Normalized defining polynomial
\( x^{16} - 4 x^{14} - 4 x^{13} + 8 x^{12} + 56 x^{11} - 2 x^{10} - 124 x^{9} + 333 x^{8} + 472 x^{7} + \cdots + 4 \)
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: | \(29960650073923649536\) \(\medspace = 2^{32}\cdot 17^{8}\) | 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: | \(16.49\) | 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: | $2^{2}17^{1/2}\approx 16.492422502470642$ | ||
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: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $16$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{6}a^{8}-\frac{1}{2}a^{4}+\frac{1}{3}$, $\frac{1}{6}a^{9}-\frac{1}{2}a^{5}+\frac{1}{3}a$, $\frac{1}{6}a^{10}-\frac{1}{2}a^{6}+\frac{1}{3}a^{2}$, $\frac{1}{18}a^{11}+\frac{1}{18}a^{8}+\frac{1}{18}a^{7}-\frac{2}{9}a^{6}-\frac{1}{9}a^{5}-\frac{7}{18}a^{4}+\frac{1}{9}a^{2}+\frac{2}{9}a+\frac{2}{9}$, $\frac{1}{18}a^{12}+\frac{1}{18}a^{9}+\frac{1}{18}a^{8}+\frac{1}{9}a^{7}-\frac{4}{9}a^{6}-\frac{1}{18}a^{5}-\frac{1}{3}a^{4}+\frac{4}{9}a^{3}-\frac{1}{9}a^{2}-\frac{4}{9}a-\frac{1}{3}$, $\frac{1}{18}a^{13}+\frac{1}{18}a^{10}+\frac{1}{18}a^{9}-\frac{1}{18}a^{8}-\frac{1}{9}a^{7}-\frac{7}{18}a^{6}-\frac{7}{18}a^{4}+\frac{2}{9}a^{3}+\frac{2}{9}a^{2}+\frac{1}{3}$, $\frac{1}{13878}a^{14}-\frac{2}{2313}a^{13}+\frac{277}{13878}a^{12}-\frac{173}{6939}a^{11}-\frac{1067}{13878}a^{10}+\frac{154}{2313}a^{9}+\frac{5}{4626}a^{8}-\frac{80}{6939}a^{7}+\frac{325}{771}a^{6}-\frac{1877}{6939}a^{5}-\frac{866}{2313}a^{4}+\frac{211}{2313}a^{3}+\frac{328}{771}a^{2}+\frac{3295}{6939}a-\frac{2797}{6939}$, $\frac{1}{24716718}a^{15}+\frac{6}{457717}a^{14}+\frac{104135}{12358359}a^{13}+\frac{164326}{12358359}a^{12}+\frac{264002}{12358359}a^{11}+\frac{127241}{4119453}a^{10}+\frac{44281}{915434}a^{9}-\frac{1904887}{24716718}a^{8}-\frac{356471}{2746302}a^{7}+\frac{3411115}{12358359}a^{6}+\frac{3115939}{8238906}a^{5}-\frac{2632133}{8238906}a^{4}+\frac{900962}{4119453}a^{3}-\frac{859088}{12358359}a^{2}-\frac{2244901}{12358359}a+\frac{1913384}{4119453}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$, $3$ |
Class group and class number
$C_{2}$, which has order $2$
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( \frac{4165333}{8238906} a^{15} + \frac{4182689}{24716718} a^{14} - \frac{16273721}{8238906} a^{13} - \frac{33173138}{12358359} a^{12} + \frac{78711025}{24716718} a^{11} + \frac{727356299}{24716718} a^{10} + \frac{36091781}{4119453} a^{9} - \frac{248511200}{4119453} a^{8} + \frac{1830340970}{12358359} a^{7} + \frac{1192383782}{4119453} a^{6} - \frac{4043947757}{24716718} a^{5} - \frac{529306745}{4119453} a^{4} + \frac{876322771}{4119453} a^{3} - \frac{521443820}{4119453} a^{2} + \frac{514763285}{12358359} a - \frac{64830371}{12358359} \) (order $8$) | 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: | $\frac{33504125}{24716718}a^{15}+\frac{11909057}{12358359}a^{14}-\frac{122591887}{24716718}a^{13}-\frac{112456808}{12358359}a^{12}+\frac{64279985}{12358359}a^{11}+\frac{2004052285}{24716718}a^{10}+\frac{444941801}{8238906}a^{9}-\frac{3535122329}{24716718}a^{8}+\frac{8434859393}{24716718}a^{7}+\frac{22407068419}{24716718}a^{6}-\frac{1279408772}{12358359}a^{5}-\frac{3343960865}{8238906}a^{4}+\frac{1688514293}{4119453}a^{3}-\frac{2243954836}{12358359}a^{2}+\frac{410894855}{12358359}a-\frac{401989}{12358359}$, $\frac{12322774}{12358359}a^{15}-\frac{660097}{12358359}a^{14}-\frac{103738273}{24716718}a^{13}-\frac{48499070}{12358359}a^{12}+\frac{221149519}{24716718}a^{11}+\frac{702184451}{12358359}a^{10}-\frac{47469547}{8238906}a^{9}-\frac{3361217585}{24716718}a^{8}+\frac{8162054315}{24716718}a^{7}+\frac{5864388911}{12358359}a^{6}-\frac{7243063079}{12358359}a^{5}-\frac{2040925781}{8238906}a^{4}+\frac{710454077}{1373151}a^{3}-\frac{4511450014}{12358359}a^{2}+\frac{1631249861}{12358359}a-\frac{255034627}{12358359}$, $\frac{75167533}{24716718}a^{15}+\frac{10735745}{12358359}a^{14}-\frac{148308958}{12358359}a^{13}-\frac{42977347}{2746302}a^{12}+\frac{27696428}{1373151}a^{11}+\frac{2182701959}{12358359}a^{10}+\frac{120906907}{2746302}a^{9}-\frac{4567025276}{12358359}a^{8}+\frac{22343715155}{24716718}a^{7}+\frac{21042011257}{12358359}a^{6}-\frac{26848941325}{24716718}a^{5}-\frac{6419782879}{8238906}a^{4}+\frac{5358657073}{4119453}a^{3}-\frac{10139783762}{12358359}a^{2}+\frac{359773259}{1373151}a-\frac{412617883}{12358359}$, $\frac{6527341}{2746302}a^{15}+\frac{4312604}{4119453}a^{14}-\frac{12547642}{1373151}a^{13}-\frac{18667351}{1373151}a^{12}+\frac{55042274}{4119453}a^{11}+\frac{1150079641}{8238906}a^{10}+\frac{464366063}{8238906}a^{9}-\frac{2268578411}{8238906}a^{8}+\frac{5492663843}{8238906}a^{7}+\frac{1304633273}{915434}a^{6}-\frac{5008767997}{8238906}a^{5}-\frac{5370130751}{8238906}a^{4}+\frac{3728988638}{4119453}a^{3}-\frac{722402621}{1373151}a^{2}+\frac{67472789}{457717}a-\frac{7045548}{457717}$, $\frac{6967019}{24716718}a^{15}+\frac{1273369}{24716718}a^{14}-\frac{27991621}{24716718}a^{13}-\frac{16465940}{12358359}a^{12}+\frac{51305941}{24716718}a^{11}+\frac{400806319}{24716718}a^{10}+\frac{1031788}{457717}a^{9}-\frac{437173369}{12358359}a^{8}+\frac{1082544760}{12358359}a^{7}+\frac{1868511167}{12358359}a^{6}-\frac{2990944399}{24716718}a^{5}-\frac{277101587}{4119453}a^{4}+\frac{189739322}{1373151}a^{3}-\frac{1016830306}{12358359}a^{2}+\frac{307983821}{12358359}a-\frac{43553353}{12358359}$, $\frac{6559246}{12358359}a^{15}+\frac{4150799}{24716718}a^{14}-\frac{51231769}{24716718}a^{13}-\frac{3812174}{1373151}a^{12}+\frac{9255347}{2746302}a^{11}+\frac{380531893}{12358359}a^{10}+\frac{11914441}{1373151}a^{9}-\frac{1561610423}{24716718}a^{8}+\frac{3877240183}{24716718}a^{7}+\frac{7414926769}{24716718}a^{6}-\frac{4372909613}{24716718}a^{5}-\frac{351671375}{2746302}a^{4}+\frac{920136772}{4119453}a^{3}-\frac{1786425991}{12358359}a^{2}+\frac{184650020}{4119453}a-\frac{69916385}{12358359}$, $\frac{10356263}{12358359}a^{15}+\frac{7795993}{24716718}a^{14}-\frac{80723327}{24716718}a^{13}-\frac{6329503}{1373151}a^{12}+\frac{20978306}{4119453}a^{11}+\frac{1213184503}{24716718}a^{10}+\frac{68673080}{4119453}a^{9}-\frac{1231167524}{12358359}a^{8}+\frac{2965630552}{12358359}a^{7}+\frac{6050469313}{12358359}a^{6}-\frac{6200402773}{24716718}a^{5}-\frac{957444542}{4119453}a^{4}+\frac{1367794445}{4119453}a^{3}-\frac{2442223304}{12358359}a^{2}+\frac{239687569}{4119453}a-\frac{78203509}{12358359}$ | 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: | \( 5277.21342233 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 5277.21342233 \cdot 2}{8\cdot\sqrt{29960650073923649536}}\cr\approx \mathstrut & 0.585474518060 \end{aligned}\]
Galois group
$C_2\times D_4$ (as 16T9):
A solvable group of order 16 |
The 10 conjugacy class representatives for $D_4\times C_2$ |
Character table for $D_4\times C_2$ |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 8 siblings: | 8.0.18939904.3, 8.4.1368408064.2, 8.0.5473632256.4, 8.0.342102016.3 |
Minimal sibling: | 8.0.18939904.3 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.2.0.1}{2} }^{8}$ | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.16.6 | $x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$ | $4$ | $2$ | $16$ | $C_2^3$ | $[2, 3]^{2}$ |
2.8.16.6 | $x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$ | $4$ | $2$ | $16$ | $C_2^3$ | $[2, 3]^{2}$ | |
\(17\) | 17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |