Normalized defining polynomial
\( x^{16} - 2 x^{15} + 2 x^{14} - x^{13} - 8 x^{12} - 46 x^{11} + 93 x^{10} - 78 x^{9} + 80 x^{8} + 271 x^{7} + 434 x^{6} - 866 x^{5} - 47 x^{4} - 776 x^{3} - 960 x^{2} - 512 x + 4096 \)
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: | \(208225350937744140625\) \(\medspace = 5^{12}\cdot 31^{8}\) | 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: | \(18.62\) | 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: | $5^{3/4}31^{1/2}\approx 18.616942190179014$ | ||
Ramified primes: | \(5\), \(31\) | 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 over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{41}a^{10}+\frac{2}{41}a^{5}-\frac{3}{41}$, $\frac{1}{41}a^{11}+\frac{2}{41}a^{6}-\frac{3}{41}a$, $\frac{1}{1025}a^{12}+\frac{11}{1025}a^{11}+\frac{1}{1025}a^{10}-\frac{6}{25}a^{8}+\frac{207}{1025}a^{7}+\frac{22}{1025}a^{6}-\frac{162}{1025}a^{5}+\frac{1}{25}a^{4}+\frac{4}{25}a^{3}-\frac{44}{1025}a^{2}-\frac{64}{205}a+\frac{161}{1025}$, $\frac{1}{155800}a^{13}-\frac{29}{77900}a^{12}+\frac{421}{77900}a^{11}-\frac{1569}{155800}a^{10}-\frac{7}{475}a^{9}-\frac{21647}{77900}a^{8}-\frac{53211}{155800}a^{7}-\frac{4563}{15580}a^{6}-\frac{6532}{19475}a^{5}-\frac{173}{760}a^{4}+\frac{4809}{15580}a^{3}-\frac{38617}{77900}a^{2}-\frac{7159}{155800}a-\frac{7873}{19475}$, $\frac{1}{1246400}a^{14}-\frac{1}{623200}a^{13}-\frac{63}{623200}a^{12}-\frac{1537}{1246400}a^{11}+\frac{83}{31160}a^{10}+\frac{13973}{124640}a^{9}+\frac{46649}{249280}a^{8}-\frac{1597}{32800}a^{7}+\frac{25889}{77900}a^{6}+\frac{434639}{1246400}a^{5}+\frac{49253}{124640}a^{4}+\frac{13077}{32800}a^{3}-\frac{537583}{1246400}a^{2}+\frac{72639}{155800}a+\frac{7209}{19475}$, $\frac{1}{49856000}a^{15}-\frac{1}{4985600}a^{14}+\frac{1}{608000}a^{13}+\frac{22383}{49856000}a^{12}-\frac{2339}{389500}a^{11}-\frac{9323}{4985600}a^{10}-\frac{2571827}{49856000}a^{9}+\frac{213789}{608000}a^{8}+\frac{17478}{97375}a^{7}+\frac{1116943}{49856000}a^{6}-\frac{719431}{4985600}a^{5}+\frac{8035687}{24928000}a^{4}+\frac{38521}{1216000}a^{3}-\frac{92129}{3116000}a^{2}+\frac{367003}{779000}a+\frac{23341}{97375}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{3}$, which has order $3$
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -\frac{7}{124640} a^{15} + \frac{21}{249280} a^{14} - \frac{7}{24928} a^{13} + \frac{43}{15580} a^{12} + \frac{7}{249280} a^{11} + \frac{189}{62320} a^{10} - \frac{7}{3280} a^{9} + \frac{3017}{249280} a^{8} - \frac{15179}{124640} a^{7} + \frac{7}{124640} a^{6} - \frac{12453}{249280} a^{5} - \frac{609}{24928} a^{4} - \frac{1099}{15580} a^{3} + \frac{192809}{249280} a^{2} + \frac{336}{3895} a + \frac{896}{3895} \) (order $10$) | 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: | $\frac{89}{1994240}a^{15}-\frac{153}{997120}a^{14}+\frac{953}{997120}a^{13}-\frac{5849}{1994240}a^{12}+\frac{39}{249280}a^{11}-\frac{1759}{997120}a^{10}+\frac{10581}{1994240}a^{9}-\frac{25359}{997120}a^{8}+\frac{15961}{124640}a^{7}-\frac{4213}{398848}a^{6}+\frac{19889}{997120}a^{5}-\frac{5609}{997120}a^{4}-\frac{505303}{1994240}a^{3}-\frac{189497}{249280}a^{2}-\frac{14}{3895}a-\frac{528}{3895}$, $\frac{9477}{24928000}a^{15}-\frac{1089}{2492800}a^{14}+\frac{2917}{12464000}a^{13}+\frac{22651}{24928000}a^{12}+\frac{3387}{3116000}a^{11}-\frac{489}{26240}a^{10}+\frac{485841}{24928000}a^{9}-\frac{9233}{656000}a^{8}-\frac{43883}{1558000}a^{7}-\frac{1555509}{24928000}a^{6}+\frac{15601}{60800}a^{5}-\frac{112119}{656000}a^{4}+\frac{2267797}{24928000}a^{3}+\frac{223899}{3116000}a^{2}+\frac{45204}{97375}a-\frac{125721}{97375}$, $\frac{4369}{49856000}a^{15}+\frac{599}{4985600}a^{14}-\frac{34871}{24928000}a^{13}-\frac{63233}{49856000}a^{12}-\frac{3793}{3116000}a^{11}-\frac{34811}{4985600}a^{10}-\frac{192323}{49856000}a^{9}+\frac{1573661}{24928000}a^{8}+\frac{68067}{1558000}a^{7}+\frac{2959487}{49856000}a^{6}+\frac{672097}{4985600}a^{5}-\frac{1453497}{24928000}a^{4}-\frac{21137711}{49856000}a^{3}-\frac{36057}{389500}a^{2}-\frac{145619}{389500}a-\frac{34426}{97375}$, $\frac{7153}{24928000}a^{15}-\frac{1823}{2492800}a^{14}+\frac{10693}{12464000}a^{13}-\frac{30361}{24928000}a^{12}+\frac{7091}{6232000}a^{11}-\frac{31019}{2492800}a^{10}+\frac{31591}{1312000}a^{9}-\frac{312313}{12464000}a^{8}+\frac{140061}{3116000}a^{7}-\frac{1696801}{24928000}a^{6}+\frac{227831}{2492800}a^{5}+\frac{523691}{12464000}a^{4}-\frac{6370327}{24928000}a^{3}+\frac{488377}{6232000}a^{2}+\frac{433463}{779000}a-\frac{35729}{97375}$, $\frac{2969}{49856000}a^{15}-\frac{241}{4985600}a^{14}+\frac{32529}{24928000}a^{13}-\frac{84233}{49856000}a^{12}-\frac{2393}{3116000}a^{11}-\frac{19411}{4985600}a^{10}+\frac{192677}{49856000}a^{9}-\frac{1350139}{24928000}a^{8}+\frac{91867}{1558000}a^{7}+\frac{1684087}{49856000}a^{6}+\frac{307817}{4985600}a^{5}-\frac{3277697}{24928000}a^{4}+\frac{15251289}{49856000}a^{3}-\frac{23457}{389500}a^{2}-\frac{67219}{389500}a+\frac{85349}{97375}$, $\frac{1277}{3116000}a^{15}-\frac{489}{623200}a^{14}+\frac{991}{779000}a^{13}-\frac{1179}{3116000}a^{12}-\frac{9629}{3116000}a^{11}-\frac{7193}{311600}a^{10}+\frac{110771}{3116000}a^{9}-\frac{187109}{3116000}a^{8}+\frac{59091}{1558000}a^{7}+\frac{385371}{3116000}a^{6}+\frac{184969}{623200}a^{5}-\frac{47643}{779000}a^{4}+\frac{488267}{3116000}a^{3}-\frac{515523}{3116000}a^{2}-\frac{737779}{779000}a-\frac{56398}{97375}$, $\frac{41517}{49856000}a^{15}-\frac{13773}{4985600}a^{14}+\frac{98037}{24928000}a^{13}-\frac{218109}{49856000}a^{12}-\frac{2541}{779000}a^{11}-\frac{26179}{997120}a^{10}+\frac{5599881}{49856000}a^{9}-\frac{3785887}{24928000}a^{8}+\frac{73829}{389500}a^{7}+\frac{4265891}{49856000}a^{6}-\frac{274907}{4985600}a^{5}-\frac{14618021}{24928000}a^{4}+\frac{12545677}{49856000}a^{3}-\frac{697553}{3116000}a^{2}-\frac{53831}{41000}a+\frac{257522}{97375}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 5085.12397787 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 5085.12397787 \cdot 3}{10\cdot\sqrt{208225350937744140625}}\cr\approx \mathstrut & 0.256799956563 \end{aligned}\]
Galois group
$C_2^2:C_4$ (as 16T10):
A solvable group of order 16 |
The 10 conjugacy class representatives for $C_2^2 : C_4$ |
Character table for $C_2^2 : C_4$ |
Intermediate fields
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 | ${\href{/padicField/2.4.0.1}{4} }^{4}$ | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.1.0.1}{1} }^{16}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 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 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
\(31\) | 31.4.2.1 | $x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
31.4.2.1 | $x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
31.4.2.1 | $x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
31.4.2.1 | $x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |