Normalized defining polynomial
\( x^{16} - 4x^{14} + 9x^{12} - 19x^{10} + 27x^{8} - 28x^{6} + 18x^{4} - 6x^{2} + 1 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[4, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(1454550543311110144\) \(\medspace = 2^{16}\cdot 43^{2}\cdot 331^{4}\) | 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: | \(13.65\) | 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: | \(2\), \(43\), \(331\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{47}a^{14}+\frac{15}{47}a^{12}+\frac{12}{47}a^{10}+\frac{21}{47}a^{8}+\frac{3}{47}a^{6}-\frac{18}{47}a^{4}+\frac{5}{47}a^{2}-\frac{5}{47}$, $\frac{1}{47}a^{15}+\frac{15}{47}a^{13}+\frac{12}{47}a^{11}+\frac{21}{47}a^{9}+\frac{3}{47}a^{7}-\frac{18}{47}a^{5}+\frac{5}{47}a^{3}-\frac{5}{47}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $9$ | 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: | $\frac{7}{47}a^{14}-\frac{36}{47}a^{12}+\frac{84}{47}a^{10}-\frac{182}{47}a^{8}+\frac{303}{47}a^{6}-\frac{314}{47}a^{4}+\frac{270}{47}a^{2}-\frac{82}{47}$, $\frac{40}{47}a^{14}-\frac{152}{47}a^{12}+\frac{339}{47}a^{10}-\frac{711}{47}a^{8}+\frac{966}{47}a^{6}-\frac{1002}{47}a^{4}+\frac{576}{47}a^{2}-\frac{153}{47}$, $\frac{82}{47}a^{15}-\frac{321}{47}a^{13}+\frac{702}{47}a^{11}-\frac{1474}{47}a^{9}+\frac{2032}{47}a^{7}-\frac{1993}{47}a^{5}+\frac{1162}{47}a^{3}-\frac{222}{47}a$, $\frac{24}{47}a^{15}-\frac{110}{47}a^{13}+\frac{241}{47}a^{11}-\frac{483}{47}a^{9}+\frac{730}{47}a^{7}-\frac{667}{47}a^{5}+\frac{402}{47}a^{3}-\frac{26}{47}a$, $\frac{33}{47}a^{15}+\frac{42}{47}a^{14}-\frac{163}{47}a^{13}-\frac{122}{47}a^{12}+\frac{396}{47}a^{11}+\frac{222}{47}a^{10}-\frac{811}{47}a^{9}-\frac{481}{47}a^{8}+\frac{1274}{47}a^{7}+\frac{455}{47}a^{6}-\frac{1346}{47}a^{5}-\frac{333}{47}a^{4}+\frac{917}{47}a^{3}-\frac{25}{47}a^{2}-\frac{259}{47}a+\frac{72}{47}$, $\frac{24}{47}a^{15}+\frac{89}{47}a^{14}-\frac{110}{47}a^{13}-\frac{310}{47}a^{12}+\frac{241}{47}a^{11}+\frac{645}{47}a^{10}-\frac{483}{47}a^{9}-\frac{1374}{47}a^{8}+\frac{730}{47}a^{7}+\frac{1724}{47}a^{6}-\frac{667}{47}a^{5}-\frac{1649}{47}a^{4}+\frac{402}{47}a^{3}+\frac{774}{47}a^{2}-\frac{26}{47}a-\frac{163}{47}$, $\frac{74}{47}a^{15}-\frac{253}{47}a^{13}+\frac{512}{47}a^{11}-\frac{1078}{47}a^{9}+\frac{1303}{47}a^{7}-\frac{1191}{47}a^{5}+\frac{464}{47}a^{3}+\frac{6}{47}a+1$, $\frac{89}{47}a^{15}+\frac{32}{47}a^{14}-\frac{310}{47}a^{13}-\frac{131}{47}a^{12}+\frac{645}{47}a^{11}+\frac{290}{47}a^{10}-\frac{1374}{47}a^{9}-\frac{597}{47}a^{8}+\frac{1724}{47}a^{7}+\frac{848}{47}a^{6}-\frac{1649}{47}a^{5}-\frac{811}{47}a^{4}+\frac{821}{47}a^{3}+\frac{442}{47}a^{2}-\frac{163}{47}a-\frac{66}{47}$, $\frac{89}{47}a^{15}-\frac{40}{47}a^{14}-\frac{310}{47}a^{13}+\frac{152}{47}a^{12}+\frac{645}{47}a^{11}-\frac{339}{47}a^{10}-\frac{1374}{47}a^{9}+\frac{711}{47}a^{8}+\frac{1724}{47}a^{7}-\frac{966}{47}a^{6}-\frac{1649}{47}a^{5}+\frac{1002}{47}a^{4}+\frac{821}{47}a^{3}-\frac{529}{47}a^{2}-\frac{163}{47}a+\frac{153}{47}$ | 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: | \( 445.363754488 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 445.363754488 \cdot 1}{2\cdot\sqrt{1454550543311110144}}\cr\approx \mathstrut & 0.181768933129 \end{aligned}\]
Galois group
$C_2^8.S_4$ (as 16T1664):
A solvable group of order 6144 |
The 105 conjugacy class representatives for $C_2^8.S_4$ |
Character table for $C_2^8.S_4$ |
Intermediate fields
4.2.331.1, 8.2.28047616.3, 8.2.4711123.1, 8.4.1206047488.2 |
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 | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.3.0.1}{3} }^{4}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{8}$ |
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 $16$ | $2$ | $8$ | $16$ | |||
\(43\) | 43.4.2.1 | $x^{4} + 84 x^{3} + 1856 x^{2} + 3864 x + 77452$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
43.6.0.1 | $x^{6} + 19 x^{3} + 28 x^{2} + 21 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
43.6.0.1 | $x^{6} + 19 x^{3} + 28 x^{2} + 21 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
\(331\) | $\Q_{331}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{331}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{331}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{331}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $4$ | $2$ | $2$ | $2$ | ||||
Deg $4$ | $2$ | $2$ | $2$ |