Normalized defining polynomial
\( x^{16} - 6x^{14} + 54x^{12} - 64x^{10} + 587x^{8} - 240x^{6} + 5870x^{4} - 6030x^{2} + 2209 \)
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: | \(2824911165797606216433664\) \(\medspace = 2^{24}\cdot 17^{14}\) | 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: | \(33.74\) | 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: | $2^{2}17^{7/8}\approx 47.72025959461256$ | ||
Ramified primes: | \(2\), \(17\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $8$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is a CM field. | |||
Reflex fields: | unavailable$^{128}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{8}a^{12}-\frac{1}{4}a^{11}+\frac{1}{8}a^{10}+\frac{1}{4}a^{7}+\frac{3}{8}a^{6}-\frac{1}{2}a^{4}-\frac{1}{4}a^{3}-\frac{3}{8}a^{2}+\frac{1}{8}$, $\frac{1}{8}a^{13}+\frac{1}{8}a^{11}-\frac{1}{4}a^{10}-\frac{1}{4}a^{8}+\frac{3}{8}a^{7}+\frac{1}{4}a^{6}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}-\frac{3}{8}a^{3}-\frac{1}{4}a^{2}+\frac{1}{8}a-\frac{1}{4}$, $\frac{1}{4291187787688}a^{14}-\frac{7322056951}{536398473461}a^{12}-\frac{955173450889}{4291187787688}a^{10}-\frac{1}{4}a^{9}+\frac{9267665395}{4291187787688}a^{8}+\frac{77406554401}{4291187787688}a^{6}+\frac{1}{4}a^{5}-\frac{701343469655}{4291187787688}a^{4}-\frac{45389648923}{1072796946922}a^{2}-\frac{1}{4}a+\frac{1729605644679}{4291187787688}$, $\frac{1}{201685826021336}a^{15}-\frac{7322056951}{25210728252667}a^{13}+\frac{35519922744459}{201685826021336}a^{11}-\frac{1}{4}a^{10}-\frac{27883452954577}{201685826021336}a^{9}+\frac{58008441688189}{201685826021336}a^{7}+\frac{1}{4}a^{6}+\frac{40064940513381}{201685826021336}a^{5}+\frac{10437075408028}{25210728252667}a^{3}-\frac{1}{4}a^{2}-\frac{94822119578301}{201685826021336}a$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{2}\times C_{2}$, which has order $4$ (assuming GRH)
Unit group
Rank: | $7$ | 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: | $\frac{6796959}{1072796946922}a^{14}-\frac{103234411}{536398473461}a^{12}+\frac{1011430602}{536398473461}a^{10}-\frac{6507679365}{536398473461}a^{8}+\frac{25954138702}{536398473461}a^{6}-\frac{36299415895}{536398473461}a^{4}+\frac{43005377887}{1072796946922}a^{2}+\frac{256637116025}{536398473461}$, $\frac{17432943}{1072796946922}a^{14}-\frac{140245485}{536398473461}a^{12}+\frac{599295837}{1072796946922}a^{10}-\frac{616936127}{536398473461}a^{8}-\frac{25768487089}{1072796946922}a^{6}-\frac{28059050816}{536398473461}a^{4}+\frac{37682957508}{536398473461}a^{2}+\frac{169310123999}{536398473461}$, $\frac{5317992}{536398473461}a^{14}-\frac{37011074}{536398473461}a^{12}-\frac{1423565367}{1072796946922}a^{10}+\frac{5890743238}{536398473461}a^{8}-\frac{77676764493}{1072796946922}a^{6}+\frac{8240365079}{536398473461}a^{4}+\frac{32360537129}{1072796946922}a^{2}-\frac{87326992026}{536398473461}$, $\frac{16860250517}{100842913010668}a^{15}-\frac{6796959}{1072796946922}a^{14}-\frac{99041826555}{100842913010668}a^{13}+\frac{103234411}{536398473461}a^{12}+\frac{462456654673}{50421456505334}a^{11}-\frac{1011430602}{536398473461}a^{10}-\frac{1182051431567}{100842913010668}a^{9}+\frac{6507679365}{536398473461}a^{8}+\frac{5659215695737}{50421456505334}a^{7}-\frac{25954138702}{536398473461}a^{6}-\frac{11084240202805}{100842913010668}a^{5}+\frac{36299415895}{536398473461}a^{4}+\frac{127672347492107}{100842913010668}a^{3}-\frac{43005377887}{1072796946922}a^{2}-\frac{100870034056993}{50421456505334}a-\frac{256637116025}{536398473461}$, $\frac{50229816175}{100842913010668}a^{15}+\frac{93399731}{1072796946922}a^{14}-\frac{80699645880}{25210728252667}a^{13}-\frac{568802527}{536398473461}a^{12}+\frac{2719206101853}{100842913010668}a^{11}+\frac{6680299995}{1072796946922}a^{10}-\frac{3885273045389}{100842913010668}a^{9}-\frac{13050474818}{536398473461}a^{8}+\frac{26791589475091}{100842913010668}a^{7}+\frac{13074453689}{1072796946922}a^{6}-\frac{30449170740267}{100842913010668}a^{5}-\frac{161703577904}{536398473461}a^{4}+\frac{66883487817813}{25210728252667}a^{3}+\frac{165064626509}{536398473461}a^{2}-\frac{415224650199637}{100842913010668}a-\frac{1159223809196}{536398473461}$, $\frac{50229816175}{100842913010668}a^{15}-\frac{17432943}{1072796946922}a^{14}-\frac{80699645880}{25210728252667}a^{13}+\frac{140245485}{536398473461}a^{12}+\frac{2719206101853}{100842913010668}a^{11}-\frac{599295837}{1072796946922}a^{10}-\frac{3885273045389}{100842913010668}a^{9}+\frac{616936127}{536398473461}a^{8}+\frac{26791589475091}{100842913010668}a^{7}+\frac{25768487089}{1072796946922}a^{6}-\frac{30449170740267}{100842913010668}a^{5}+\frac{28059050816}{536398473461}a^{4}+\frac{66883487817813}{25210728252667}a^{3}-\frac{37682957508}{536398473461}a^{2}-\frac{415224650199637}{100842913010668}a-\frac{1242107070921}{536398473461}$, $\frac{16860250517}{100842913010668}a^{15}+\frac{82763747}{1072796946922}a^{14}-\frac{99041826555}{100842913010668}a^{13}-\frac{531791453}{536398473461}a^{12}+\frac{462456654673}{50421456505334}a^{11}+\frac{4051932681}{536398473461}a^{10}-\frac{1182051431567}{100842913010668}a^{9}-\frac{18941218056}{536398473461}a^{8}+\frac{5659215695737}{50421456505334}a^{7}+\frac{45375609091}{536398473461}a^{6}-\frac{11084240202805}{100842913010668}a^{5}-\frac{169943942983}{536398473461}a^{4}+\frac{127672347492107}{100842913010668}a^{3}+\frac{297768715889}{1072796946922}a^{2}-\frac{100870034056993}{50421456505334}a-\frac{535498343709}{536398473461}$ (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: | \( 131202.513168 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 131202.513168 \cdot 4}{2\cdot\sqrt{2824911165797606216433664}}\cr\approx \mathstrut & 0.379235250506 \end{aligned}\] (assuming GRH)
Galois group
$C_2^2:C_8$ (as 16T24):
A solvable group of order 32 |
The 20 conjugacy class representatives for $C_2^2 : C_8$ |
Character table for $C_2^2 : C_8$ |
Intermediate fields
\(\Q(\sqrt{17}) \), 4.4.4913.1, 4.0.39304.1, 4.0.2312.1, 8.0.105046700288.1, 8.8.420186801152.1, 8.0.1544804416.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 32 |
Degree 16 sibling: | deg 16 |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.1.0.1}{1} }^{16}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.4.2 | $x^{4} + 4 x^{3} + 4 x^{2} + 12$ | $2$ | $2$ | $4$ | $C_4$ | $[2]^{2}$ |
2.4.4.2 | $x^{4} + 4 x^{3} + 4 x^{2} + 12$ | $2$ | $2$ | $4$ | $C_4$ | $[2]^{2}$ | |
2.8.16.3 | $x^{8} + 8 x^{7} + 16 x^{6} + 8 x^{5} + 36 x^{4} - 32 x^{3} + 88 x^{2} - 32 x + 124$ | $4$ | $2$ | $16$ | $C_4\times C_2$ | $[2, 3]^{2}$ | |
\(17\) | 17.16.14.1 | $x^{16} + 128 x^{15} + 7192 x^{14} + 232064 x^{13} + 4716796 x^{12} + 62185088 x^{11} + 525781480 x^{10} + 2696730752 x^{9} + 7365142088 x^{8} + 8090194432 x^{7} + 4732152320 x^{6} + 1682759680 x^{5} + 456414056 x^{4} + 996830464 x^{3} + 7439529968 x^{2} + 33582546688 x + 66368009604$ | $8$ | $2$ | $14$ | $C_8\times C_2$ | $[\ ]_{8}^{2}$ |