Normalized defining polynomial
\( x^{16} + x^{14} + 4x^{12} - 5x^{10} + 7x^{8} - 5x^{6} + 4x^{4} + x^{2} + 1 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(200779471797682176\) \(\medspace = 2^{16}\cdot 3^{12}\cdot 7^{8}\) | 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: | \(12.06\) | 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: | $2\cdot 3^{3/4}7^{1/2}\approx 12.06201756903818$ | ||
Ramified primes: | \(2\), \(3\), \(7\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{6}-\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}a^{7}-\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{6}-\frac{1}{3}a^{4}+\frac{1}{3}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{7}-\frac{1}{3}a^{5}+\frac{1}{3}a$, $\frac{1}{3}a^{12}+\frac{1}{3}a^{6}+\frac{1}{3}$, $\frac{1}{3}a^{13}+\frac{1}{3}a^{7}+\frac{1}{3}a$, $\frac{1}{45}a^{14}+\frac{1}{9}a^{12}-\frac{2}{15}a^{10}+\frac{1}{45}a^{8}-\frac{19}{45}a^{6}-\frac{7}{15}a^{4}-\frac{4}{9}a^{2}-\frac{4}{45}$, $\frac{1}{45}a^{15}+\frac{1}{9}a^{13}-\frac{2}{15}a^{11}+\frac{1}{45}a^{9}-\frac{19}{45}a^{7}-\frac{7}{15}a^{5}-\frac{4}{9}a^{3}-\frac{4}{45}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( \frac{2}{45} a^{15} + \frac{2}{9} a^{13} + \frac{2}{5} a^{11} + \frac{32}{45} a^{9} - \frac{8}{45} a^{7} + \frac{7}{5} a^{5} - \frac{14}{9} a^{3} + \frac{52}{45} a \) (order $12$) | 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: | $\frac{8}{45}a^{14}+\frac{2}{9}a^{12}+\frac{14}{15}a^{10}-\frac{22}{45}a^{8}+\frac{73}{45}a^{6}-\frac{26}{15}a^{4}+\frac{10}{9}a^{2}-\frac{2}{45}$, $\frac{2}{5}a^{15}+\frac{1}{3}a^{13}+\frac{19}{15}a^{11}-\frac{13}{5}a^{9}+\frac{31}{15}a^{7}-\frac{16}{15}a^{5}+a^{3}+\frac{7}{5}a$, $\frac{14}{45}a^{14}+\frac{5}{9}a^{12}+\frac{22}{15}a^{10}-\frac{31}{45}a^{8}+\frac{34}{45}a^{6}+\frac{2}{15}a^{4}-\frac{2}{9}a^{2}+\frac{49}{45}$, $\frac{23}{45}a^{15}+\frac{2}{9}a^{13}+\frac{8}{5}a^{11}-\frac{172}{45}a^{9}+\frac{208}{45}a^{7}-\frac{17}{5}a^{5}+\frac{28}{9}a^{3}-\frac{2}{45}a-1$, $\frac{2}{45}a^{15}-\frac{7}{15}a^{14}+\frac{2}{9}a^{13}-\frac{1}{3}a^{12}+\frac{2}{5}a^{11}-\frac{28}{15}a^{10}+\frac{32}{45}a^{9}+\frac{38}{15}a^{8}-\frac{8}{45}a^{7}-\frac{67}{15}a^{6}+\frac{7}{5}a^{5}+\frac{52}{15}a^{4}-\frac{14}{9}a^{3}-\frac{5}{3}a^{2}+\frac{52}{45}a-\frac{4}{5}$, $\frac{7}{45}a^{15}+\frac{4}{15}a^{14}+\frac{1}{9}a^{13}+\frac{11}{15}a^{11}+\frac{11}{15}a^{10}-\frac{38}{45}a^{9}-\frac{12}{5}a^{8}+\frac{77}{45}a^{7}+\frac{44}{15}a^{6}-\frac{29}{15}a^{5}-\frac{44}{15}a^{4}+\frac{17}{9}a^{3}+\frac{4}{3}a^{2}+\frac{17}{45}a-\frac{11}{15}$, $\frac{1}{5}a^{15}+\frac{13}{45}a^{14}-\frac{1}{3}a^{13}+\frac{1}{9}a^{12}+\frac{2}{15}a^{11}+\frac{14}{15}a^{10}-\frac{47}{15}a^{9}-\frac{92}{45}a^{8}+\frac{58}{15}a^{7}+\frac{128}{45}a^{6}-\frac{53}{15}a^{5}-\frac{26}{15}a^{4}+\frac{7}{3}a^{3}+\frac{5}{9}a^{2}-\frac{17}{15}a+\frac{38}{45}$ | 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: | \( 548.305383135 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 548.305383135 \cdot 1}{12\cdot\sqrt{200779471797682176}}\cr\approx \mathstrut & 0.247696891754 \end{aligned}\]
Galois group
$C_2\times D_4$ (as 16T9):
A solvable group of order 16 |
The 10 conjugacy class representatives for $D_4\times C_2$ |
Character table for $D_4\times C_2$ |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 8 siblings: | 8.4.448084224.2, 8.0.9144576.3, 8.0.448084224.5, 8.0.448084224.7 |
Minimal sibling: | 8.0.9144576.3 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | R | ${\href{/padicField/5.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.8.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ |
2.8.8.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ | |
\(3\) | 3.8.6.2 | $x^{8} + 6 x^{5} + 6 x^{4} + 18 x^{2} + 18 x + 9$ | $4$ | $2$ | $6$ | $D_4$ | $[\ ]_{4}^{2}$ |
3.8.6.2 | $x^{8} + 6 x^{5} + 6 x^{4} + 18 x^{2} + 18 x + 9$ | $4$ | $2$ | $6$ | $D_4$ | $[\ ]_{4}^{2}$ | |
\(7\) | 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |