Normalized defining polynomial
\( x^{16} - 6x^{12} + 35x^{8} - 6x^{4} + 1 \)
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: | \(115422332637413376\) \(\medspace = 2^{44}\cdot 3^{8}\) | 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: | \(11.65\) | 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: | $2^{11/4}3^{1/2}\approx 11.651802520975762$ | ||
Ramified primes: | \(2\), \(3\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{140}a^{12}-\frac{1}{4}a^{10}-\frac{1}{4}a^{6}-\frac{1}{4}a^{2}-\frac{41}{140}$, $\frac{1}{140}a^{13}-\frac{1}{4}a^{11}-\frac{1}{4}a^{7}-\frac{1}{4}a^{3}-\frac{41}{140}a$, $\frac{1}{140}a^{14}-\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{4}a^{6}-\frac{1}{4}a^{4}+\frac{16}{35}a^{2}-\frac{1}{4}$, $\frac{1}{140}a^{15}-\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{1}{4}a^{5}+\frac{16}{35}a^{3}-\frac{1}{4}a$
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: | \( -\frac{1}{140} a^{14} - \frac{17}{140} a^{12} + \frac{3}{4} a^{8} - \frac{17}{4} a^{4} - \frac{239}{140} a^{2} + \frac{51}{70} \) (order $24$) | 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: | $\frac{1}{35}a^{15}+\frac{204}{35}a^{3}$, $\frac{29}{140}a^{15}-\frac{1}{70}a^{14}-\frac{3}{14}a^{13}-\frac{3}{35}a^{12}-\frac{5}{4}a^{11}+\frac{5}{4}a^{9}+\frac{1}{2}a^{8}+\frac{29}{4}a^{7}-\frac{29}{4}a^{5}-3a^{4}-\frac{87}{70}a^{3}-\frac{102}{35}a^{2}+\frac{1}{28}a+\frac{18}{35}$, $\frac{3}{10}a^{14}-\frac{1}{4}a^{12}-\frac{7}{4}a^{10}+\frac{3}{2}a^{8}+\frac{41}{4}a^{6}-\frac{17}{2}a^{4}-\frac{1}{20}a^{2}+\frac{3}{4}$, $\frac{3}{10}a^{15}+\frac{5}{28}a^{13}-\frac{7}{4}a^{11}-a^{9}+\frac{41}{4}a^{7}+6a^{5}-\frac{1}{20}a^{3}+\frac{19}{28}a$, $\frac{1}{5}a^{15}+\frac{29}{140}a^{14}-\frac{1}{140}a^{13}-\frac{3}{70}a^{12}-\frac{5}{4}a^{11}-\frac{5}{4}a^{10}+\frac{1}{4}a^{8}+\frac{29}{4}a^{7}+\frac{29}{4}a^{6}-\frac{5}{4}a^{4}-\frac{49}{20}a^{3}-\frac{87}{70}a^{2}-\frac{169}{140}a+\frac{1}{140}$, $\frac{1}{70}a^{15}-\frac{3}{35}a^{14}+\frac{3}{35}a^{13}+\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-3a^{6}+3a^{5}+\frac{102}{35}a^{3}+\frac{1}{70}a^{2}-\frac{18}{35}a+\frac{1}{2}$, $\frac{3}{14}a^{15}-\frac{3}{14}a^{14}+\frac{1}{140}a^{13}+\frac{11}{140}a^{12}-\frac{5}{4}a^{11}+\frac{5}{4}a^{10}-\frac{1}{2}a^{8}+\frac{29}{4}a^{7}-\frac{29}{4}a^{6}+3a^{4}-\frac{1}{28}a^{3}-\frac{13}{28}a^{2}+\frac{169}{140}a-\frac{101}{140}$ | 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: | \( 844.712913494 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 844.712913494 \cdot 1}{24\cdot\sqrt{115422332637413376}}\cr\approx \mathstrut & 0.251647091441 \end{aligned}\]
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: | 8.0.84934656.2, 8.4.339738624.2, 8.0.339738624.7, 8.0.21233664.2 |
Minimal sibling: | 8.0.21233664.2 |
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 | ${\href{/padicField/5.2.0.1}{2} }^{8}$ | ${\href{/padicField/7.2.0.1}{2} }^{8}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.16.44.2 | $x^{16} + 8 x^{15} + 20 x^{14} + 16 x^{13} + 16 x^{12} + 8 x^{11} + 72 x^{10} + 136 x^{9} + 136 x^{8} + 160 x^{7} + 136 x^{6} + 208 x^{5} + 240 x^{4} + 208 x^{3} + 160 x^{2} + 48 x + 36$ | $8$ | $2$ | $44$ | $D_4\times C_2$ | $[2, 3, 7/2]^{2}$ |
\(3\) | 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |