Normalized defining polynomial
\( x^{16} - 3 x^{15} - 3 x^{14} + 12 x^{13} - 19 x^{12} + 100 x^{11} - 70 x^{10} + 95 x^{9} - 90 x^{8} - 1065 x^{7} + 3906 x^{6} - 6078 x^{5} + 6817 x^{4} - 7738 x^{3} + 11971 x^{2} + \cdots + 7295 \)
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: | \(3455222492240908603515625\) \(\medspace = 5^{10}\cdot 29^{12}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(34.17\) | 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: | $5^{3/4}29^{3/4}\approx 41.78553833475025$ | ||
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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{15}a^{10}+\frac{1}{15}a^{9}-\frac{2}{5}a^{7}+\frac{2}{5}a^{6}+\frac{2}{5}a^{4}-\frac{1}{5}a^{3}-\frac{1}{15}a^{2}+\frac{1}{3}a$, $\frac{1}{90}a^{11}-\frac{1}{30}a^{10}-\frac{7}{45}a^{9}+\frac{7}{45}a^{8}+\frac{1}{3}a^{7}-\frac{1}{10}a^{6}+\frac{7}{30}a^{5}-\frac{7}{15}a^{4}+\frac{13}{45}a^{3}-\frac{2}{5}a^{2}+\frac{7}{18}a-\frac{1}{18}$, $\frac{1}{270}a^{12}-\frac{1}{270}a^{11}-\frac{1}{135}a^{10}-\frac{13}{135}a^{9}-\frac{16}{135}a^{8}+\frac{11}{90}a^{7}-\frac{23}{90}a^{6}+\frac{1}{3}a^{5}+\frac{5}{27}a^{4}-\frac{19}{135}a^{3}+\frac{7}{54}a^{2}+\frac{19}{54}a+\frac{8}{27}$, $\frac{1}{810}a^{13}-\frac{1}{270}a^{11}-\frac{1}{81}a^{10}-\frac{4}{81}a^{9}+\frac{1}{810}a^{8}+\frac{7}{45}a^{7}+\frac{133}{270}a^{6}-\frac{13}{81}a^{5}-\frac{5}{27}a^{4}-\frac{109}{270}a^{3}+\frac{56}{405}a^{2}-\frac{55}{162}a-\frac{19}{81}$, $\frac{1}{36231300}a^{14}+\frac{421}{36231300}a^{13}+\frac{2917}{6038550}a^{12}+\frac{3683}{5175900}a^{11}-\frac{155333}{36231300}a^{10}-\frac{151253}{1341900}a^{9}+\frac{4341553}{36231300}a^{8}-\frac{104687}{3019275}a^{7}-\frac{12706993}{36231300}a^{6}+\frac{7116239}{36231300}a^{5}-\frac{1480939}{4025700}a^{4}-\frac{10042343}{36231300}a^{3}-\frac{5364983}{18115650}a^{2}-\frac{4339}{115020}a+\frac{2087761}{7246260}$, $\frac{1}{18559048649400}a^{15}+\frac{27613}{9279524324700}a^{14}-\frac{82056103}{261395051400}a^{13}-\frac{29330143909}{18559048649400}a^{12}+\frac{733165981}{257764564575}a^{11}-\frac{295738418}{39320018325}a^{10}-\frac{203432990143}{1325646332100}a^{9}-\frac{77483822099}{18559048649400}a^{8}+\frac{4678994671067}{18559048649400}a^{7}+\frac{2895889593797}{9279524324700}a^{6}+\frac{187156274968}{2319881081175}a^{5}-\frac{1247677378249}{9279524324700}a^{4}-\frac{216431730283}{687372172200}a^{3}+\frac{262324849369}{3711809729880}a^{2}-\frac{56060675653}{463976216235}a-\frac{268168794079}{742361945976}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$, $3$, $5$ |
Class group and class number
$C_{3}\times C_{12}$, which has order $36$ (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: | $\frac{43002619}{314560146600}a^{15}-\frac{4703803}{11234290950}a^{14}-\frac{40346629}{62912029320}a^{13}+\frac{697970633}{314560146600}a^{12}-\frac{2331163}{2097067644}a^{11}+\frac{50308423}{4493716380}a^{10}-\frac{72469196}{7864003665}a^{9}-\frac{494493907}{62912029320}a^{8}-\frac{255157603}{12582405864}a^{7}-\frac{213729949}{1572800733}a^{6}+\frac{12864434131}{22468581900}a^{5}-\frac{3076459669}{5617145475}a^{4}+\frac{5394615697}{20970676440}a^{3}-\frac{272064137257}{314560146600}a^{2}+\frac{8760643561}{6291202932}a-\frac{80883951443}{62912029320}$, $\frac{3278631151}{3711809729880}a^{15}-\frac{1193242691}{662823166050}a^{14}-\frac{86751197473}{18559048649400}a^{13}+\frac{115487433109}{18559048649400}a^{12}-\frac{27855683893}{3093174774900}a^{11}+\frac{1805234669}{22468581900}a^{10}+\frac{49281095746}{2319881081175}a^{9}+\frac{1556654274881}{18559048649400}a^{8}-\frac{888759001943}{18559048649400}a^{7}-\frac{5035169650369}{4639762162350}a^{6}+\frac{3000190293527}{1325646332100}a^{5}-\frac{946304012657}{331411583025}a^{4}+\frac{16999226875133}{6186349549800}a^{3}-\frac{64155151625837}{18559048649400}a^{2}+\frac{12051838181503}{1855904864940}a-\frac{18398004265363}{3711809729880}$, $\frac{29705009}{157280073300}a^{15}-\frac{137908217}{157280073300}a^{14}-\frac{18929332}{39320018325}a^{13}+\frac{644617567}{157280073300}a^{12}-\frac{123824321}{52426691100}a^{11}+\frac{3885354029}{157280073300}a^{10}-\frac{783216391}{22468581900}a^{9}-\frac{2364131597}{78640036650}a^{8}-\frac{20663563283}{157280073300}a^{7}-\frac{57873903271}{157280073300}a^{6}+\frac{153124611217}{157280073300}a^{5}-\frac{248389508461}{157280073300}a^{4}+\frac{22611399362}{13106672775}a^{3}-\frac{270814325459}{157280073300}a^{2}+\frac{76150566911}{31456014660}a-\frac{23271911884}{7864003665}$, $\frac{102412637}{314560146600}a^{15}-\frac{203761459}{157280073300}a^{14}-\frac{50452543}{44937163800}a^{13}+\frac{1987205767}{314560146600}a^{12}-\frac{15175283}{4368890925}a^{11}+\frac{2823074417}{78640036650}a^{10}-\frac{6931898657}{157280073300}a^{9}-\frac{11928995923}{314560146600}a^{8}-\frac{47706066641}{314560146600}a^{7}-\frac{11320985453}{22468581900}a^{6}+\frac{121587825067}{78640036650}a^{5}-\frac{334530379193}{157280073300}a^{4}+\frac{69288091127}{34951127400}a^{3}-\frac{32547711527}{12582405864}a^{2}+\frac{29988446179}{7864003665}a-\frac{65994255167}{12582405864}$, $\frac{22087141961}{18559048649400}a^{15}-\frac{542779756}{331411583025}a^{14}-\frac{24756334979}{3711809729880}a^{13}+\frac{72419889907}{18559048649400}a^{12}-\frac{9156529537}{618634954980}a^{11}+\frac{433321759}{4493716380}a^{10}+\frac{79081862413}{927952432470}a^{9}+\frac{777849970411}{3711809729880}a^{8}+\frac{760146206123}{3711809729880}a^{7}-\frac{519307171687}{463976216235}a^{6}+\frac{3493790813009}{1325646332100}a^{5}-\frac{1773731756227}{662823166050}a^{4}+\frac{700077629323}{247453981992}a^{3}-\frac{46422988848083}{18559048649400}a^{2}+\frac{2348884354079}{371180972988}a-\frac{12317624273617}{3711809729880}$, $\frac{4205992087}{6186349549800}a^{15}+\frac{1732753151}{3093174774900}a^{14}-\frac{9539028937}{2062116516600}a^{13}-\frac{47960312863}{6186349549800}a^{12}-\frac{9313619002}{773293693725}a^{11}+\frac{59624401}{1456296975}a^{10}+\frac{514987056053}{3093174774900}a^{9}+\frac{667375564789}{2062116516600}a^{8}+\frac{2973212177849}{6186349549800}a^{7}-\frac{827460480761}{3093174774900}a^{6}+\frac{1118840561}{9546835725}a^{5}+\frac{1730565481937}{3093174774900}a^{4}+\frac{6705146463413}{6186349549800}a^{3}+\frac{10415581909}{9164962296}a^{2}+\frac{259450602584}{154658738745}a+\frac{80774564455}{27494886888}$, $\frac{52192844}{331411583025}a^{15}-\frac{228570382}{463976216235}a^{14}-\frac{2977924139}{4639762162350}a^{13}+\frac{6272142862}{2319881081175}a^{12}-\frac{232409273}{73647018450}a^{11}+\frac{558164884}{39320018325}a^{10}-\frac{14672145343}{2319881081175}a^{9}-\frac{36827104817}{4639762162350}a^{8}+\frac{7404692188}{2319881081175}a^{7}-\frac{1054397714813}{4639762162350}a^{6}+\frac{317454292247}{463976216235}a^{5}-\frac{2246066604511}{2319881081175}a^{4}+\frac{236123974063}{515529129150}a^{3}+\frac{1039605804238}{2319881081175}a^{2}-\frac{106588020491}{132564633210}a+\frac{102791134157}{463976216235}$ (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: | \( 73371.5821514 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 73371.5821514 \cdot 36}{2\cdot\sqrt{3455222492240908603515625}}\cr\approx \mathstrut & 1.72584316534 \end{aligned}\] (assuming GRH)
Galois group
$C_2^3:C_4$ (as 16T33):
A solvable group of order 32 |
The 11 conjugacy class representatives for $C_2^3:C_4$ |
Character table for $C_2^3:C_4$ |
Intermediate fields
\(\Q(\sqrt{5}) \), \(\Q(\sqrt{145}) \), \(\Q(\sqrt{29}) \), 4.0.121945.1 x2, \(\Q(\sqrt{5}, \sqrt{29})\), 4.0.609725.1 x2, 8.0.371764575625.3 x2, 8.4.64097340625.2 x2, 8.0.371764575625.5 |
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 | ${\href{/padicField/2.4.0.1}{4} }^{4}$ | ${\href{/padicField/3.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.4.3.1 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
5.4.3.1 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
\(29\) | 29.8.6.1 | $x^{8} + 96 x^{7} + 3464 x^{6} + 55872 x^{5} + 345682 x^{4} + 114528 x^{3} + 113384 x^{2} + 1587648 x + 9488961$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
29.8.6.1 | $x^{8} + 96 x^{7} + 3464 x^{6} + 55872 x^{5} + 345682 x^{4} + 114528 x^{3} + 113384 x^{2} + 1587648 x + 9488961$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |