Normalized defining polynomial
\( x^{16} + 12x^{12} + 156x^{8} - 144x^{4} + 144 \)
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: | \(1346286087882789617664\) \(\medspace = 2^{48}\cdot 3^{14}\) | 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: | \(20.92\) | 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: | $2^{3}3^{7/8}\approx 20.920453028921656$ | ||
Ramified primes: | \(2\), \(3\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{24}a^{8}-\frac{1}{4}a^{4}-\frac{1}{2}$, $\frac{1}{48}a^{9}-\frac{1}{4}a^{7}-\frac{1}{8}a^{5}-\frac{1}{2}a^{3}-\frac{1}{4}a$, $\frac{1}{48}a^{10}-\frac{1}{8}a^{6}-\frac{1}{4}a^{2}$, $\frac{1}{48}a^{11}-\frac{1}{8}a^{7}-\frac{1}{4}a^{3}$, $\frac{1}{1872}a^{12}+\frac{1}{6}a^{4}+\frac{11}{26}$, $\frac{1}{3744}a^{13}-\frac{1}{4}a^{7}-\frac{1}{6}a^{5}+\frac{11}{52}a$, $\frac{1}{3744}a^{14}-\frac{1}{6}a^{6}+\frac{11}{52}a^{2}$, $\frac{1}{3744}a^{15}-\frac{1}{6}a^{7}+\frac{11}{52}a^{3}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $13$ |
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: | \( \frac{1}{234} a^{12} + \frac{1}{24} a^{8} + \frac{7}{12} a^{4} - \frac{29}{26} \) (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: | $\frac{1}{468}a^{12}+\frac{1}{24}a^{8}+\frac{5}{12}a^{4}+\frac{5}{26}$, $\frac{5}{624}a^{14}-\frac{17}{1872}a^{12}+\frac{5}{48}a^{10}-\frac{1}{8}a^{8}+\frac{11}{8}a^{6}-\frac{19}{12}a^{4}+\frac{5}{52}a^{2}+\frac{4}{13}$, $\frac{5}{936}a^{14}+\frac{1}{468}a^{12}+\frac{1}{16}a^{10}+\frac{1}{24}a^{8}+\frac{19}{24}a^{6}+\frac{5}{12}a^{4}-\frac{79}{52}a^{2}+\frac{31}{26}$, $\frac{1}{234}a^{15}+\frac{5}{1872}a^{14}-\frac{1}{156}a^{13}-\frac{7}{936}a^{12}+\frac{1}{24}a^{11}+\frac{1}{48}a^{10}-\frac{1}{12}a^{9}-\frac{1}{8}a^{8}+\frac{7}{12}a^{7}+\frac{5}{24}a^{6}-a^{5}-\frac{19}{12}a^{4}-\frac{55}{26}a^{3}-\frac{111}{52}a^{2}-\frac{14}{13}a-\frac{11}{26}$, $\frac{5}{936}a^{15}+\frac{11}{1248}a^{14}+\frac{3}{416}a^{13}-\frac{25}{1872}a^{12}+\frac{1}{16}a^{11}+\frac{5}{48}a^{10}+\frac{5}{48}a^{9}-\frac{1}{6}a^{8}+\frac{19}{24}a^{7}+\frac{11}{8}a^{6}+\frac{11}{8}a^{5}-\frac{13}{6}a^{4}-\frac{53}{52}a^{3}-\frac{33}{26}a^{2}+\frac{19}{13}a+\frac{37}{26}$, $\frac{17}{3744}a^{15}-\frac{1}{416}a^{14}-\frac{3}{416}a^{13}+\frac{1}{234}a^{12}+\frac{1}{16}a^{11}-\frac{1}{48}a^{10}-\frac{1}{16}a^{9}+\frac{1}{12}a^{8}+\frac{19}{24}a^{7}-\frac{3}{8}a^{6}-\frac{9}{8}a^{5}+\frac{1}{3}a^{4}-\frac{17}{26}a^{3}-\frac{17}{26}a^{2}+\frac{27}{26}a+\frac{5}{13}$, $\frac{5}{3744}a^{15}-\frac{5}{936}a^{14}-\frac{1}{1248}a^{13}-\frac{1}{624}a^{12}+\frac{1}{48}a^{11}-\frac{1}{16}a^{10}+\frac{7}{24}a^{7}-\frac{19}{24}a^{6}+\frac{21}{26}a^{3}+\frac{27}{52}a^{2}-\frac{33}{52}a-\frac{7}{26}$ (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: | \( 114888.403605 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 114888.403605 \cdot 1}{12\cdot\sqrt{1346286087882789617664}}\cr\approx \mathstrut & 0.633818945745 \end{aligned}\] (assuming GRH)
Galois group
$\SD_{16}$ (as 16T12):
A solvable group of order 16 |
The 7 conjugacy class representatives for $QD_{16}$ |
Character table for $QD_{16}$ |
Intermediate fields
\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{12})\), 4.2.6912.1 x2, 4.0.1728.1 x2, 8.0.47775744.1, 8.2.36691771392.3 x4 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 8 sibling: | 8.2.36691771392.3 |
Minimal sibling: | 8.2.36691771392.3 |
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 | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.1.0.1}{1} }^{16}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.16.48.9 | $x^{16} + 8 x^{15} + 60 x^{14} + 280 x^{13} + 1058 x^{12} + 3072 x^{11} + 7248 x^{10} + 13816 x^{9} + 21669 x^{8} + 27752 x^{7} + 28060 x^{6} + 20544 x^{5} + 4332 x^{4} - 8696 x^{3} - 4532 x^{2} + 1680 x + 1677$ | $8$ | $2$ | $48$ | $QD_{16}$ | $[2, 3, 4]^{2}$ |
\(3\) | 3.16.14.1 | $x^{16} + 16 x^{15} + 128 x^{14} + 672 x^{13} + 2576 x^{12} + 7616 x^{11} + 17920 x^{10} + 34182 x^{9} + 53410 x^{8} + 68544 x^{7} + 71344 x^{6} + 57904 x^{5} + 34832 x^{4} + 16128 x^{3} + 7241 x^{2} + 2966 x + 634$ | $8$ | $2$ | $14$ | $QD_{16}$ | $[\ ]_{8}^{2}$ |