Normalized defining polynomial
\( x^{16} + 23x^{12} + 73x^{8} + 23x^{4} + 1 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(1186620610969600000000\) \(\medspace = 2^{32}\cdot 5^{8}\cdot 29^{4}\) | 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: | \(20.76\) | 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))
| |
Galois root discriminant: | $2^{2}5^{1/2}29^{1/2}\approx 48.16637831516918$ | ||
Ramified primes: | \(2\), \(5\), \(29\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
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 a CM field. | |||
Reflex fields: | unavailable$^{128}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{7}a^{8}+\frac{1}{7}a^{4}+\frac{1}{7}$, $\frac{1}{7}a^{9}+\frac{1}{7}a^{5}+\frac{1}{7}a$, $\frac{1}{7}a^{10}+\frac{1}{7}a^{6}+\frac{1}{7}a^{2}$, $\frac{1}{7}a^{11}+\frac{1}{7}a^{7}+\frac{1}{7}a^{3}$, $\frac{1}{77}a^{12}+\frac{2}{77}a^{8}+\frac{9}{77}a^{4}-\frac{34}{77}$, $\frac{1}{77}a^{13}+\frac{2}{77}a^{9}+\frac{9}{77}a^{5}-\frac{34}{77}a$, $\frac{1}{77}a^{14}+\frac{2}{77}a^{10}+\frac{9}{77}a^{6}-\frac{34}{77}a^{2}$, $\frac{1}{77}a^{15}+\frac{2}{77}a^{11}+\frac{9}{77}a^{7}-\frac{34}{77}a^{3}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{3}$, which has order $3$
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -\frac{18}{77} a^{13} - \frac{410}{77} a^{9} - \frac{1229}{77} a^{5} - \frac{32}{11} a \) (order $8$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
Fundamental units: | $\frac{1}{7}a^{12}+\frac{23}{7}a^{8}+\frac{72}{7}a^{4}+\frac{8}{7}$, $a$, $\frac{17}{77}a^{12}-\frac{386}{77}a^{8}-\frac{1121}{77}a^{4}-\frac{5}{77}$, $\frac{59}{77}a^{15}-\frac{12}{77}a^{14}-\frac{1350}{77}a^{11}-\frac{277}{77}a^{10}-\frac{4150}{77}a^{7}-\frac{900}{77}a^{6}-\frac{920}{77}a^{3}-\frac{384}{77}a^{2}+1$, $\frac{12}{77}a^{14}+\frac{12}{77}a^{13}+\frac{277}{77}a^{10}+\frac{277}{77}a^{9}+\frac{900}{77}a^{6}+\frac{900}{77}a^{5}+\frac{384}{77}a^{2}+\frac{307}{77}a+1$, $\frac{59}{77}a^{15}-\frac{18}{77}a^{13}+\frac{6}{77}a^{12}+\frac{1350}{77}a^{11}-\frac{410}{77}a^{9}+\frac{19}{11}a^{8}+\frac{4150}{77}a^{7}-\frac{1229}{77}a^{5}+\frac{47}{11}a^{4}+\frac{920}{77}a^{3}-\frac{32}{11}a-\frac{83}{77}$, $\frac{6}{77}a^{15}-\frac{6}{77}a^{14}-\frac{6}{77}a^{13}+\frac{144}{77}a^{11}-\frac{144}{77}a^{10}-\frac{19}{11}a^{9}+\frac{571}{77}a^{7}-\frac{571}{77}a^{6}-\frac{47}{11}a^{5}+\frac{467}{77}a^{3}-\frac{467}{77}a^{2}+\frac{83}{77}a$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 9192.496540646083 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 9192.496540646083 \cdot 3}{8\cdot\sqrt{1186620610969600000000}}\cr\approx \mathstrut & 0.243079170651684 \end{aligned}\]
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
Galois closure: | data not computed |
Degree 16 siblings: | data not computed |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/7.2.0.1}{2} }^{8}$ | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | Deg $16$ | $4$ | $4$ | $32$ | |||
\(5\) | 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(29\) | 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |