Normalized defining polynomial
\( x^{16} + 2x^{12} - 5x^{8} + 2x^{4} + 1 \)
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: | \(42238838692642816\) \(\medspace = 2^{44}\cdot 7^{4}\) | 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: | \(10.94\) | 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: | $2^{11/4}7^{1/2}\approx 17.798422345016238$ | ||
Ramified primes: | \(2\), \(7\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{10}-\frac{1}{4}a^{6}-\frac{1}{4}a^{2}-\frac{1}{4}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{11}-\frac{1}{4}a^{7}-\frac{1}{4}a^{3}-\frac{1}{4}a$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{1}{2}a^{2}-\frac{1}{4}$, $\frac{1}{4}a^{15}-\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{2}a^{3}-\frac{1}{4}a$
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: | \( \frac{1}{2} a^{14} + \frac{1}{4} a^{12} + \frac{5}{4} a^{10} + \frac{1}{2} a^{8} - \frac{7}{4} a^{6} - \frac{3}{2} a^{4} + \frac{3}{4} a^{2} + \frac{1}{4} \) (order $8$) | 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: | $\frac{1}{2}a^{15}+\frac{1}{4}a^{13}+\frac{5}{4}a^{11}+\frac{1}{2}a^{9}-\frac{7}{4}a^{7}-\frac{3}{2}a^{5}+\frac{3}{4}a^{3}+\frac{1}{4}a$, $\frac{1}{4}a^{15}-\frac{1}{2}a^{13}+\frac{3}{4}a^{11}-\frac{5}{4}a^{9}-\frac{1}{4}a^{7}+\frac{7}{4}a^{5}+\frac{1}{2}a^{3}-\frac{3}{4}a$, $\frac{1}{2}a^{13}+a^{9}-2a^{5}+\frac{3}{2}a$, $\frac{1}{4}a^{15}+\frac{1}{2}a^{14}+\frac{1}{4}a^{11}+a^{10}+\frac{1}{4}a^{9}-\frac{7}{4}a^{7}-\frac{5}{2}a^{6}+\frac{3}{4}a^{5}+\frac{3}{2}a^{3}+\frac{1}{2}a^{2}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{12}+\frac{1}{4}a^{11}-\frac{1}{4}a^{10}+a^{9}-\frac{1}{2}a^{8}+\frac{3}{4}a^{7}-\frac{3}{4}a^{6}+a^{4}-\frac{3}{4}a^{3}+\frac{1}{4}a^{2}-\frac{3}{4}a-\frac{1}{4}$, $\frac{1}{2}a^{14}+\frac{1}{4}a^{13}+\frac{1}{4}a^{12}+\frac{1}{4}a^{11}+\frac{5}{4}a^{10}+\frac{1}{2}a^{9}+\frac{1}{2}a^{8}+\frac{3}{4}a^{7}-\frac{7}{4}a^{6}-a^{5}-a^{4}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}+\frac{5}{4}a+\frac{3}{4}$, $\frac{1}{4}a^{15}-\frac{1}{2}a^{14}+\frac{1}{4}a^{11}-\frac{3}{2}a^{10}-\frac{1}{4}a^{9}-\frac{7}{4}a^{7}+a^{6}-\frac{3}{4}a^{5}+\frac{3}{2}a^{3}+\frac{1}{4}a-\frac{1}{2}$ | 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: | \( 152.597792892 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 152.597792892 \cdot 1}{8\cdot\sqrt{42238838692642816}}\cr\approx \mathstrut & 0.225445321661 \end{aligned}\]
Galois group
$C_2^2\wr C_2$ (as 16T39):
A solvable group of order 32 |
The 14 conjugacy class representatives for $C_2^2\wr C_2$ |
Character table for $C_2^2\wr C_2$ |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 | $x^{16} + 8 x^{15} + 20 x^{14} + 16 x^{13} + 16 x^{12} + 8 x^{11} + 72 x^{10} + 136 x^{9} + 136 x^{8} + 160 x^{7} + 136 x^{6} + 208 x^{5} + 240 x^{4} + 208 x^{3} + 160 x^{2} + 48 x + 36$ | $8$ | $2$ | $44$ | $D_4\times C_2$ | $[2, 3, 7/2]^{2}$ |
\(7\) | 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
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 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |