Normalized defining polynomial
\( x^{16} - 3 x^{15} + 9 x^{14} - 12 x^{13} + 18 x^{12} - 9 x^{11} + 21 x^{9} - 2 x^{8} - 51 x^{7} + \cdots + 1 \)
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: | \(57585340506288295936\) \(\medspace = 2^{12}\cdot 10889^{4}\) | 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: | \(17.18\) | 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: | $2^{15/8}10889^{1/2}\approx 382.75884973591815$ | ||
Ramified primes: | \(2\), \(10889\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{29568007}a^{15}-\frac{1240636}{29568007}a^{14}+\frac{11358212}{29568007}a^{13}+\frac{307817}{29568007}a^{12}+\frac{12450269}{29568007}a^{11}-\frac{856502}{4224001}a^{10}-\frac{1941798}{4224001}a^{9}+\frac{1083811}{4224001}a^{8}+\frac{13117746}{29568007}a^{7}+\frac{11183552}{29568007}a^{6}-\frac{14655602}{29568007}a^{5}-\frac{1500523}{29568007}a^{4}-\frac{3369607}{29568007}a^{3}+\frac{362788}{4224001}a^{2}+\frac{11632270}{29568007}a-\frac{2578397}{29568007}$
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: | \( -1 \) (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: | $\frac{3933514}{4224001}a^{15}-\frac{7359589}{4224001}a^{14}+\frac{26059873}{4224001}a^{13}-\frac{15855714}{4224001}a^{12}+\frac{45517220}{4224001}a^{11}+\frac{21023008}{4224001}a^{10}+\frac{8562614}{4224001}a^{9}+\frac{87944062}{4224001}a^{8}+\frac{84459833}{4224001}a^{7}-\frac{132468756}{4224001}a^{6}+\frac{191154345}{4224001}a^{5}-\frac{97838514}{4224001}a^{4}+\frac{41516889}{4224001}a^{3}-\frac{12625651}{4224001}a^{2}+\frac{8624470}{4224001}a-\frac{599979}{4224001}$, $\frac{599979}{4224001}a^{15}+\frac{2133577}{4224001}a^{14}-\frac{1959778}{4224001}a^{13}+\frac{18860125}{4224001}a^{12}-\frac{5056092}{4224001}a^{11}+\frac{40117409}{4224001}a^{10}+\frac{21023008}{4224001}a^{9}+\frac{21162173}{4224001}a^{8}+\frac{86744104}{4224001}a^{7}+\frac{53860904}{4224001}a^{6}-\frac{77270688}{4224001}a^{5}+\frac{139556151}{4224001}a^{4}-\frac{65439648}{4224001}a^{3}+\frac{25317456}{4224001}a^{2}-\frac{4825924}{4224001}a+\frac{5624575}{4224001}$, $\frac{39822045}{29568007}a^{15}-\frac{71084460}{29568007}a^{14}+\frac{251903315}{29568007}a^{13}-\frac{135804232}{29568007}a^{12}+\frac{427676483}{29568007}a^{11}+\frac{31752146}{4224001}a^{10}+\frac{11289520}{4224001}a^{9}+\frac{113667823}{4224001}a^{8}+\frac{892162375}{29568007}a^{7}-\frac{1351294146}{29568007}a^{6}+\frac{1567180925}{29568007}a^{5}-\frac{766555438}{29568007}a^{4}+\frac{444260980}{29568007}a^{3}-\frac{39966757}{4224001}a^{2}+\frac{132256197}{29568007}a-\frac{8157861}{29568007}$, $\frac{34270913}{4224001}a^{15}-\frac{79058939}{4224001}a^{14}+\frac{249982158}{4224001}a^{13}-\frac{230630569}{4224001}a^{12}+\frac{432711972}{4224001}a^{11}+\frac{8391586}{4224001}a^{10}-\frac{33937250}{4224001}a^{9}+\frac{682032691}{4224001}a^{8}+\frac{405716974}{4224001}a^{7}-\frac{1538649763}{4224001}a^{6}+\frac{2020298278}{4224001}a^{5}-\frac{1397941521}{4224001}a^{4}+\frac{712661870}{4224001}a^{3}-\frac{342274024}{4224001}a^{2}+\frac{169313557}{4224001}a-\frac{34570962}{4224001}$, $\frac{97266301}{29568007}a^{15}-\frac{248023302}{29568007}a^{14}+\frac{739683821}{29568007}a^{13}-\frac{786433388}{29568007}a^{12}+\frac{1242320241}{29568007}a^{11}-\frac{30352391}{4224001}a^{10}-\frac{47587321}{4224001}a^{9}+\frac{257260197}{4224001}a^{8}+\frac{664558652}{29568007}a^{7}-\frac{5082380312}{29568007}a^{6}+\frac{6250808623}{29568007}a^{5}-\frac{4502007864}{29568007}a^{4}+\frac{2273166955}{29568007}a^{3}-\frac{160394789}{4224001}a^{2}+\frac{547436485}{29568007}a-\frac{152478645}{29568007}$, $\frac{26081352}{29568007}a^{15}-\frac{70575506}{29568007}a^{14}+\frac{216602702}{29568007}a^{13}-\frac{258276112}{29568007}a^{12}+\frac{417680841}{29568007}a^{11}-\frac{20606185}{4224001}a^{10}-\frac{1161146}{4224001}a^{9}+\frac{76764433}{4224001}a^{8}+\frac{69108285}{29568007}a^{7}-\frac{1235739513}{29568007}a^{6}+\frac{2022870484}{29568007}a^{5}-\frac{1849490463}{29568007}a^{4}+\frac{1169641457}{29568007}a^{3}-\frac{73286698}{4224001}a^{2}+\frac{131484910}{29568007}a-\frac{23896280}{29568007}$, $\frac{18074649}{4224001}a^{15}-\frac{42768049}{4224001}a^{14}+\frac{135397431}{4224001}a^{13}-\frac{131869958}{4224001}a^{12}+\frac{242487499}{4224001}a^{11}-\frac{14781574}{4224001}a^{10}-\frac{8741162}{4224001}a^{9}+\frac{363259801}{4224001}a^{8}+\frac{185817901}{4224001}a^{7}-\frac{807777756}{4224001}a^{6}+\frac{1127974406}{4224001}a^{5}-\frac{860632840}{4224001}a^{4}+\frac{464673834}{4224001}a^{3}-\frac{222280847}{4224001}a^{2}+\frac{105116362}{4224001}a-\frac{27644608}{4224001}$ | 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: | \( 2284.86538364 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 2284.86538364 \cdot 2}{2\cdot\sqrt{57585340506288295936}}\cr\approx \mathstrut & 0.731380867568 \end{aligned}\]
Galois group
$C_2\wr C_2^3.S_4$ (as 16T1852):
A solvable group of order 49152 |
The 116 conjugacy class representatives for $C_2\wr C_2^3.S_4$ |
Character table for $C_2\wr C_2^3.S_4$ |
Intermediate fields
4.4.10889.1, 8.2.474281284.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 16 siblings: | data not computed |
Degree 32 siblings: | data not computed |
Minimal sibling: | 16.6.921365448100612734976.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{3}{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.4.0.1}{4} }$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.6.8 | $x^{4} + 2 x^{3} + 2$ | $4$ | $1$ | $6$ | $D_{4}$ | $[2, 2]^{2}$ |
2.6.6.1 | $x^{6} + 6 x^{5} + 34 x^{4} + 80 x^{3} + 204 x^{2} + 216 x + 216$ | $2$ | $3$ | $6$ | $A_4$ | $[2, 2]^{3}$ | |
2.6.0.1 | $x^{6} + x^{4} + x^{3} + x + 1$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
\(10889\) | $\Q_{10889}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{10889}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{10889}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{10889}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{10889}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{10889}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{10889}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{10889}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $8$ | $2$ | $4$ | $4$ |