Normalized defining polynomial
\( x^{16} + 24x^{14} + 216x^{12} + 936x^{10} + 2094x^{8} + 2376x^{6} + 1224x^{4} + 216x^{2} + 9 \)
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: | \(1378596953991976568487936\) \(\medspace = 2^{58}\cdot 3^{14}\) | 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: | \(32.26\) | 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: | $2^{29/8}3^{7/8}\approx 32.263749133641326$ | ||
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 a CM field. | |||
Reflex fields: | unavailable$^{128}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{6}a^{8}-\frac{1}{2}$, $\frac{1}{6}a^{9}-\frac{1}{2}a$, $\frac{1}{30}a^{10}-\frac{1}{30}a^{8}+\frac{1}{5}a^{6}-\frac{2}{5}a^{4}-\frac{1}{10}a^{2}+\frac{1}{10}$, $\frac{1}{60}a^{11}-\frac{1}{60}a^{10}-\frac{1}{60}a^{9}+\frac{1}{60}a^{8}+\frac{1}{10}a^{7}-\frac{1}{10}a^{6}+\frac{3}{10}a^{5}-\frac{3}{10}a^{4}-\frac{1}{20}a^{3}+\frac{1}{20}a^{2}+\frac{1}{20}a-\frac{1}{20}$, $\frac{1}{60}a^{12}-\frac{1}{12}a^{8}-\frac{1}{10}a^{6}-\frac{1}{4}a^{4}-\frac{1}{2}a^{2}+\frac{1}{20}$, $\frac{1}{60}a^{13}-\frac{1}{12}a^{9}-\frac{1}{10}a^{7}-\frac{1}{4}a^{5}-\frac{1}{2}a^{3}+\frac{1}{20}a$, $\frac{1}{180}a^{14}-\frac{1}{60}a^{10}+\frac{1}{15}a^{8}-\frac{11}{60}a^{6}+\frac{1}{5}a^{4}+\frac{3}{20}a^{2}+\frac{1}{5}$, $\frac{1}{180}a^{15}-\frac{1}{60}a^{10}+\frac{1}{20}a^{9}+\frac{1}{60}a^{8}-\frac{1}{12}a^{7}-\frac{1}{10}a^{6}-\frac{1}{2}a^{5}-\frac{3}{10}a^{4}+\frac{1}{10}a^{3}+\frac{1}{20}a^{2}+\frac{1}{4}a-\frac{1}{20}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{3}\times C_{6}$, which has order $18$ (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: | $\frac{1}{45}a^{14}+\frac{29}{60}a^{12}+\frac{56}{15}a^{10}+\frac{773}{60}a^{8}+\frac{65}{3}a^{6}+\frac{409}{20}a^{4}+\frac{66}{5}a^{2}+\frac{53}{20}$, $\frac{1}{90}a^{14}+\frac{1}{4}a^{12}+\frac{61}{30}a^{10}+\frac{449}{60}a^{8}+\frac{188}{15}a^{6}+\frac{137}{20}a^{4}-\frac{7}{5}a^{2}+\frac{7}{20}$, $\frac{1}{18}a^{14}+\frac{77}{60}a^{12}+\frac{109}{10}a^{10}+\frac{867}{20}a^{8}+\frac{1288}{15}a^{6}+\frac{1639}{20}a^{4}+\frac{164}{5}a^{2}+\frac{71}{20}$, $\frac{2}{45}a^{14}+\frac{31}{30}a^{12}+\frac{133}{15}a^{10}+\frac{538}{15}a^{8}+\frac{220}{3}a^{6}+\frac{751}{10}a^{4}+\frac{171}{5}a^{2}+\frac{11}{5}$, $\frac{1}{36}a^{14}+\frac{37}{60}a^{12}+\frac{97}{20}a^{10}+\frac{979}{60}a^{8}+\frac{1229}{60}a^{6}-\frac{79}{20}a^{4}-\frac{401}{20}a^{2}-\frac{57}{20}$, $\frac{1}{36}a^{14}+\frac{13}{20}a^{12}+\frac{67}{12}a^{10}+\frac{265}{12}a^{8}+\frac{2441}{60}a^{6}+\frac{111}{4}a^{4}-\frac{13}{4}a^{2}-\frac{111}{20}$, $\frac{13}{180}a^{14}+\frac{33}{20}a^{12}+\frac{823}{60}a^{10}+\frac{3131}{60}a^{8}+\frac{5629}{60}a^{6}+\frac{1423}{20}a^{4}+\frac{283}{20}a^{2}+\frac{7}{20}$ (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: | \( 20146.2577766 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 20146.2577766 \cdot 18}{2\cdot\sqrt{1378596953991976568487936}}\cr\approx \mathstrut & 0.375108741970 \end{aligned}\] (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
\(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}, \sqrt{3})\), 4.4.27648.1 x2, 4.4.13824.1 x2, 8.8.3057647616.1, 8.0.587068342272.1 x4 |
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 | ${\href{/padicField/5.2.0.1}{2} }^{8}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.16.58.4 | $x^{16} + 12 x^{14} + 8 x^{13} + 8 x^{11} + 10 x^{8} + 20 x^{4} + 16 x^{2} + 16 x + 30$ | $16$ | $1$ | $58$ | $QD_{16}$ | $[2, 3, 7/2, 9/2]$ |
\(3\) | 3.16.14.2 | $x^{16} + 16 x^{15} + 128 x^{14} + 672 x^{13} + 2576 x^{12} + 7616 x^{11} + 17920 x^{10} + 34188 x^{9} + 53458 x^{8} + 68592 x^{7} + 71008 x^{6} + 56896 x^{5} + 33488 x^{4} + 14784 x^{3} + 6308 x^{2} + 2732 x + 661$ | $8$ | $2$ | $14$ | $QD_{16}$ | $[\ ]_{8}^{2}$ |