Normalized defining polynomial
\( x^{16} - x^{14} - 24x^{12} - 19x^{10} + 91x^{8} + 57x^{6} - 216x^{4} + 27x^{2} + 81 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[4, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(2499114735283240471761\) \(\medspace = 3^{12}\cdot 7^{8}\cdot 13^{8}\) | 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: | \(21.75\) | 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: | $3^{3/4}7^{1/2}13^{1/2}\approx 21.745111415357325$ | ||
Ramified primes: | \(3\), \(7\), \(13\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{6}a^{10}-\frac{1}{6}a^{8}-\frac{1}{2}a^{7}-\frac{1}{6}a^{4}-\frac{1}{2}a^{3}-\frac{1}{3}a^{2}-\frac{1}{2}a$, $\frac{1}{6}a^{11}-\frac{1}{6}a^{9}-\frac{1}{2}a^{7}+\frac{1}{3}a^{5}+\frac{1}{6}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{126}a^{12}-\frac{2}{63}a^{10}-\frac{5}{21}a^{8}-\frac{1}{2}a^{7}-\frac{23}{63}a^{6}-\frac{25}{63}a^{4}+\frac{3}{14}a^{2}+\frac{2}{7}$, $\frac{1}{378}a^{13}-\frac{25}{378}a^{11}+\frac{1}{7}a^{9}+\frac{17}{378}a^{7}-\frac{1}{2}a^{6}-\frac{155}{378}a^{5}-\frac{1}{2}a^{4}+\frac{1}{63}a^{3}+\frac{2}{21}a-\frac{1}{2}$, $\frac{1}{97902}a^{14}-\frac{122}{48951}a^{12}+\frac{554}{16317}a^{10}+\frac{491}{13986}a^{8}+\frac{1579}{13986}a^{6}-\frac{1}{2}a^{5}-\frac{2416}{16317}a^{4}+\frac{179}{3626}a^{2}-\frac{1}{2}a-\frac{769}{1813}$, $\frac{1}{293706}a^{15}-\frac{122}{146853}a^{13}-\frac{4331}{97902}a^{11}-\frac{4171}{41958}a^{9}+\frac{15565}{41958}a^{7}+\frac{8462}{48951}a^{5}+\frac{4163}{32634}a^{3}-\frac{1}{2}a^{2}+\frac{275}{10878}a$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $9$ | 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: | $\frac{346}{48951}a^{15}+\frac{34}{1813}a^{14}-\frac{83}{16317}a^{13}-\frac{8}{1813}a^{12}-\frac{7885}{48951}a^{11}-\frac{1629}{3626}a^{10}-\frac{1276}{6993}a^{9}-\frac{178}{259}a^{8}+\frac{836}{2331}a^{7}+\frac{258}{259}a^{6}+\frac{16673}{97902}a^{5}+\frac{2862}{1813}a^{4}-\frac{19387}{16317}a^{3}-\frac{3510}{1813}a^{2}+\frac{2104}{5439}a-\frac{2481}{3626}$, $\frac{346}{48951}a^{15}+\frac{34}{1813}a^{14}-\frac{83}{16317}a^{13}-\frac{8}{1813}a^{12}-\frac{7885}{48951}a^{11}-\frac{1629}{3626}a^{10}-\frac{1276}{6993}a^{9}-\frac{178}{259}a^{8}+\frac{836}{2331}a^{7}+\frac{258}{259}a^{6}+\frac{16673}{97902}a^{5}+\frac{2862}{1813}a^{4}-\frac{19387}{16317}a^{3}-\frac{3510}{1813}a^{2}+\frac{2104}{5439}a+\frac{1145}{3626}$, $\frac{52}{5439}a^{14}+\frac{1}{1813}a^{12}-\frac{400}{1813}a^{10}-\frac{368}{777}a^{8}+\frac{88}{259}a^{6}+\frac{2556}{1813}a^{4}-\frac{2504}{5439}a^{2}-\frac{1836}{1813}$, $\frac{274}{48951}a^{14}-\frac{811}{48951}a^{12}-\frac{676}{5439}a^{10}+\frac{668}{6993}a^{8}+\frac{5629}{6993}a^{6}+\frac{11954}{16317}a^{4}-\frac{4636}{5439}a^{2}+\frac{240}{1813}$, $\frac{2090}{146853}a^{15}-\frac{211}{13986}a^{14}-\frac{3356}{146853}a^{13}+\frac{101}{6993}a^{12}-\frac{31855}{97902}a^{11}+\frac{1643}{4662}a^{10}-\frac{3229}{41958}a^{9}+\frac{314}{999}a^{8}+\frac{26942}{20979}a^{7}-\frac{2167}{1998}a^{6}+\frac{1327}{32634}a^{5}-\frac{3757}{4662}a^{4}-\frac{14908}{5439}a^{3}+\frac{1492}{777}a^{2}+\frac{25699}{10878}a-\frac{527}{518}$, $\frac{2090}{146853}a^{15}+\frac{211}{13986}a^{14}-\frac{3356}{146853}a^{13}-\frac{101}{6993}a^{12}-\frac{31855}{97902}a^{11}-\frac{1643}{4662}a^{10}-\frac{3229}{41958}a^{9}-\frac{314}{999}a^{8}+\frac{26942}{20979}a^{7}+\frac{2167}{1998}a^{6}+\frac{1327}{32634}a^{5}+\frac{3757}{4662}a^{4}-\frac{14908}{5439}a^{3}-\frac{1492}{777}a^{2}+\frac{25699}{10878}a+\frac{527}{518}$, $\frac{550}{146853}a^{15}+\frac{179}{32634}a^{14}-\frac{556}{146853}a^{13}-\frac{82}{16317}a^{12}-\frac{1304}{16317}a^{11}-\frac{183}{1813}a^{10}-\frac{1691}{41958}a^{9}-\frac{689}{4662}a^{8}+\frac{1415}{41958}a^{7}-\frac{482}{2331}a^{6}-\frac{72263}{97902}a^{5}-\frac{970}{5439}a^{4}-\frac{12511}{32634}a^{3}+\frac{17}{1813}a^{2}+\frac{3881}{3626}a-\frac{991}{3626}$, $\frac{334}{146853}a^{15}-\frac{433}{32634}a^{14}-\frac{2242}{146853}a^{13}-\frac{10}{16317}a^{12}-\frac{2111}{48951}a^{11}+\frac{1263}{3626}a^{10}+\frac{9973}{41958}a^{9}+\frac{2515}{4662}a^{8}+\frac{20141}{41958}a^{7}-\frac{2804}{2331}a^{6}-\frac{8606}{48951}a^{5}-\frac{9556}{5439}a^{4}-\frac{1553}{32634}a^{3}+\frac{3527}{1813}a^{2}+\frac{8875}{10878}a+\frac{2558}{1813}$, $\frac{11720}{146853}a^{15}+\frac{2788}{146853}a^{13}-\frac{93481}{48951}a^{11}-\frac{81971}{20979}a^{9}+\frac{59012}{20979}a^{7}+\frac{459812}{48951}a^{5}-\frac{74393}{16317}a^{3}-\frac{25637}{5439}a$ (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: | \( 55788.699551 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 55788.699551 \cdot 1}{2\cdot\sqrt{2499114735283240471761}}\cr\approx \mathstrut & 0.54931611205 \end{aligned}\] (assuming GRH)
Galois group
$C_2^2\wr C_2$ (as 16T46):
A solvable group of order 32 |
The 14 conjugacy class representatives for $C_2^2\wr C_2$ |
Character table for $C_2^2\wr C_2$ |
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 | ${\href{/padicField/2.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/17.2.0.1}{2} }^{6}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.4.3.2 | $x^{4} + 6$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ |
3.4.3.2 | $x^{4} + 6$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
3.8.6.2 | $x^{8} + 6 x^{5} + 6 x^{4} + 18 x^{2} + 18 x + 9$ | $4$ | $2$ | $6$ | $D_4$ | $[\ ]_{4}^{2}$ | |
\(7\) | 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 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(13\) | 13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |