Normalized defining polynomial
\( x^{16} - 3x^{12} + 8x^{8} - 3x^{4} + 1 \)
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: | \(11007531417600000000\) \(\medspace = 2^{32}\cdot 3^{8}\cdot 5^{8}\) | 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: | \(15.49\) | 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: | $2^{2}3^{1/2}5^{1/2}\approx 15.491933384829668$ | ||
Ramified primes: | \(2\), \(3\), \(5\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{8}a^{12}-\frac{3}{8}$, $\frac{1}{8}a^{13}-\frac{3}{8}a$, $\frac{1}{8}a^{14}-\frac{3}{8}a^{2}$, $\frac{1}{8}a^{15}-\frac{3}{8}a^{3}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( \frac{5}{8} a^{14} - 2 a^{10} + 5 a^{6} - \frac{15}{8} a^{2} \) (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: | $\frac{1}{8}a^{13}+\frac{13}{8}a$, $\frac{1}{8}a^{14}+\frac{13}{8}a^{2}-1$, $\frac{3}{4}a^{15}+\frac{5}{8}a^{14}+\frac{5}{8}a^{13}+\frac{5}{8}a^{12}-2a^{11}-2a^{10}-2a^{9}-2a^{8}+5a^{7}+5a^{6}+5a^{5}+5a^{4}-\frac{1}{4}a^{3}-\frac{15}{8}a^{2}-\frac{15}{8}a-\frac{15}{8}$, $\frac{9}{8}a^{15}-\frac{1}{8}a^{14}-\frac{3}{8}a^{13}-3a^{11}+a^{9}+8a^{7}-3a^{5}-\frac{3}{8}a^{3}-\frac{21}{8}a^{2}+\frac{9}{8}a+1$, $\frac{5}{8}a^{15}+\frac{9}{8}a^{14}+\frac{1}{8}a^{13}-\frac{3}{8}a^{12}-2a^{11}-3a^{10}+a^{8}+5a^{7}+8a^{6}-3a^{4}-\frac{15}{8}a^{3}-\frac{3}{8}a^{2}+\frac{13}{8}a+\frac{9}{8}$, $\frac{3}{4}a^{15}+\frac{1}{8}a^{14}-\frac{3}{4}a^{13}-\frac{1}{8}a^{12}-2a^{11}+2a^{9}+5a^{7}-5a^{5}-\frac{1}{4}a^{3}+\frac{13}{8}a^{2}+\frac{1}{4}a-\frac{5}{8}$, $\frac{3}{4}a^{15}-\frac{3}{4}a^{14}+\frac{5}{8}a^{13}-\frac{1}{4}a^{12}-2a^{11}+2a^{10}-2a^{9}+a^{8}+5a^{7}-5a^{6}+5a^{5}-2a^{4}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{15}{8}a+\frac{3}{4}$ (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: | \( 5686.37968405 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 5686.37968405 \cdot 1}{12\cdot\sqrt{11007531417600000000}}\cr\approx \mathstrut & 0.346935378095 \end{aligned}\] (assuming GRH)
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: | 8.4.3317760000.2, 8.0.207360000.4, 8.0.132710400.4, 8.0.3317760000.11 |
Minimal sibling: | 8.0.132710400.4 |
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 | R | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.16.4 | $x^{8} + 12 x^{7} + 58 x^{6} + 160 x^{5} + 329 x^{4} + 500 x^{3} + 408 x^{2} + 68 x + 61$ | $4$ | $2$ | $16$ | $D_4$ | $[2, 3]^{2}$ |
2.8.16.4 | $x^{8} + 12 x^{7} + 58 x^{6} + 160 x^{5} + 329 x^{4} + 500 x^{3} + 408 x^{2} + 68 x + 61$ | $4$ | $2$ | $16$ | $D_4$ | $[2, 3]^{2}$ | |
\(3\) | 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 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
\(5\) | 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |