Normalized defining polynomial
\( x^{16} - 6x^{14} + 27x^{12} - 74x^{10} + 108x^{8} - 74x^{6} + 27x^{4} - 6x^{2} + 1 \)
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: | \(6338465731314712576\) \(\medspace = 2^{40}\cdot 7^{8}\) | 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: | \(14.97\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3}a^{10}+\frac{1}{3}$, $\frac{1}{3}a^{11}+\frac{1}{3}a$, $\frac{1}{36}a^{12}+\frac{1}{9}a^{10}-\frac{1}{3}a^{8}-\frac{1}{6}a^{6}-\frac{1}{3}a^{4}-\frac{2}{9}a^{2}-\frac{5}{36}$, $\frac{1}{36}a^{13}+\frac{1}{9}a^{11}-\frac{1}{3}a^{9}-\frac{1}{6}a^{7}-\frac{1}{3}a^{5}-\frac{2}{9}a^{3}-\frac{5}{36}a$, $\frac{1}{108}a^{14}-\frac{1}{108}a^{12}+\frac{1}{27}a^{10}-\frac{1}{6}a^{8}+\frac{1}{6}a^{6}+\frac{4}{27}a^{4}-\frac{1}{108}a^{2}+\frac{25}{108}$, $\frac{1}{108}a^{15}-\frac{1}{108}a^{13}+\frac{1}{27}a^{11}-\frac{1}{6}a^{9}+\frac{1}{6}a^{7}+\frac{4}{27}a^{5}-\frac{1}{108}a^{3}+\frac{25}{108}a$
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: | \( -1 \) (order $2$) | 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: | $\frac{49}{108}a^{15}-\frac{125}{54}a^{13}+\frac{271}{27}a^{11}-\frac{143}{6}a^{9}+\frac{73}{3}a^{7}-\frac{92}{27}a^{5}-\frac{169}{108}a^{3}+\frac{71}{54}a$, $\frac{1}{18}a^{15}-\frac{7}{12}a^{13}+\frac{25}{9}a^{11}-\frac{29}{3}a^{9}+\frac{115}{6}a^{7}-\frac{160}{9}a^{5}+\frac{19}{6}a^{3}+\frac{25}{36}a$, $\frac{95}{108}a^{15}-\frac{503}{108}a^{13}+\frac{551}{27}a^{11}-\frac{101}{2}a^{9}+\frac{117}{2}a^{7}-\frac{610}{27}a^{5}+\frac{901}{108}a^{3}-\frac{229}{108}a$, $\frac{49}{108}a^{15}-\frac{125}{54}a^{13}+\frac{271}{27}a^{11}-\frac{143}{6}a^{9}+\frac{73}{3}a^{7}-\frac{92}{27}a^{5}-\frac{169}{108}a^{3}+\frac{71}{54}a+1$, $\frac{49}{108}a^{15}-\frac{71}{108}a^{14}-\frac{125}{54}a^{13}+\frac{377}{108}a^{12}+\frac{271}{27}a^{11}-\frac{413}{27}a^{10}-\frac{143}{6}a^{9}+\frac{227}{6}a^{8}+\frac{73}{3}a^{7}-\frac{263}{6}a^{6}-\frac{92}{27}a^{5}+\frac{418}{27}a^{4}-\frac{169}{108}a^{3}-\frac{433}{108}a^{2}+\frac{71}{54}a+\frac{43}{108}$, $\frac{55}{54}a^{15}+\frac{77}{108}a^{14}-\frac{158}{27}a^{13}-\frac{110}{27}a^{12}+\frac{695}{27}a^{11}+\frac{488}{27}a^{10}-\frac{202}{3}a^{9}-\frac{95}{2}a^{8}+\frac{259}{3}a^{7}+63a^{6}-\frac{991}{27}a^{5}-\frac{898}{27}a^{4}-\frac{127}{54}a^{3}+\frac{775}{108}a^{2}+\frac{71}{27}a+\frac{8}{27}$, $\frac{1}{2}a^{15}+\frac{1}{2}a^{14}-\frac{5}{2}a^{13}-\frac{5}{2}a^{12}+11a^{11}+11a^{10}-26a^{9}-26a^{8}+28a^{7}+28a^{6}-9a^{5}-9a^{4}+\frac{9}{2}a^{3}+\frac{9}{2}a^{2}+\frac{3}{2}a+\frac{1}{2}$ (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: | \( 747.271213234 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 747.271213234 \cdot 1}{2\cdot\sqrt{6338465731314712576}}\cr\approx \mathstrut & 0.360491584123 \end{aligned}\] (assuming GRH)
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 | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{8}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.20.1 | $x^{8} + 8 x^{7} + 36 x^{6} + 80 x^{5} + 104 x^{4} - 32 x^{3} - 8 x^{2} + 124$ | $4$ | $2$ | $20$ | $D_4\times C_2$ | $[2, 3, 7/2]^{2}$ |
2.8.20.1 | $x^{8} + 8 x^{7} + 36 x^{6} + 80 x^{5} + 104 x^{4} - 32 x^{3} - 8 x^{2} + 124$ | $4$ | $2$ | $20$ | $D_4\times C_2$ | $[2, 3, 7/2]^{2}$ | |
\(7\) | 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 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |