Normalized defining polynomial
\( x^{16} - 4x^{14} + 15x^{12} - 16x^{10} + 21x^{8} + 38x^{6} + 32x^{4} + 24x^{2} + 16 \)
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: | \(634562281237118976\) \(\medspace = 2^{24}\cdot 3^{8}\cdot 7^{8}\) | 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: | \(12.96\) | 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: | $2^{3/2}3^{1/2}7^{1/2}\approx 12.96148139681572$ | ||
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: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $16$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{10}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{7}-\frac{1}{2}a^{4}$, $\frac{1}{44608}a^{14}-\frac{1689}{22304}a^{12}+\frac{43}{44608}a^{10}-\frac{61}{22304}a^{8}-\frac{975}{44608}a^{6}-\frac{19}{5576}a^{4}-\frac{1}{2}a^{3}-\frac{7}{2788}a^{2}+\frac{2631}{5576}$, $\frac{1}{44608}a^{15}-\frac{1689}{22304}a^{13}+\frac{43}{44608}a^{11}-\frac{61}{22304}a^{9}-\frac{975}{44608}a^{7}-\frac{19}{5576}a^{5}-\frac{1}{2}a^{4}-\frac{7}{2788}a^{3}+\frac{2631}{5576}a$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
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: | \( \frac{1613}{44608} a^{14} - \frac{3269}{22304} a^{12} + \frac{24751}{44608} a^{10} - \frac{14753}{22304} a^{8} + \frac{44365}{44608} a^{6} + \frac{4203}{5576} a^{4} + \frac{4043}{2788} a^{2} + \frac{6043}{5576} \) (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: | $\frac{1081}{22304}a^{15}-\frac{3}{697}a^{14}-\frac{2457}{11152}a^{13}+\frac{55}{1394}a^{12}+\frac{18603}{22304}a^{11}-\frac{129}{697}a^{10}-\frac{12969}{11152}a^{9}+\frac{366}{697}a^{8}+\frac{33345}{22304}a^{7}-\frac{560}{697}a^{6}+\frac{3159}{2788}a^{5}+\frac{456}{697}a^{4}+\frac{747}{697}a^{3}+\frac{336}{697}a^{2}+\frac{351}{2788}a+\frac{283}{697}$, $\frac{129}{44608}a^{14}-\frac{417}{22304}a^{12}+\frac{5547}{44608}a^{10}-\frac{7869}{22304}a^{8}+\frac{41505}{44608}a^{6}-\frac{5239}{5576}a^{4}+\frac{3279}{2788}a^{2}+\frac{4839}{5576}$, $\frac{1081}{22304}a^{15}-\frac{43}{44608}a^{14}-\frac{2457}{11152}a^{13}+\frac{139}{22304}a^{12}+\frac{18603}{22304}a^{11}-\frac{1849}{44608}a^{10}-\frac{12969}{11152}a^{9}+\frac{2623}{22304}a^{8}+\frac{33345}{22304}a^{7}-\frac{13835}{44608}a^{6}+\frac{3159}{2788}a^{5}+\frac{817}{5576}a^{4}+\frac{747}{697}a^{3}+\frac{301}{2788}a^{2}+\frac{351}{2788}a-\frac{1613}{5576}$, $\frac{1441}{44608}a^{14}-\frac{2713}{22304}a^{12}+\frac{17355}{44608}a^{10}-\frac{4261}{22304}a^{8}-\frac{10975}{44608}a^{6}+\frac{13047}{5576}a^{4}-\frac{329}{2788}a^{2}+\frac{5167}{5576}$, $\frac{1081}{22304}a^{15}+\frac{1633}{44608}a^{14}-\frac{2457}{11152}a^{13}-\frac{3593}{22304}a^{12}+\frac{18603}{22304}a^{11}+\frac{25611}{44608}a^{10}-\frac{12969}{11152}a^{9}-\frac{15973}{22304}a^{8}+\frac{33345}{22304}a^{7}+\frac{24865}{44608}a^{6}+\frac{3159}{2788}a^{5}+\frac{9399}{5576}a^{4}+\frac{747}{697}a^{3}-\frac{1673}{2788}a^{2}+\frac{351}{2788}a+\frac{2903}{5576}$, $\frac{337}{22304}a^{14}-\frac{441}{11152}a^{12}+\frac{3339}{22304}a^{10}+\frac{1747}{11152}a^{8}-\frac{5167}{22304}a^{6}+\frac{4749}{2788}a^{4}+\frac{563}{697}a^{2}+\frac{63}{2788}$, $\frac{727}{22304}a^{15}+\frac{3087}{44608}a^{14}-\frac{1183}{11152}a^{13}-\frac{5959}{22304}a^{12}+\frac{8957}{22304}a^{11}+\frac{43525}{44608}a^{10}-\frac{2527}{11152}a^{9}-\frac{21027}{22304}a^{8}+\frac{10479}{22304}a^{7}+\frac{45823}{44608}a^{6}+\frac{903}{697}a^{5}+\frac{16623}{5576}a^{4}+\frac{3275}{1394}a^{3}+\frac{4877}{2788}a^{2}+\frac{2957}{2788}a+\frac{3241}{5576}$ | 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: | \( 903.286579457 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 903.286579457 \cdot 1}{6\cdot\sqrt{634562281237118976}}\cr\approx \mathstrut & 0.459067055692 \end{aligned}\]
Galois group
$C_2\times D_4$ (as 16T9):
A solvable group of order 16 |
The 10 conjugacy class representatives for $D_4\times C_2$ |
Character table for $D_4\times C_2$ |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 8 siblings: | 8.0.16257024.1, 8.0.12446784.1, 8.4.796594176.2, 8.0.796594176.13 |
Minimal sibling: | 8.0.12446784.1 |
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 | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.6.1 | $x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ |
2.4.6.1 | $x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
2.4.6.1 | $x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
2.4.6.1 | $x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
\(3\) | 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 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |