Normalized defining polynomial
\( x^{16} - 3 x^{15} - 9 x^{14} + 14 x^{13} + 49 x^{12} - 22 x^{11} - 148 x^{10} + x^{9} + 224 x^{8} + \cdots + 1 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[8, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(164338763007891015625\) \(\medspace = 5^{8}\cdot 29^{10}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(18.34\) | 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: | $5^{1/2}29^{3/4}\approx 27.943673649584856$ | ||
Ramified primes: | \(5\), \(29\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{11}a^{14}-\frac{4}{11}a^{13}-\frac{1}{11}a^{12}-\frac{1}{11}a^{11}+\frac{2}{11}a^{10}+\frac{5}{11}a^{9}-\frac{2}{11}a^{8}+\frac{1}{11}a^{7}-\frac{5}{11}a^{6}-\frac{3}{11}a^{5}-\frac{3}{11}a^{4}+\frac{4}{11}a^{3}-\frac{3}{11}a^{2}+\frac{3}{11}a-\frac{3}{11}$, $\frac{1}{46080899557}a^{15}+\frac{510937146}{46080899557}a^{14}+\frac{18954549379}{46080899557}a^{13}+\frac{4043445055}{46080899557}a^{12}+\frac{3300643526}{46080899557}a^{11}-\frac{22917630174}{46080899557}a^{10}+\frac{18534738617}{46080899557}a^{9}-\frac{507513879}{6582985651}a^{8}+\frac{324765561}{6582985651}a^{7}+\frac{730644535}{2425310503}a^{6}+\frac{1765531836}{6582985651}a^{5}-\frac{4325365650}{46080899557}a^{4}+\frac{184698172}{598453241}a^{3}+\frac{1630425605}{6582985651}a^{2}+\frac{687700040}{6582985651}a+\frac{13068647840}{46080899557}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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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)
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Fundamental units: | $\frac{46273090}{220482773}a^{15}-\frac{165261834}{220482773}a^{14}-\frac{334214515}{220482773}a^{13}+\frac{890065831}{220482773}a^{12}+\frac{1802145178}{220482773}a^{11}-\frac{2261232786}{220482773}a^{10}-\frac{5833058950}{220482773}a^{9}+\frac{561398879}{31497539}a^{8}+\frac{1274164927}{31497539}a^{7}-\frac{4620092381}{220482773}a^{6}-\frac{765616427}{31497539}a^{5}+\frac{4859221894}{220482773}a^{4}+\frac{420403553}{31497539}a^{3}-\frac{130387159}{31497539}a^{2}-\frac{45976199}{31497539}a-\frac{74760467}{220482773}$, $\frac{5046572820}{6582985651}a^{15}-\frac{19685363838}{6582985651}a^{14}-\frac{27126014814}{6582985651}a^{13}+\frac{93443135867}{6582985651}a^{12}+\frac{157760676020}{6582985651}a^{11}-\frac{244892830533}{6582985651}a^{10}-\frac{497653651591}{6582985651}a^{9}+\frac{439563365698}{6582985651}a^{8}+\frac{58765165368}{598453241}a^{7}-\frac{21899914866}{346472929}a^{6}-\frac{302991935542}{6582985651}a^{5}+\frac{36476851209}{598453241}a^{4}+\frac{207536189637}{6582985651}a^{3}-\frac{25773969785}{6582985651}a^{2}-\frac{1698545043}{6582985651}a-\frac{5927583674}{6582985651}$, $\frac{87612765841}{46080899557}a^{15}-\frac{27818886737}{4189172687}a^{14}-\frac{628778050701}{46080899557}a^{13}+\frac{1504514456047}{46080899557}a^{12}+\frac{3490724598699}{46080899557}a^{11}-\frac{3495156722703}{46080899557}a^{10}-\frac{10900853868537}{46080899557}a^{9}+\frac{729485448784}{6582985651}a^{8}+\frac{2291774412476}{6582985651}a^{7}-\frac{241916160106}{2425310503}a^{6}-\frac{1488128500984}{6582985651}a^{5}+\frac{6861160299410}{46080899557}a^{4}+\frac{1065242131230}{6582985651}a^{3}-\frac{92818147068}{6582985651}a^{2}-\frac{126746303257}{6582985651}a+\frac{82594044460}{46080899557}$, $\frac{18512471612}{46080899557}a^{15}-\frac{81120535830}{46080899557}a^{14}-\frac{70493443481}{46080899557}a^{13}+\frac{413782325525}{46080899557}a^{12}+\frac{441400625852}{46080899557}a^{11}-\frac{1287051245890}{46080899557}a^{10}-\frac{1551240804770}{46080899557}a^{9}+\frac{397915697125}{6582985651}a^{8}+\frac{299287715015}{6582985651}a^{7}-\frac{170410240675}{2425310503}a^{6}-\frac{135263781208}{6582985651}a^{5}+\frac{2588689243946}{46080899557}a^{4}+\frac{2331129917}{598453241}a^{3}-\frac{135662972705}{6582985651}a^{2}-\frac{14365151365}{6582985651}a+\frac{22255524383}{46080899557}$, $\frac{1909828542}{2425310503}a^{15}-\frac{6118320529}{2425310503}a^{14}-\frac{15716540164}{2425310503}a^{13}+\frac{29126879006}{2425310503}a^{12}+\frac{86199329393}{2425310503}a^{11}-\frac{55981945616}{2425310503}a^{10}-\frac{262914838693}{2425310503}a^{9}+\frac{6818400967}{346472929}a^{8}+\frac{56106148771}{346472929}a^{7}-\frac{9048488005}{2425310503}a^{6}-\frac{39125324536}{346472929}a^{5}+\frac{98647128164}{2425310503}a^{4}+\frac{30685797427}{346472929}a^{3}+\frac{2460887705}{346472929}a^{2}-\frac{3302450306}{346472929}a+\frac{50137879}{220482773}$, $\frac{95415136090}{46080899557}a^{15}-\frac{357217202077}{46080899557}a^{14}-\frac{589430583949}{46080899557}a^{13}+\frac{1759726814663}{46080899557}a^{12}+\frac{3351019569140}{46080899557}a^{11}-\frac{411866874289}{4189172687}a^{10}-\frac{10659902624879}{46080899557}a^{9}+\frac{1124878960762}{6582985651}a^{8}+\frac{2179063943534}{6582985651}a^{7}-\frac{422840764572}{2425310503}a^{6}-\frac{1326555405420}{6582985651}a^{5}+\frac{9010806036270}{46080899557}a^{4}+\frac{888320855669}{6582985651}a^{3}-\frac{253786087441}{6582985651}a^{2}-\frac{103151815155}{6582985651}a+\frac{229961472618}{46080899557}$, $\frac{81786303559}{46080899557}a^{15}-\frac{302211430096}{46080899557}a^{14}-\frac{529128219320}{46080899557}a^{13}+\frac{1525800734750}{46080899557}a^{12}+\frac{2959190746970}{46080899557}a^{11}-\frac{355642727544}{4189172687}a^{10}-\frac{9460663271403}{46080899557}a^{9}+\frac{971273388728}{6582985651}a^{8}+\frac{1974051927300}{6582985651}a^{7}-\frac{380958690550}{2425310503}a^{6}-\frac{1196896851160}{6582985651}a^{5}+\frac{7959574185422}{46080899557}a^{4}+\frac{755946578873}{6582985651}a^{3}-\frac{221001985352}{6582985651}a^{2}-\frac{86379695375}{6582985651}a+\frac{143912383502}{46080899557}$, $\frac{1243443293}{598453241}a^{15}-\frac{4490811922}{598453241}a^{14}-\frac{8287951072}{598453241}a^{13}+\frac{21932316210}{598453241}a^{12}+\frac{46434847708}{598453241}a^{11}-\frac{53247341495}{598453241}a^{10}-\frac{145550307716}{598453241}a^{9}+\frac{84828226226}{598453241}a^{8}+\frac{209039930827}{598453241}a^{7}-\frac{4241838244}{31497539}a^{6}-\frac{129080256460}{598453241}a^{5}+\frac{102232416710}{598453241}a^{4}+\frac{92768524209}{598453241}a^{3}-\frac{9644920066}{598453241}a^{2}-\frac{9375350365}{598453241}a+\frac{1092043609}{598453241}$, $\frac{87828774043}{46080899557}a^{15}-\frac{315787588210}{46080899557}a^{14}-\frac{601249939029}{46080899557}a^{13}+\frac{143757966899}{4189172687}a^{12}+\frac{3364390937377}{46080899557}a^{11}-\frac{3911140711668}{46080899557}a^{10}-\frac{10675589671696}{46080899557}a^{9}+\frac{908166527255}{6582985651}a^{8}+\frac{2274066292656}{6582985651}a^{7}-\frac{339233810754}{2425310503}a^{6}-\frac{137057966039}{598453241}a^{5}+\frac{8116988522757}{46080899557}a^{4}+\frac{1027984740641}{6582985651}a^{3}-\frac{230806096888}{6582985651}a^{2}-\frac{145991204322}{6582985651}a+\frac{212351046168}{46080899557}$, $\frac{24443342283}{46080899557}a^{15}-\frac{87911561216}{46080899557}a^{14}-\frac{165515674546}{46080899557}a^{13}+\frac{435404715393}{46080899557}a^{12}+\frac{919616945941}{46080899557}a^{11}-\frac{1068998100738}{46080899557}a^{10}-\frac{2881165035635}{46080899557}a^{9}+\frac{22787140856}{598453241}a^{8}+\frac{595169817570}{6582985651}a^{7}-\frac{8912011092}{220482773}a^{6}-\frac{366435009690}{6582985651}a^{5}+\frac{2443423789893}{46080899557}a^{4}+\frac{252874223868}{6582985651}a^{3}-\frac{70692494326}{6582985651}a^{2}-\frac{15777241301}{6582985651}a+\frac{134546336528}{46080899557}$, $\frac{186101785}{331517263}a^{15}-\frac{33976135}{30137933}a^{14}-\frac{2417778105}{331517263}a^{13}+\frac{1741890673}{331517263}a^{12}+\frac{12486456475}{331517263}a^{11}+\frac{1140842672}{331517263}a^{10}-\frac{36475813543}{331517263}a^{9}-\frac{2300717464}{47359609}a^{8}+\frac{8273465753}{47359609}a^{7}+\frac{1320937215}{17448277}a^{6}-\frac{6308046623}{47359609}a^{5}-\frac{2733893247}{331517263}a^{4}+\frac{5405418309}{47359609}a^{3}+\frac{1605355680}{47359609}a^{2}-\frac{547679841}{47359609}a-\frac{14520353}{331517263}$ (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: | \( 10281.9321931 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 10281.9321931 \cdot 1}{2\cdot\sqrt{164338763007891015625}}\cr\approx \mathstrut & 0.160005227527 \end{aligned}\] (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
\(\Q(\sqrt{29}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{145}) \), 4.4.4205.1 x2, 4.4.725.1 x2, \(\Q(\sqrt{5}, \sqrt{29})\), 8.4.15243125.1, 8.4.12819468125.1, 8.8.442050625.1 |
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: | 8.4.15243125.1, 8.4.12819468125.1 |
Degree 16 sibling: | 16.0.5528355987585453765625.2 |
Minimal sibling: | 8.4.15243125.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/7.2.0.1}{2} }^{8}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 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 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
\(29\) | 29.8.4.1 | $x^{8} + 2784 x^{7} + 2906616 x^{6} + 1348864734 x^{5} + 234834277018 x^{4} + 41857830864 x^{3} + 492109772617 x^{2} + 3561769809750 x + 616658760166$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
29.8.6.2 | $x^{8} - 1914 x^{4} - 2069701$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |