Normalized defining polynomial
\( x^{16} - x^{14} + 2x^{12} + 3x^{10} - x^{8} + 3x^{6} + 2x^{4} - x^{2} + 1 \)
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: | \(350749278894882816\) \(\medspace = 2^{16}\cdot 3^{8}\cdot 13^{8}\) | 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: | \(12.49\) | 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: | $2^{3/2}3^{1/2}13^{1/2}\approx 17.663521732655695$ | ||
Ramified primes: | \(2\), \(3\), \(13\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3}a^{12}+\frac{1}{3}a^{10}-\frac{1}{3}a^{6}+\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{3}a^{13}+\frac{1}{3}a^{11}-\frac{1}{3}a^{7}+\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{10}-\frac{1}{3}a^{8}+\frac{1}{3}a^{6}+\frac{1}{3}a^{4}-\frac{1}{3}$, $\frac{1}{3}a^{15}-\frac{1}{3}a^{11}-\frac{1}{3}a^{9}+\frac{1}{3}a^{7}+\frac{1}{3}a^{5}-\frac{1}{3}a$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
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: | \( \frac{2}{3} a^{14} + \frac{1}{3} a^{10} + \frac{10}{3} a^{8} + \frac{2}{3} a^{6} + \frac{2}{3} a^{4} + 2 a^{2} + \frac{1}{3} \) (order $6$) | 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: | $\frac{2}{3}a^{15}+\frac{1}{3}a^{11}+\frac{10}{3}a^{9}+\frac{2}{3}a^{7}+\frac{2}{3}a^{5}+2a^{3}+\frac{1}{3}a$, $\frac{1}{3}a^{15}-\frac{1}{3}a^{13}+\frac{1}{3}a^{11}+\frac{5}{3}a^{9}-\frac{4}{3}a^{7}+\frac{1}{3}a^{5}+\frac{5}{3}a^{3}-\frac{5}{3}a$, $a^{11}-a^{9}+2a^{7}+2a^{5}+a$, $\frac{1}{3}a^{15}-\frac{1}{3}a^{14}-a^{13}+\frac{1}{3}a^{12}+\frac{2}{3}a^{11}-\frac{4}{3}a^{10}+\frac{2}{3}a^{9}+\frac{1}{3}a^{8}-\frac{11}{3}a^{7}-\frac{5}{3}a^{6}+\frac{1}{3}a^{5}-\frac{4}{3}a^{4}+\frac{4}{3}a^{2}-\frac{4}{3}a-\frac{4}{3}$, $\frac{2}{3}a^{15}-\frac{2}{3}a^{14}-\frac{1}{3}a^{13}+\frac{4}{3}a^{12}+a^{11}-2a^{10}+\frac{7}{3}a^{9}-\frac{1}{3}a^{8}+2a^{6}+\frac{5}{3}a^{5}-\frac{5}{3}a^{4}+\frac{2}{3}a^{3}+\frac{1}{3}a^{2}-a+1$, $\frac{2}{3}a^{14}+\frac{1}{3}a^{13}+\frac{1}{3}a^{11}+\frac{4}{3}a^{10}+\frac{7}{3}a^{8}+\frac{8}{3}a^{7}+\frac{8}{3}a^{6}+a^{5}+\frac{8}{3}a^{4}+\frac{4}{3}a^{3}+2a^{2}+\frac{7}{3}a+\frac{4}{3}$, $\frac{2}{3}a^{15}-\frac{1}{3}a^{14}+\frac{1}{3}a^{13}+\frac{1}{3}a^{12}+\frac{2}{3}a^{11}-\frac{4}{3}a^{10}+\frac{10}{3}a^{9}+\frac{1}{3}a^{8}+\frac{7}{3}a^{7}-\frac{5}{3}a^{6}+\frac{8}{3}a^{5}-\frac{4}{3}a^{4}+\frac{4}{3}a^{3}+\frac{4}{3}a^{2}+\frac{2}{3}a-\frac{4}{3}$ | 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: | \( 392.232344738 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 392.232344738 \cdot 1}{6\cdot\sqrt{350749278894882816}}\cr\approx \mathstrut & 0.268122158619 \end{aligned}\]
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 | R | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{8}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.8.2 | $x^{8} + 8 x^{7} + 56 x^{6} + 240 x^{5} + 816 x^{4} + 2048 x^{3} + 3776 x^{2} + 4928 x + 3760$ | $2$ | $4$ | $8$ | $C_2^2:C_4$ | $[2, 2]^{4}$ |
2.8.8.2 | $x^{8} + 8 x^{7} + 56 x^{6} + 240 x^{5} + 816 x^{4} + 2048 x^{3} + 3776 x^{2} + 4928 x + 3760$ | $2$ | $4$ | $8$ | $C_2^2:C_4$ | $[2, 2]^{4}$ | |
\(3\) | 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(13\) | 13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |