Normalized defining polynomial
\( x^{16} + 30x^{14} + 364x^{12} + 2288x^{10} + 7920x^{8} + 14784x^{6} + 13440x^{4} + 4608x^{2} + 256 \)
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: | $2^{3/2}17^{7/8}\approx 33.74331915933294$ | ||
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 and abelian over $\Q$. | |||
Conductor: | \(136=2^{3}\cdot 17\) | ||
Dirichlet character group: | $\lbrace$$\chi_{136}(1,·)$, $\chi_{136}(83,·)$, $\chi_{136}(67,·)$, $\chi_{136}(9,·)$, $\chi_{136}(81,·)$, $\chi_{136}(19,·)$, $\chi_{136}(25,·)$, $\chi_{136}(89,·)$, $\chi_{136}(33,·)$, $\chi_{136}(35,·)$, $\chi_{136}(43,·)$, $\chi_{136}(49,·)$, $\chi_{136}(115,·)$, $\chi_{136}(59,·)$, $\chi_{136}(121,·)$, $\chi_{136}(123,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{128}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{4}a^{5}$, $\frac{1}{8}a^{6}$, $\frac{1}{8}a^{7}$, $\frac{1}{16}a^{8}$, $\frac{1}{16}a^{9}$, $\frac{1}{32}a^{10}$, $\frac{1}{32}a^{11}$, $\frac{1}{64}a^{12}$, $\frac{1}{64}a^{13}$, $\frac{1}{128}a^{14}$, $\frac{1}{128}a^{15}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{3}\times C_{24}$, which has order $72$ (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: | $\frac{1}{8}a^{6}+\frac{3}{2}a^{4}+\frac{9}{2}a^{2}+2$, $\frac{1}{128}a^{14}+\frac{13}{64}a^{12}+\frac{33}{16}a^{10}+\frac{83}{8}a^{8}+\frac{217}{8}a^{6}+\frac{141}{4}a^{4}+\frac{39}{2}a^{2}+3$, $\frac{1}{128}a^{14}+\frac{13}{64}a^{12}+\frac{65}{32}a^{10}+\frac{155}{16}a^{8}+\frac{175}{8}a^{6}+\frac{77}{4}a^{4}+3a^{2}-1$, $\frac{1}{64}a^{12}+\frac{11}{32}a^{10}+\frac{45}{16}a^{8}+\frac{21}{2}a^{6}+\frac{35}{2}a^{4}+11a^{2}+2$, $\frac{1}{16}a^{8}+a^{6}+5a^{4}+8a^{2}+3$, $\frac{1}{4}a^{4}+2a^{2}+3$, $\frac{1}{32}a^{10}+\frac{9}{16}a^{8}+\frac{7}{2}a^{6}+\frac{35}{4}a^{4}+\frac{15}{2}a^{2}+1$ (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: | \( 3640.01221338 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 3640.01221338 \cdot 72}{2\cdot\sqrt{2824911165797606216433664}}\cr\approx \mathstrut & 0.189383392014 \end{aligned}\] (assuming GRH)
Galois group
$C_2\times C_8$ (as 16T5):
An abelian group of order 16 |
The 16 conjugacy class representatives for $C_8\times C_2$ |
Character table for $C_8\times C_2$ |
Intermediate fields
\(\Q(\sqrt{-2}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-34}) \), \(\Q(\sqrt{-2}, \sqrt{17})\), 4.4.4913.1, 4.0.314432.2, 8.0.98867482624.1, \(\Q(\zeta_{17})^+\), 8.0.1680747204608.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
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 | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.12.2 | $x^{8} + 16 x^{7} + 120 x^{6} + 546 x^{5} + 1646 x^{4} + 3352 x^{3} + 4457 x^{2} + 3470 x + 1203$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ |
2.8.12.2 | $x^{8} + 16 x^{7} + 120 x^{6} + 546 x^{5} + 1646 x^{4} + 3352 x^{3} + 4457 x^{2} + 3470 x + 1203$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ | |
\(17\) | 17.8.7.3 | $x^{8} + 17$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |
17.8.7.3 | $x^{8} + 17$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |