Normalized defining polynomial
\( x^{16} - 3 x^{15} + 6 x^{14} - 21 x^{13} + 28 x^{12} + 77 x^{11} - 224 x^{10} + 25 x^{9} + 548 x^{8} + \cdots + 64 \)
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: | \(145383138964293121969\) \(\medspace = 7^{14}\cdot 11^{8}\) | 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: | \(18.20\) | 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: | $7^{7/8}11^{1/2}\approx 18.203593448567553$ | ||
Ramified primes: | \(7\), \(11\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{8}a^{8}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}+\frac{3}{8}a^{2}+\frac{1}{4}a$, $\frac{1}{8}a^{9}-\frac{1}{4}a^{5}+\frac{3}{8}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{10}-\frac{1}{4}a^{6}-\frac{1}{8}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{32}a^{11}+\frac{1}{32}a^{10}-\frac{1}{32}a^{9}-\frac{1}{16}a^{7}-\frac{3}{16}a^{6}-\frac{7}{32}a^{5}+\frac{1}{32}a^{4}-\frac{1}{32}a^{3}-\frac{5}{16}a^{2}+\frac{1}{4}a$, $\frac{1}{32}a^{12}-\frac{1}{16}a^{10}+\frac{1}{32}a^{9}-\frac{1}{16}a^{8}-\frac{1}{8}a^{7}-\frac{1}{32}a^{6}-\frac{1}{4}a^{5}-\frac{1}{16}a^{4}-\frac{9}{32}a^{3}+\frac{1}{16}a^{2}-\frac{1}{4}a$, $\frac{1}{32}a^{13}-\frac{1}{32}a^{10}+\frac{3}{32}a^{7}-\frac{1}{8}a^{6}-\frac{3}{32}a^{4}+\frac{3}{8}a^{3}-\frac{1}{2}a^{2}+\frac{1}{4}a$, $\frac{1}{1408}a^{14}+\frac{5}{352}a^{13}-\frac{21}{1408}a^{12}+\frac{1}{176}a^{11}-\frac{3}{128}a^{10}-\frac{23}{704}a^{9}-\frac{83}{1408}a^{8}+\frac{29}{704}a^{7}-\frac{289}{1408}a^{6}-\frac{141}{704}a^{5}-\frac{125}{1408}a^{4}-\frac{123}{352}a^{3}+\frac{73}{352}a^{2}+\frac{39}{88}a-\frac{7}{22}$, $\frac{1}{656425088}a^{15}+\frac{32613}{328212544}a^{14}+\frac{6978339}{656425088}a^{13}-\frac{3578363}{328212544}a^{12}+\frac{3581771}{656425088}a^{11}+\frac{4923333}{82053136}a^{10}+\frac{12606141}{656425088}a^{9}-\frac{3405891}{82053136}a^{8}-\frac{18465885}{656425088}a^{7}+\frac{16540231}{82053136}a^{6}-\frac{117719229}{656425088}a^{5}-\frac{2153629}{29837504}a^{4}-\frac{41834}{466211}a^{3}-\frac{235363}{20513284}a^{2}-\frac{8498411}{20513284}a-\frac{938172}{5128321}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$ (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: | $\frac{4030201}{656425088}a^{15}-\frac{1529479}{82053136}a^{14}+\frac{2134849}{59675008}a^{13}-\frac{648694}{5128321}a^{12}+\frac{115060491}{656425088}a^{11}+\frac{164836937}{328212544}a^{10}-\frac{915759211}{656425088}a^{9}+\frac{34745}{29837504}a^{8}+\frac{2284603823}{656425088}a^{7}-\frac{1513093817}{328212544}a^{6}+\frac{1608510279}{656425088}a^{5}-\frac{78061943}{164106272}a^{4}+\frac{230619273}{164106272}a^{3}-\frac{1454843}{932422}a^{2}+\frac{13413781}{20513284}a-\frac{4367683}{5128321}$, $\frac{2920055}{656425088}a^{15}+\frac{928967}{656425088}a^{14}+\frac{84241}{656425088}a^{13}-\frac{2717425}{59675008}a^{12}-\frac{72661291}{656425088}a^{11}+\frac{293199895}{656425088}a^{10}+\frac{268871025}{656425088}a^{9}-\frac{1031703763}{656425088}a^{8}-\frac{74272745}{656425088}a^{7}+\frac{1687737043}{656425088}a^{6}-\frac{740857157}{656425088}a^{5}-\frac{768653247}{656425088}a^{4}+\frac{22857963}{41026568}a^{3}+\frac{76654991}{164106272}a^{2}+\frac{14122565}{41026568}a+\frac{6606573}{10256642}$, $\frac{1789843}{59675008}a^{15}-\frac{1260099}{20513284}a^{14}+\frac{79823371}{656425088}a^{13}-\frac{43010647}{82053136}a^{12}+\frac{236460759}{656425088}a^{11}+\frac{78116601}{29837504}a^{10}-\frac{2670298663}{656425088}a^{9}-\frac{1047500249}{328212544}a^{8}+\frac{8151745991}{656425088}a^{7}-\frac{3671785449}{328212544}a^{6}+\frac{2680378787}{656425088}a^{5}-\frac{275804713}{82053136}a^{4}+\frac{49765989}{10256642}a^{3}-\frac{291584325}{82053136}a^{2}+\frac{17982086}{5128321}a-\frac{12471372}{5128321}$, $\frac{8673213}{656425088}a^{15}-\frac{8434149}{328212544}a^{14}+\frac{32498219}{656425088}a^{13}-\frac{75046469}{328212544}a^{12}+\frac{83970771}{656425088}a^{11}+\frac{194250117}{164106272}a^{10}-\frac{1062869883}{656425088}a^{9}-\frac{255164229}{164106272}a^{8}+\frac{3321347555}{656425088}a^{7}-\frac{758856447}{164106272}a^{6}+\frac{1337164995}{656425088}a^{5}-\frac{26501915}{29837504}a^{4}+\frac{9367571}{14918752}a^{3}-\frac{54874187}{82053136}a^{2}+\frac{11083836}{5128321}a-\frac{2248924}{5128321}$, $\frac{5576955}{328212544}a^{15}-\frac{12186039}{328212544}a^{14}+\frac{23295203}{328212544}a^{13}-\frac{95784795}{328212544}a^{12}+\frac{74938333}{328212544}a^{11}+\frac{499943303}{328212544}a^{10}-\frac{871988113}{328212544}a^{9}-\frac{595291717}{328212544}a^{8}+\frac{2771785473}{328212544}a^{7}-\frac{2331328193}{328212544}a^{6}+\frac{135375215}{328212544}a^{5}+\frac{267977285}{328212544}a^{4}+\frac{434613953}{164106272}a^{3}-\frac{18505816}{5128321}a^{2}+\frac{41962163}{20513284}a-\frac{2531129}{5128321}$, $\frac{12696833}{656425088}a^{15}-\frac{13322525}{328212544}a^{14}+\frac{58910631}{656425088}a^{13}-\frac{116139803}{328212544}a^{12}+\frac{184105719}{656425088}a^{11}+\frac{254146351}{164106272}a^{10}-\frac{1780654523}{656425088}a^{9}-\frac{180327629}{164106272}a^{8}+\frac{4760742127}{656425088}a^{7}-\frac{735056717}{82053136}a^{6}+\frac{4493463767}{656425088}a^{5}-\frac{1559956913}{328212544}a^{4}+\frac{168003551}{82053136}a^{3}-\frac{6275855}{20513284}a^{2}+\frac{12965467}{20513284}a-\frac{5140315}{5128321}$, $\frac{19072953}{656425088}a^{15}-\frac{25134781}{328212544}a^{14}+\frac{89065063}{656425088}a^{13}-\frac{16170597}{29837504}a^{12}+\frac{378901079}{656425088}a^{11}+\frac{431399475}{164106272}a^{10}-\frac{3697349203}{656425088}a^{9}-\frac{371071237}{164106272}a^{8}+\frac{10644178159}{656425088}a^{7}-\frac{664208103}{41026568}a^{6}+\frac{2012680599}{656425088}a^{5}+\frac{426039975}{328212544}a^{4}+\frac{388723587}{82053136}a^{3}-\frac{128715623}{20513284}a^{2}+\frac{86009813}{20513284}a-\frac{9193174}{5128321}$ (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)]
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Regulator: | \( 13134.7691742 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 13134.7691742 \cdot 1}{2\cdot\sqrt{145383138964293121969}}\cr\approx \mathstrut & 1.32304426783 \end{aligned}\] (assuming GRH)
Galois group
A solvable group of order 16 |
The 7 conjugacy class representatives for $D_{8}$ |
Character table for $D_{8}$ |
Intermediate fields
\(\Q(\sqrt{-11}) \), \(\Q(\sqrt{77}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-7}, \sqrt{-11})\), 4.2.41503.1 x2, 4.0.3773.1 x2, 8.0.1722499009.1, 8.2.12057493063.1 x4, 8.0.1096135733.1 x4 |
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 | ${\href{/padicField/2.2.0.1}{2} }^{8}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | R | R | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.16.14.1 | $x^{16} + 28 x^{9} + 14 x^{8} - 98 x^{2} + 196 x + 49$ | $8$ | $2$ | $14$ | $D_{8}$ | $[\ ]_{8}^{2}$ |
\(11\) | 11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |