Normalized defining polynomial
\( x^{16} - 8x^{14} + 26x^{12} - 41x^{10} + 24x^{8} + 9x^{6} - 10x^{4} - x^{2} + 1 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[8, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(26907088989407739904\) \(\medspace = 2^{16}\cdot 11^{2}\cdot 23^{2}\cdot 283^{4}\) | 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: | \(16.38\) | 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: | not computed | ||
Ramified primes: | \(2\), \(11\), \(23\), \(283\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{17}a^{14}-\frac{8}{17}a^{10}-\frac{3}{17}a^{8}-\frac{8}{17}a^{4}-\frac{6}{17}a^{2}+\frac{2}{17}$, $\frac{1}{17}a^{15}-\frac{8}{17}a^{11}-\frac{3}{17}a^{9}-\frac{8}{17}a^{5}-\frac{6}{17}a^{3}+\frac{2}{17}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $11$ | 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: | $\frac{18}{17}a^{14}-8a^{12}+\frac{417}{17}a^{10}-\frac{615}{17}a^{8}+19a^{6}+\frac{111}{17}a^{4}-\frac{74}{17}a^{2}-\frac{15}{17}$, $a^{15}-8a^{13}+26a^{11}-41a^{9}+24a^{7}+9a^{5}-10a^{3}-a$, $\frac{18}{17}a^{14}-8a^{12}+\frac{417}{17}a^{10}-\frac{615}{17}a^{8}+19a^{6}+\frac{111}{17}a^{4}-\frac{91}{17}a^{2}+\frac{2}{17}$, $\frac{9}{17}a^{15}-4a^{13}+\frac{200}{17}a^{11}-\frac{248}{17}a^{9}+\frac{251}{17}a^{5}-\frac{88}{17}a^{3}-\frac{50}{17}a$, $\frac{21}{17}a^{15}-10a^{13}+\frac{563}{17}a^{11}-\frac{930}{17}a^{9}+39a^{7}-\frac{15}{17}a^{5}-\frac{92}{17}a^{3}-\frac{9}{17}a$, $\frac{1}{17}a^{15}+\frac{10}{17}a^{14}-4a^{12}-\frac{25}{17}a^{11}+\frac{175}{17}a^{10}+\frac{82}{17}a^{9}-\frac{166}{17}a^{8}-5a^{7}-5a^{6}-\frac{42}{17}a^{5}+\frac{209}{17}a^{4}+\frac{79}{17}a^{3}-\frac{9}{17}a^{2}+\frac{19}{17}a-\frac{31}{17}$, $\frac{21}{17}a^{15}-\frac{3}{17}a^{14}-10a^{13}+a^{12}+\frac{563}{17}a^{11}-\frac{27}{17}a^{10}-\frac{930}{17}a^{9}-\frac{25}{17}a^{8}+39a^{7}+7a^{6}-\frac{15}{17}a^{5}-\frac{78}{17}a^{4}-\frac{92}{17}a^{3}-\frac{50}{17}a^{2}-\frac{9}{17}a+\frac{28}{17}$, $\frac{5}{17}a^{15}+\frac{8}{17}a^{14}-2a^{13}-4a^{12}+\frac{79}{17}a^{11}+\frac{242}{17}a^{10}-\frac{32}{17}a^{9}-\frac{449}{17}a^{8}-10a^{7}+24a^{6}+\frac{249}{17}a^{5}-\frac{98}{17}a^{4}-\frac{47}{17}a^{3}-\frac{82}{17}a^{2}-\frac{24}{17}a+\frac{33}{17}$, $\frac{6}{17}a^{15}-\frac{10}{17}a^{14}-2a^{13}+4a^{12}+\frac{54}{17}a^{11}-\frac{175}{17}a^{10}+\frac{50}{17}a^{9}+\frac{166}{17}a^{8}-15a^{7}+5a^{6}+\frac{207}{17}a^{5}-\frac{209}{17}a^{4}+\frac{49}{17}a^{3}+\frac{9}{17}a^{2}-\frac{39}{17}a+\frac{31}{17}$, $\frac{10}{17}a^{15}-\frac{10}{17}a^{14}-4a^{13}+4a^{12}+\frac{175}{17}a^{11}-\frac{175}{17}a^{10}-\frac{166}{17}a^{9}+\frac{166}{17}a^{8}-5a^{7}+5a^{6}+\frac{209}{17}a^{5}-\frac{209}{17}a^{4}-\frac{9}{17}a^{3}+\frac{9}{17}a^{2}-\frac{31}{17}a+\frac{31}{17}$, $\frac{2}{17}a^{15}+\frac{18}{17}a^{14}-a^{13}-8a^{12}+\frac{69}{17}a^{11}+\frac{417}{17}a^{10}-\frac{159}{17}a^{9}-\frac{615}{17}a^{8}+12a^{7}+19a^{6}-\frac{118}{17}a^{5}+\frac{111}{17}a^{4}-\frac{12}{17}a^{3}-\frac{91}{17}a^{2}+\frac{4}{17}a+\frac{2}{17}$ (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: | \( 4639.55424287 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 4639.55424287 \cdot 1}{2\cdot\sqrt{26907088989407739904}}\cr\approx \mathstrut & 0.178431835137 \end{aligned}\] (assuming GRH)
Galois group
$C_2^8.S_4$ (as 16T1664):
A solvable group of order 6144 |
The 105 conjugacy class representatives for $C_2^8.S_4$ are not computed |
Character table for $C_2^8.S_4$ is not computed |
Intermediate fields
4.2.283.1, 8.4.20262517.1, 8.4.225530624.2, 8.4.471564032.1 |
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: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{6}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | Deg $16$ | $2$ | $8$ | $16$ | |||
\(11\) | 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
11.6.0.1 | $x^{6} + 3 x^{4} + 4 x^{3} + 6 x^{2} + 7 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
11.6.0.1 | $x^{6} + 3 x^{4} + 4 x^{3} + 6 x^{2} + 7 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
\(23\) | 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
23.6.0.1 | $x^{6} + x^{4} + 9 x^{3} + 9 x^{2} + x + 5$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
23.6.0.1 | $x^{6} + x^{4} + 9 x^{3} + 9 x^{2} + x + 5$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
\(283\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |