cdots\!68}a^{5}+\frac{98\!\cdots\!63}{32\!\cdots\!96}a^{4}+\frac{15\!\cdots\!05}{64\!\cdots\!92}a^{3}+\frac{20\!\cdots\!73}{16\!\cdots\!48}a^{2}+\frac{12\!\cdots\!49}{16\!\cdots\!48}a+\frac{27\!\cdots\!09}{50\!\cdots\!89}$, $\frac{10\!\cdots\!81}{15\!\cdots\!32}a^{15}+\frac{61\!\cdots\!79}{76\!\cdots\!16}a^{14}-\frac{17\!\cdots\!29}{15\!\cdots\!32}a^{13}-\frac{34\!\cdots\!05}{38\!\cdots\!08}a^{12}+\frac{16\!\cdots\!33}{15\!\cdots\!32}a^{11}+\frac{67\!\cdots\!07}{76\!\cdots\!16}a^{10}-\frac{67\!\cdots\!31}{15\!\cdots\!32}a^{9}-\frac{45\!\cdots\!19}{11\!\cdots\!44}a^{8}+\frac{15\!\cdots\!35}{15\!\cdots\!32}a^{7}-\frac{25\!\cdots\!35}{76\!\cdots\!16}a^{6}-\frac{24\!\cdots\!63}{15\!\cdots\!32}a^{5}+\frac{53\!\cdots\!05}{19\!\cdots\!04}a^{4}+\frac{15\!\cdots\!91}{38\!\cdots\!08}a^{3}+\frac{22\!\cdots\!83}{95\!\cdots\!52}a^{2}+\frac{36\!\cdots\!87}{95\!\cdots\!52}a-\frac{19\!\cdots\!26}{29\!\cdots\!11}$, $\frac{61\!\cdots\!15}{15\!\cdots\!32}a^{15}-\frac{44\!\cdots\!47}{76\!\cdots\!16}a^{14}-\frac{93\!\cdots\!27}{15\!\cdots\!32}a^{13}+\frac{23\!\cdots\!11}{38\!\cdots\!08}a^{12}+\frac{90\!\cdots\!99}{15\!\cdots\!32}a^{11}-\frac{51\!\cdots\!47}{76\!\cdots\!16}a^{10}-\frac{37\!\cdots\!65}{15\!\cdots\!32}a^{9}+\frac{34\!\cdots\!59}{95\!\cdots\!52}a^{8}+\frac{11\!\cdots\!21}{15\!\cdots\!32}a^{7}-\frac{13\!\cdots\!05}{76\!\cdots\!16}a^{6}-\frac{13\!\cdots\!65}{15\!\cdots\!32}a^{5}+\frac{57\!\cdots\!77}{19\!\cdots\!04}a^{4}+\frac{57\!\cdots\!09}{38\!\cdots\!08}a^{3}-\frac{31\!\cdots\!31}{95\!\cdots\!52}a^{2}-\frac{60\!\cdots\!67}{95\!\cdots\!52}a-\frac{18\!\cdots\!42}{29\!\cdots\!11}$, $\frac{14\!\cdots\!51}{12\!\cdots\!84}a^{15}-\frac{39\!\cdots\!03}{64\!\cdots\!92}a^{14}-\frac{22\!\cdots\!43}{12\!\cdots\!84}a^{13}-\frac{49\!\cdots\!39}{16\!\cdots\!48}a^{12}+\frac{20\!\cdots\!79}{12\!\cdots\!84}a^{11}+\frac{20\!\cdots\!61}{64\!\cdots\!92}a^{10}-\frac{70\!\cdots\!33}{12\!\cdots\!84}a^{9}-\frac{34\!\cdots\!87}{16\!\cdots\!48}a^{8}+\frac{13\!\cdots\!53}{12\!\cdots\!84}a^{7}-\frac{70\!\cdots\!83}{64\!\cdots\!92}a^{6}-\frac{67\!\cdots\!09}{12\!\cdots\!84}a^{5}+\frac{20\!\cdots\!09}{80\!\cdots\!24}a^{4}+\frac{11\!\cdots\!89}{32\!\cdots\!96}a^{3}+\frac{42\!\cdots\!75}{80\!\cdots\!24}a^{2}+\frac{17\!\cdots\!81}{80\!\cdots\!24}a-\frac{35\!\cdots\!15}{50\!\cdots\!89}$, $\frac{14\!\cdots\!39}{76\!\cdots\!16}a^{15}-\frac{12\!\cdots\!99}{38\!\cdots\!08}a^{14}-\frac{23\!\cdots\!83}{76\!\cdots\!16}a^{13}-\frac{12\!\cdots\!35}{95\!\cdots\!52}a^{12}+\frac{20\!\cdots\!99}{76\!\cdots\!16}a^{11}+\frac{32\!\cdots\!41}{38\!\cdots\!08}a^{10}-\frac{77\!\cdots\!17}{76\!\cdots\!16}a^{9}-\frac{56\!\cdots\!79}{29\!\cdots\!11}a^{8}+\frac{19\!\cdots\!57}{76\!\cdots\!16}a^{7}-\frac{11\!\cdots\!31}{38\!\cdots\!08}a^{6}-\frac{28\!\cdots\!33}{76\!\cdots\!16}a^{5}+\frac{54\!\cdots\!17}{95\!\cdots\!52}a^{4}+\frac{90\!\cdots\!57}{19\!\cdots\!04}a^{3}-\frac{89\!\cdots\!63}{47\!\cdots\!76}a^{2}-\frac{19\!\cdots\!95}{47\!\cdots\!76}a-\frac{29\!\cdots\!27}{29\!\cdots\!11}$, $\frac{41\!\cdots\!03}{15\!\cdots\!32}a^{15}+\frac{93\!\cdots\!09}{76\!\cdots\!16}a^{14}-\frac{23\!\cdots\!75}{15\!\cdots\!32}a^{13}-\frac{48\!\cdots\!75}{38\!\cdots\!08}a^{12}-\frac{46\!\cdots\!81}{15\!\cdots\!32}a^{11}+\frac{11\!\cdots\!37}{76\!\cdots\!16}a^{10}+\frac{37\!\cdots\!27}{15\!\cdots\!32}a^{9}-\frac{94\!\cdots\!37}{11\!\cdots\!44}a^{8}-\frac{19\!\cdots\!71}{15\!\cdots\!32}a^{7}+\frac{39\!\cdots\!39}{76\!\cdots\!16}a^{6}-\frac{71\!\cdots\!09}{15\!\cdots\!32}a^{5}-\frac{54\!\cdots\!45}{19\!\cdots\!04}a^{4}+\frac{11\!\cdots\!81}{38\!\cdots\!08}a^{3}-\frac{79\!\cdots\!99}{95\!\cdots\!52}a^{2}+\frac{15\!\cdots\!73}{95\!\cdots\!52}a-\frac{74\!\cdots\!48}{29\!\cdots\!11}$ (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: | \( 95443.616698 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 95443.616698 \cdot 37}{2\cdot\sqrt{12180359347643040469140625}}\cr\approx \mathstrut & 1.2289310371 \end{aligned}\] (assuming GRH)
Galois group
$C_4\wr C_2$ (as 16T42):
A solvable group of order 32 |
The 14 conjugacy class representatives for $C_4\wr C_2$ |
Character table for $C_4\wr C_2$ |
Intermediate fields
\(\Q(\sqrt{89}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{445}) \), 4.4.39605.1 x2, 4.4.2225.1 x2, \(\Q(\sqrt{5}, \sqrt{89})\), 8.0.3490037155625.1, 8.0.440605625.1, 8.8.39213900625.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 32 |
Degree 8 siblings: | 8.0.440605625.1, 8.0.3490037155625.1 |
Degree 16 sibling: | 16.0.3859225055707220942242515625.1 |
Minimal sibling: | 8.0.440605625.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.2.0.1}{2} }^{8}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
\(89\) | 89.8.4.1 | $x^{8} + 24208 x^{7} + 219760588 x^{6} + 886665859360 x^{5} + 1341555590998180 x^{4} + 82475717140288 x^{3} + 5431695147417296 x^{2} + 96749624686259840 x + 5446086119176448$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
89.8.6.2 | $x^{8} - 35778 x^{4} - 331264141$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |