Normalized defining polynomial
\( x^{16} - x^{15} + 5 x^{14} - 14 x^{13} + 14 x^{12} + 31 x^{11} - 30 x^{10} + 434 x^{9} - 1059 x^{8} + 2170 x^{7} - 750 x^{6} + 3875 x^{5} + 8750 x^{4} - 43750 x^{3} + 78125 x^{2} + \cdots + 390625 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(27204384060547119140625\) \(\medspace = 3^{8}\cdot 5^{12}\cdot 19^{8}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(25.24\) | 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))
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Galois root discriminant: | $3^{1/2}5^{3/4}19^{1/2}\approx 25.24439291382227$ | ||
Ramified primes: | \(3\), \(5\), \(19\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $16$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois and abelian over $\Q$. | |||
Conductor: | \(285=3\cdot 5\cdot 19\) | ||
Dirichlet character group: | $\lbrace$$\chi_{285}(1,·)$, $\chi_{285}(134,·)$, $\chi_{285}(77,·)$, $\chi_{285}(208,·)$, $\chi_{285}(248,·)$, $\chi_{285}(151,·)$, $\chi_{285}(284,·)$, $\chi_{285}(94,·)$, $\chi_{285}(37,·)$, $\chi_{285}(227,·)$, $\chi_{285}(229,·)$, $\chi_{285}(172,·)$, $\chi_{285}(113,·)$, $\chi_{285}(56,·)$, $\chi_{285}(58,·)$, $\chi_{285}(191,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{128}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5}a^{9}-\frac{1}{5}a^{8}+\frac{1}{5}a^{6}-\frac{1}{5}a^{5}+\frac{1}{5}a^{4}-\frac{1}{5}a^{2}+\frac{1}{5}a$, $\frac{1}{275}a^{10}-\frac{1}{25}a^{9}-\frac{2}{5}a^{8}+\frac{1}{25}a^{7}-\frac{11}{25}a^{6}-\frac{109}{275}a^{5}+\frac{2}{5}a^{4}-\frac{6}{25}a^{3}-\frac{9}{25}a^{2}+\frac{2}{5}a+\frac{4}{11}$, $\frac{1}{1375}a^{11}-\frac{1}{1375}a^{10}+\frac{1}{25}a^{9}+\frac{51}{125}a^{8}+\frac{24}{125}a^{7}+\frac{331}{1375}a^{6}+\frac{134}{275}a^{5}+\frac{44}{125}a^{4}+\frac{6}{125}a^{3}+\frac{9}{25}a^{2}-\frac{18}{55}a-\frac{3}{11}$, $\frac{1}{27500}a^{12}+\frac{1}{6875}a^{11}+\frac{1}{2500}a^{9}+\frac{226}{625}a^{8}-\frac{756}{6875}a^{7}+\frac{1}{220}a^{6}-\frac{164}{625}a^{5}-\frac{131}{625}a^{4}+\frac{1}{4}a^{3}+\frac{1}{275}a^{2}+\frac{14}{55}a-\frac{1}{4}$, $\frac{1}{243237500}a^{13}+\frac{629}{243237500}a^{12}-\frac{72}{486475}a^{11}+\frac{3501}{22112500}a^{10}-\frac{820971}{22112500}a^{9}+\frac{14323994}{60809375}a^{8}+\frac{327069}{1945900}a^{7}+\frac{48514909}{243237500}a^{6}+\frac{1234244}{5528125}a^{5}-\frac{33807}{176900}a^{4}-\frac{2004361}{9729500}a^{3}+\frac{217994}{486475}a^{2}+\frac{134413}{389180}a+\frac{1803}{7076}$, $\frac{1}{1216187500}a^{14}-\frac{1}{1216187500}a^{13}+\frac{499}{60809375}a^{12}-\frac{278489}{1216187500}a^{11}+\frac{318389}{1216187500}a^{10}-\frac{21552561}{304046875}a^{9}+\frac{84062849}{243237500}a^{8}+\frac{323550659}{1216187500}a^{7}-\frac{18720796}{304046875}a^{6}+\frac{41316639}{243237500}a^{5}+\frac{13932769}{48647500}a^{4}+\frac{784951}{2432375}a^{3}-\frac{304223}{1945900}a^{2}-\frac{61173}{389180}a-\frac{4705}{19459}$, $\frac{1}{6080937500}a^{15}-\frac{1}{6080937500}a^{14}+\frac{1}{1216187500}a^{13}+\frac{40493}{3040468750}a^{12}+\frac{1195889}{6080937500}a^{11}+\frac{2850031}{6080937500}a^{10}+\frac{33321597}{608093750}a^{9}+\frac{1328617559}{6080937500}a^{8}+\frac{2921404941}{6080937500}a^{7}+\frac{173040217}{608093750}a^{6}+\frac{15754301}{48647500}a^{5}+\frac{224859}{48647500}a^{4}+\frac{633277}{4864750}a^{3}-\frac{930101}{1945900}a^{2}+\frac{35505}{77836}a+\frac{1139}{77836}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{8}$, which has order $8$ (assuming GRH)
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( \frac{1}{389180} a^{15} - \frac{1}{77836} a^{14} + \frac{7}{194590} a^{13} - \frac{7}{194590} a^{12} - \frac{31}{389180} a^{11} + \frac{3}{38918} a^{10} - \frac{217}{194590} a^{9} + \frac{1059}{389180} a^{8} - \frac{217}{38918} a^{7} + \frac{75}{38918} a^{6} - \frac{775}{77836} a^{5} - \frac{875}{38918} a^{4} + \frac{4375}{38918} a^{3} - \frac{15625}{77836} a^{2} - \frac{1253}{194590} a - \frac{78125}{77836} \) (order $30$) | 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: | $\frac{3024}{1520234375}a^{15}-\frac{6804}{1520234375}a^{14}+\frac{35789}{1520234375}a^{12}+\frac{756}{1520234375}a^{11}+\frac{305424}{1520234375}a^{10}-\frac{756}{2432375}a^{9}+\frac{76356}{138203125}a^{8}-\frac{4275936}{1520234375}a^{7}+\frac{756}{486475}a^{6}+\frac{76356}{12161875}a^{5}-\frac{42336}{2432375}a^{4}+\frac{756}{44225}a^{3}-\frac{76356}{486475}a^{2}+\frac{3024}{19459}a-\frac{15679}{19459}$, $\frac{2601}{1216187500}a^{15}-\frac{8201}{243237500}a^{14}+\frac{63559}{1216187500}a^{13}-\frac{28546}{304046875}a^{12}+\frac{79309}{243237500}a^{11}-\frac{101}{884500}a^{10}-\frac{714826}{304046875}a^{9}+\frac{5121579}{1216187500}a^{8}-\frac{2289829}{243237500}a^{7}+\frac{8003464}{304046875}a^{6}-\frac{969701}{22112500}a^{5}-\frac{2023559}{48647500}a^{4}+\frac{12796}{97295}a^{3}-\frac{28405}{77836}a^{2}+\frac{459531}{389180}a-\frac{203611}{77836}$, $\frac{559}{6080937500}a^{15}+\frac{38783}{3040468750}a^{14}+\frac{559}{1216187500}a^{13}-\frac{3913}{3040468750}a^{12}+\frac{3913}{3040468750}a^{11}+\frac{17329}{6080937500}a^{10}-\frac{1677}{608093750}a^{9}+\frac{121303}{3040468750}a^{8}-\frac{591981}{6080937500}a^{7}+\frac{121303}{608093750}a^{6}-\frac{1677}{24323750}a^{5}+\frac{17329}{48647500}a^{4}+\frac{3913}{4864750}a^{3}-\frac{3913}{972950}a^{2}+\frac{559}{77836}a-\frac{78395}{77836}$, $\frac{29233}{3040468750}a^{15}-\frac{7533}{3040468750}a^{14}-\frac{35217}{608093750}a^{13}-\frac{1917}{276406250}a^{12}-\frac{1035163}{3040468750}a^{11}+\frac{1987723}{3040468750}a^{10}-\frac{129573}{608093750}a^{9}-\frac{9491303}{3040468750}a^{8}-\frac{295677}{276406250}a^{7}-\frac{8715053}{608093750}a^{6}+\frac{7620353}{121618750}a^{5}-\frac{103977}{24323750}a^{4}+\frac{17009}{972950}a^{3}-\frac{135}{3538}a^{2}-\frac{160463}{194590}a+\frac{32880}{19459}$, $\frac{2923}{6080937500}a^{15}+\frac{175937}{6080937500}a^{14}-\frac{26789}{1216187500}a^{13}+\frac{131039}{3040468750}a^{12}-\frac{3634993}{6080937500}a^{11}+\frac{940453}{6080937500}a^{10}+\frac{237267}{608093750}a^{9}+\frac{15134957}{6080937500}a^{8}+\frac{24804783}{6080937500}a^{7}-\frac{6597713}{608093750}a^{6}+\frac{584003}{48647500}a^{5}-\frac{127547}{48647500}a^{4}-\frac{256559}{4864750}a^{3}-\frac{69451}{1945900}a^{2}-\frac{184153}{389180}a+\frac{5767}{7076}$, $\frac{118367}{6080937500}a^{15}-\frac{249301}{3040468750}a^{14}+\frac{103727}{608093750}a^{13}-\frac{2047763}{6080937500}a^{12}+\frac{50141}{104843750}a^{11}+\frac{3987581}{3040468750}a^{10}-\frac{7515949}{1216187500}a^{9}+\frac{47140639}{3040468750}a^{8}-\frac{120655509}{3040468750}a^{7}+\frac{85126661}{1216187500}a^{6}-\frac{429351}{24323750}a^{5}-\frac{917319}{4864750}a^{4}+\frac{6364219}{9729500}a^{3}-\frac{1645189}{972950}a^{2}+\frac{111371}{38918}a-\frac{39731}{19459}$, $\frac{12217}{1216187500}a^{15}+\frac{53237}{1216187500}a^{14}+\frac{41231}{1216187500}a^{13}+\frac{189207}{1216187500}a^{12}-\frac{759893}{1216187500}a^{11}-\frac{784567}{1216187500}a^{10}+\frac{572789}{1216187500}a^{9}+\frac{4026333}{1216187500}a^{8}+\frac{37636683}{1216187500}a^{7}+\frac{6717379}{1216187500}a^{6}-\frac{2406409}{243237500}a^{5}-\frac{7392507}{48647500}a^{4}-\frac{2352231}{9729500}a^{3}+\frac{1243101}{1945900}a^{2}+\frac{315511}{389180}a+\frac{53553}{19459}$ (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: | \( 49344.1781525 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 49344.1781525 \cdot 8}{30\cdot\sqrt{27204384060547119140625}}\cr\approx \mathstrut & 0.193786781781 \end{aligned}\] (assuming GRH)
Galois group
$C_2^2\times C_4$ (as 16T2):
An abelian group of order 16 |
The 16 conjugacy class representatives for $C_4\times C_2^2$ |
Character table for $C_4\times C_2^2$ |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{4}$ | R | R | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
\(5\) | 5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
\(19\) | 19.4.2.1 | $x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
19.4.2.1 | $x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
19.4.2.1 | $x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
19.4.2.1 | $x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |