Normalized defining polynomial
\( x^{16} + 27x^{12} + 227x^{8} - 39x^{4} + 16 \)
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: | \(162447943996702457856\) \(\medspace = 2^{32}\cdot 3^{8}\cdot 7^{8}\) | 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: | \(18.33\) | 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: | $2^{2}3^{1/2}7^{1/2}\approx 18.33030277982336$ | ||
Ramified primes: | \(2\), \(3\), \(7\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{6}+\frac{1}{4}a^{2}$, $\frac{1}{4}a^{7}+\frac{1}{4}a^{3}$, $\frac{1}{28}a^{8}-\frac{11}{28}a^{4}+\frac{1}{7}$, $\frac{1}{56}a^{9}-\frac{1}{8}a^{7}-\frac{11}{56}a^{5}-\frac{1}{8}a^{3}+\frac{1}{14}a$, $\frac{1}{56}a^{10}-\frac{1}{56}a^{8}+\frac{3}{56}a^{6}-\frac{17}{56}a^{4}+\frac{9}{28}a^{2}+\frac{3}{7}$, $\frac{1}{56}a^{11}-\frac{1}{14}a^{7}+\frac{11}{56}a^{3}$, $\frac{1}{448}a^{12}-\frac{1}{112}a^{11}-\frac{1}{112}a^{9}-\frac{3}{224}a^{8}+\frac{11}{112}a^{7}-\frac{17}{112}a^{5}+\frac{201}{448}a^{4}+\frac{13}{28}a^{3}+\frac{3}{14}a-\frac{11}{28}$, $\frac{1}{448}a^{13}-\frac{1}{112}a^{10}+\frac{1}{224}a^{9}+\frac{1}{112}a^{8}-\frac{1}{8}a^{7}+\frac{11}{112}a^{6}-\frac{111}{448}a^{5}-\frac{39}{112}a^{4}-\frac{1}{8}a^{3}+\frac{13}{28}a^{2}+\frac{5}{28}a+\frac{2}{7}$, $\frac{1}{448}a^{14}-\frac{1}{112}a^{11}+\frac{1}{224}a^{10}-\frac{1}{112}a^{9}-\frac{1}{56}a^{8}-\frac{3}{112}a^{7}+\frac{1}{448}a^{6}-\frac{17}{112}a^{5}-\frac{17}{56}a^{4}+\frac{19}{56}a^{3}+\frac{3}{7}a^{2}+\frac{3}{14}a+\frac{3}{7}$, $\frac{1}{448}a^{15}+\frac{1}{224}a^{11}-\frac{1}{112}a^{10}-\frac{1}{112}a^{8}-\frac{55}{448}a^{7}+\frac{11}{112}a^{6}+\frac{39}{112}a^{4}+\frac{17}{56}a^{3}+\frac{13}{28}a^{2}-\frac{2}{7}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{4}$, which has order $4$ (assuming GRH)
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -\frac{3}{112} a^{14} - \frac{41}{56} a^{10} - \frac{101}{16} a^{6} - \frac{17}{28} a^{2} \) (order $4$) | 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: | $\frac{1}{448}a^{15}-\frac{9}{448}a^{14}+\frac{5}{448}a^{13}+\frac{15}{224}a^{11}-\frac{121}{224}a^{10}+\frac{67}{224}a^{9}+\frac{309}{448}a^{7}-\frac{2025}{448}a^{6}+\frac{1105}{448}a^{5}+\frac{87}{56}a^{3}+\frac{8}{7}a^{2}-\frac{3}{4}a$, $\frac{9}{448}a^{14}-\frac{1}{56}a^{13}+\frac{5}{448}a^{12}+\frac{121}{224}a^{10}-\frac{27}{56}a^{9}+\frac{69}{224}a^{8}+\frac{2025}{448}a^{6}-\frac{115}{28}a^{5}+\frac{1173}{448}a^{4}-\frac{8}{7}a^{2}-\frac{3}{14}a+\frac{1}{28}$, $\frac{1}{224}a^{15}-\frac{1}{56}a^{13}+\frac{15}{112}a^{11}-\frac{27}{56}a^{9}+\frac{309}{224}a^{7}-\frac{115}{28}a^{5}+\frac{87}{28}a^{3}-\frac{3}{14}a$, $\frac{1}{224}a^{15}+\frac{5}{448}a^{14}+\frac{3}{448}a^{13}+\frac{1}{224}a^{12}+\frac{1}{8}a^{11}+\frac{67}{224}a^{10}+\frac{41}{224}a^{9}+\frac{1}{7}a^{8}+\frac{247}{224}a^{7}+\frac{1105}{448}a^{6}+\frac{105}{64}a^{5}+\frac{41}{32}a^{4}+\frac{11}{56}a^{3}-\frac{3}{4}a^{2}-\frac{1}{28}a-\frac{5}{14}$, $\frac{1}{448}a^{15}-\frac{1}{112}a^{14}+\frac{3}{448}a^{12}+\frac{15}{224}a^{11}-\frac{27}{112}a^{10}-\frac{1}{112}a^{9}+\frac{37}{224}a^{8}+\frac{309}{448}a^{7}-\frac{115}{56}a^{6}-\frac{17}{112}a^{5}+\frac{599}{448}a^{4}+\frac{87}{56}a^{3}+\frac{11}{28}a^{2}-\frac{11}{14}a-\frac{17}{28}$, $\frac{1}{448}a^{15}-\frac{3}{448}a^{14}-\frac{1}{224}a^{12}+\frac{15}{224}a^{11}-\frac{41}{224}a^{10}+\frac{1}{112}a^{9}-\frac{1}{8}a^{8}+\frac{309}{448}a^{7}-\frac{105}{64}a^{6}+\frac{17}{112}a^{5}-\frac{219}{224}a^{4}+\frac{87}{56}a^{3}-\frac{27}{28}a^{2}+\frac{11}{14}a-\frac{1}{14}$, $\frac{1}{112}a^{14}+\frac{3}{448}a^{13}-\frac{1}{224}a^{12}+\frac{27}{112}a^{10}+\frac{39}{224}a^{9}-\frac{1}{8}a^{8}+\frac{115}{56}a^{6}+\frac{667}{448}a^{5}-\frac{219}{224}a^{4}-\frac{11}{28}a^{2}-\frac{23}{28}a-\frac{1}{14}$ (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: | \( 5538.01231421 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 5538.01231421 \cdot 4}{4\cdot\sqrt{162447943996702457856}}\cr\approx \mathstrut & 1.05544535863 \end{aligned}\] (assuming GRH)
Galois group
$C_2^2\times D_4$ (as 16T25):
A solvable group of order 32 |
The 20 conjugacy class representatives for $C_2^2 \times D_4$ |
Character table for $C_2^2 \times D_4$ |
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 | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.8.1 | $x^{4} + 2 x^{2} + 4 x + 10$ | $4$ | $1$ | $8$ | $C_2^2$ | $[2, 3]$ |
2.4.8.1 | $x^{4} + 2 x^{2} + 4 x + 10$ | $4$ | $1$ | $8$ | $C_2^2$ | $[2, 3]$ | |
2.8.16.6 | $x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$ | $4$ | $2$ | $16$ | $C_2^3$ | $[2, 3]^{2}$ | |
\(3\) | 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 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
\(7\) | 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 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |