Normalized defining polynomial
\( x^{16} - 4 x^{15} + 2 x^{14} - 6 x^{13} + 86 x^{12} - 218 x^{11} + 226 x^{10} - 288 x^{9} + 724 x^{8} - 930 x^{7} + 340 x^{6} + 248 x^{5} - 223 x^{4} + 30 x^{3} + 20 x^{2} - 8 x + 1 \)
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: | \(9440732714731831296\) \(\medspace = 2^{24}\cdot 3^{14}\cdot 7^{6}\) | 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: | \(15.34\) | 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: | $2^{3/2}3^{7/8}7^{1/2}\approx 19.5692919025805$ | ||
Ramified primes: | \(2\), \(3\), \(7\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{5}a^{11}-\frac{2}{5}a^{10}+\frac{1}{5}a^{9}+\frac{2}{5}a^{7}+\frac{1}{5}a^{6}+\frac{1}{5}a^{4}+\frac{2}{5}a^{3}+\frac{2}{5}a^{2}+\frac{1}{5}a-\frac{1}{5}$, $\frac{1}{5}a^{12}+\frac{2}{5}a^{10}+\frac{2}{5}a^{9}+\frac{2}{5}a^{8}+\frac{2}{5}a^{6}+\frac{1}{5}a^{5}-\frac{1}{5}a^{4}+\frac{1}{5}a^{3}+\frac{1}{5}a-\frac{2}{5}$, $\frac{1}{5}a^{13}+\frac{1}{5}a^{10}-\frac{2}{5}a^{7}-\frac{1}{5}a^{6}-\frac{1}{5}a^{5}-\frac{1}{5}a^{4}+\frac{1}{5}a^{3}+\frac{2}{5}a^{2}+\frac{1}{5}a+\frac{2}{5}$, $\frac{1}{25}a^{14}+\frac{2}{25}a^{13}+\frac{1}{25}a^{11}+\frac{2}{25}a^{10}-\frac{12}{25}a^{8}+\frac{1}{5}a^{7}-\frac{8}{25}a^{6}-\frac{3}{25}a^{5}-\frac{11}{25}a^{4}-\frac{1}{25}a^{3}+\frac{4}{25}a+\frac{4}{25}$, $\frac{1}{27649375}a^{15}-\frac{63461}{27649375}a^{14}+\frac{1295679}{27649375}a^{13}-\frac{1720809}{27649375}a^{12}-\frac{1064826}{27649375}a^{11}-\frac{339172}{2126875}a^{10}+\frac{1803703}{27649375}a^{9}+\frac{5301066}{27649375}a^{8}+\frac{3611562}{27649375}a^{7}-\frac{4281139}{27649375}a^{6}+\frac{7598613}{27649375}a^{5}+\frac{8855357}{27649375}a^{4}+\frac{5978253}{27649375}a^{3}-\frac{7045716}{27649375}a^{2}+\frac{4076607}{27649375}a-\frac{12757657}{27649375}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
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: | \( \frac{81910604}{27649375} a^{15} - \frac{290760919}{27649375} a^{14} + \frac{27288541}{27649375} a^{13} - \frac{458947636}{27649375} a^{12} + \frac{6832608871}{27649375} a^{11} - \frac{1133996663}{2126875} a^{10} + \frac{11404450362}{27649375} a^{9} - \frac{17403695936}{27649375} a^{8} + \frac{50465170823}{27649375} a^{7} - \frac{51895498531}{27649375} a^{6} + \frac{690056552}{27649375} a^{5} + \frac{24877512228}{27649375} a^{4} - \frac{8532347713}{27649375} a^{3} - \frac{1926095839}{27649375} a^{2} + \frac{1210380228}{27649375} a - \frac{175251103}{27649375} \) (order $12$) | 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: | $\frac{74129726}{27649375}a^{15}-\frac{259941161}{27649375}a^{14}+\frac{22174879}{27649375}a^{13}-\frac{435327209}{27649375}a^{12}+\frac{6143576224}{27649375}a^{11}-\frac{1011386522}{2126875}a^{10}+\frac{10396441628}{27649375}a^{9}-\frac{16134636759}{27649375}a^{8}+\frac{45210026637}{27649375}a^{7}-\frac{46521591364}{27649375}a^{6}+\frac{2216244713}{27649375}a^{5}+\frac{21239026282}{27649375}a^{4}-\frac{8514255972}{27649375}a^{3}-\frac{1198643566}{27649375}a^{2}+\frac{1268153032}{27649375}a-\frac{220965132}{27649375}$, $\frac{30432676}{27649375}a^{15}-\frac{93123311}{27649375}a^{14}-\frac{38900971}{27649375}a^{13}-\frac{176561384}{27649375}a^{12}+\frac{2446122449}{27649375}a^{11}-\frac{327242722}{2126875}a^{10}+\frac{1864681603}{27649375}a^{9}-\frac{4850388409}{27649375}a^{8}+\frac{15600905287}{27649375}a^{7}-\frac{10547805689}{27649375}a^{6}-\frac{7401049712}{27649375}a^{5}+\frac{8763719007}{27649375}a^{4}-\frac{400557422}{27649375}a^{3}-\frac{1118555141}{27649375}a^{2}+\frac{302021007}{27649375}a-\frac{18116582}{27649375}$, $\frac{87228112}{27649375}a^{15}-\frac{288938132}{27649375}a^{14}-\frac{35270077}{27649375}a^{13}-\frac{507460983}{27649375}a^{12}+\frac{7144112613}{27649375}a^{11}-\frac{1079612514}{2126875}a^{10}+\frac{9133588111}{27649375}a^{9}-\frac{16763973108}{27649375}a^{8}+\frac{49779951944}{27649375}a^{7}-\frac{44079205818}{27649375}a^{6}-\frac{8011458469}{27649375}a^{5}+\frac{24204991734}{27649375}a^{4}-\frac{4675374039}{27649375}a^{3}-\frac{2508270317}{27649375}a^{2}+\frac{837084359}{27649375}a-\frac{95022334}{27649375}$, $a-1$, $\frac{28607393}{27649375}a^{15}-\frac{99766323}{27649375}a^{14}+\frac{6034672}{27649375}a^{13}-\frac{171087687}{27649375}a^{12}+\frac{2380167982}{27649375}a^{11}-\frac{385623321}{2126875}a^{10}+\frac{3914222279}{27649375}a^{9}-\frac{6432352137}{27649375}a^{8}+\frac{17754582991}{27649375}a^{7}-\frac{17748159552}{27649375}a^{6}+\frac{1216415359}{27649375}a^{5}+\frac{6385929326}{27649375}a^{4}-\frac{2031535421}{27649375}a^{3}-\frac{306632513}{27649375}a^{2}+\frac{170046076}{27649375}a-\frac{30432676}{27649375}$, $\frac{127958729}{27649375}a^{15}-\frac{460909894}{27649375}a^{14}+\frac{72462341}{27649375}a^{13}-\frac{734764886}{27649375}a^{12}+\frac{10697392896}{27649375}a^{11}-\frac{1818000688}{2126875}a^{10}+\frac{19483848862}{27649375}a^{9}-\frac{28754076486}{27649375}a^{8}+\frac{80435689073}{27649375}a^{7}-\frac{86271731231}{27649375}a^{6}+\frac{8056461977}{27649375}a^{5}+\frac{37670925703}{27649375}a^{4}-\frac{16577194488}{27649375}a^{3}-\frac{1807856964}{27649375}a^{2}+\frac{2330939953}{27649375}a-\frac{419555753}{27649375}$, $\frac{6140834}{2126875}a^{15}-\frac{21682724}{2126875}a^{14}+\frac{1736911}{2126875}a^{13}-\frac{34537831}{2126875}a^{12}+\frac{510969241}{2126875}a^{11}-\frac{1096166699}{2126875}a^{10}+\frac{841587427}{2126875}a^{9}-\frac{1290817831}{2126875}a^{8}+\frac{3740795958}{2126875}a^{7}-\frac{3819596426}{2126875}a^{6}-\frac{85508}{2126875}a^{5}+\frac{1919799363}{2126875}a^{4}-\frac{707898748}{2126875}a^{3}-\frac{129774519}{2126875}a^{2}+\frac{110372488}{2126875}a-\frac{17625063}{2126875}$ | 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: | \( 2306.51987946 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 2306.51987946 \cdot 2}{12\cdot\sqrt{9440732714731831296}}\cr\approx \mathstrut & 0.303907940522 \end{aligned}\]
Galois group
A solvable group of order 32 |
The 11 conjugacy class representatives for $D_8:C_2$ |
Character table for $D_8:C_2$ |
Intermediate fields
\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), 4.0.3024.2, 4.0.189.1, \(\Q(\zeta_{12})\), 8.0.9144576.3 |
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 | R | R | ${\href{/padicField/5.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{6}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | Deg $16$ | $4$ | $4$ | $24$ | |||
\(3\) | 3.16.14.1 | $x^{16} + 16 x^{15} + 128 x^{14} + 672 x^{13} + 2576 x^{12} + 7616 x^{11} + 17920 x^{10} + 34182 x^{9} + 53410 x^{8} + 68544 x^{7} + 71344 x^{6} + 57904 x^{5} + 34832 x^{4} + 16128 x^{3} + 7241 x^{2} + 2966 x + 634$ | $8$ | $2$ | $14$ | $QD_{16}$ | $[\ ]_{8}^{2}$ |
\(7\) | 7.4.0.1 | $x^{4} + 5 x^{2} + 4 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
7.4.2.2 | $x^{4} - 42 x^{2} + 147$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
7.8.4.1 | $x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |