Normalized defining polynomial
\( x^{16} - 16x^{14} + 108x^{12} - 416x^{10} + 972x^{8} - 1216x^{6} + 496x^{4} + 96x^{2} + 144 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[8, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(2599167103947239325696\) \(\medspace = 2^{36}\cdot 3^{8}\cdot 7^{8}\) | 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: | \(21.80\) | 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: | $2^{9/4}3^{1/2}7^{1/2}\approx 21.798526485920096$ | ||
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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $8$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{8}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{4}-\frac{1}{2}$, $\frac{1}{8}a^{9}-\frac{1}{4}a^{5}-\frac{1}{2}a$, $\frac{1}{8}a^{10}-\frac{1}{4}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{11}-\frac{1}{4}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{16}a^{12}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{16}a^{13}-\frac{1}{2}a$, $\frac{1}{68016}a^{14}+\frac{571}{34008}a^{12}-\frac{157}{2834}a^{10}-\frac{1117}{34008}a^{8}-\frac{1}{4}a^{7}-\frac{29}{1417}a^{6}+\frac{557}{17004}a^{4}-\frac{511}{8502}a^{2}-\frac{279}{2834}$, $\frac{1}{204048}a^{15}+\frac{571}{102024}a^{13}-\frac{157}{8502}a^{11}-\frac{1117}{102024}a^{9}-\frac{1475}{8502}a^{7}-\frac{7945}{51012}a^{5}-\frac{9013}{25506}a^{3}+\frac{2555}{8502}a$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $11$ | 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: | $\frac{1237}{204048}a^{15}-\frac{7841}{102024}a^{13}+\frac{6925}{17004}a^{11}-\frac{131935}{102024}a^{9}+\frac{20359}{8502}a^{7}-\frac{110179}{51012}a^{5}+\frac{22547}{25506}a^{3}+\frac{6293}{8502}a$, $\frac{475}{34008}a^{14}-\frac{5929}{34008}a^{12}+\frac{2469}{2834}a^{10}-\frac{20855}{8502}a^{8}+\frac{5041}{1417}a^{6}-\frac{7489}{8502}a^{4}-\frac{4669}{4251}a^{2}-\frac{744}{1417}$, $\frac{31}{1872}a^{15}-\frac{25}{117}a^{13}+\frac{43}{39}a^{11}-\frac{365}{117}a^{9}+\frac{167}{39}a^{7}+\frac{56}{117}a^{5}-\frac{1177}{234}a^{3}-\frac{2}{39}a-1$, $\frac{209}{204048}a^{15}+\frac{27}{2834}a^{14}-\frac{3629}{204048}a^{13}-\frac{170}{1417}a^{12}+\frac{1195}{8502}a^{11}+\frac{1705}{2834}a^{10}-\frac{8458}{12753}a^{9}-\frac{18803}{11336}a^{8}+\frac{33853}{17004}a^{7}+\frac{6343}{2834}a^{6}-\frac{96955}{25506}a^{5}-\frac{133}{5668}a^{4}+\frac{105751}{25506}a^{3}-\frac{1342}{1417}a^{2}-\frac{18635}{8502}a-\frac{6501}{2834}$, $\frac{77}{17004}a^{15}+\frac{165}{11336}a^{14}-\frac{1337}{17004}a^{13}-\frac{4313}{22672}a^{12}+\frac{6373}{11336}a^{11}+\frac{11679}{11336}a^{10}-\frac{76223}{34008}a^{9}-\frac{4629}{1417}a^{8}+\frac{30871}{5668}a^{7}+\frac{33945}{5668}a^{6}-\frac{121763}{17004}a^{5}-\frac{7185}{1417}a^{4}+\frac{14828}{4251}a^{3}+\frac{5661}{2834}a^{2}+\frac{505}{2834}a-\frac{2729}{2834}$, $\frac{77}{17004}a^{15}-\frac{165}{11336}a^{14}-\frac{1337}{17004}a^{13}+\frac{4313}{22672}a^{12}+\frac{6373}{11336}a^{11}-\frac{11679}{11336}a^{10}-\frac{76223}{34008}a^{9}+\frac{4629}{1417}a^{8}+\frac{30871}{5668}a^{7}-\frac{33945}{5668}a^{6}-\frac{121763}{17004}a^{5}+\frac{7185}{1417}a^{4}+\frac{14828}{4251}a^{3}-\frac{5661}{2834}a^{2}+\frac{505}{2834}a+\frac{2729}{2834}$, $\frac{209}{204048}a^{15}-\frac{27}{2834}a^{14}-\frac{3629}{204048}a^{13}+\frac{170}{1417}a^{12}+\frac{1195}{8502}a^{11}-\frac{1705}{2834}a^{10}-\frac{8458}{12753}a^{9}+\frac{18803}{11336}a^{8}+\frac{33853}{17004}a^{7}-\frac{6343}{2834}a^{6}-\frac{96955}{25506}a^{5}+\frac{133}{5668}a^{4}+\frac{105751}{25506}a^{3}+\frac{1342}{1417}a^{2}-\frac{18635}{8502}a+\frac{6501}{2834}$, $\frac{77}{17004}a^{15}-\frac{17}{34008}a^{14}-\frac{1337}{17004}a^{13}-\frac{569}{68016}a^{12}+\frac{6373}{11336}a^{11}+\frac{757}{5668}a^{10}-\frac{76223}{34008}a^{9}-\frac{2692}{4251}a^{8}+\frac{30871}{5668}a^{7}+\frac{2403}{1417}a^{6}-\frac{121763}{17004}a^{5}-\frac{17971}{8502}a^{4}+\frac{14828}{4251}a^{3}+\frac{185}{4251}a^{2}-\frac{2329}{2834}a-\frac{3267}{2834}$, $\frac{77}{51012}a^{15}+\frac{475}{68016}a^{14}-\frac{1337}{51012}a^{13}-\frac{5929}{68016}a^{12}+\frac{6373}{34008}a^{11}+\frac{2469}{5668}a^{10}-\frac{76223}{102024}a^{9}-\frac{20855}{17004}a^{8}+\frac{30871}{17004}a^{7}+\frac{5041}{2834}a^{6}-\frac{113261}{51012}a^{5}-\frac{1619}{8502}a^{4}+\frac{2075}{12753}a^{3}-\frac{13171}{8502}a^{2}+\frac{9007}{8502}a+\frac{673}{2834}$, $\frac{331}{15696}a^{15}+\frac{17}{34008}a^{14}-\frac{1147}{3924}a^{13}+\frac{569}{68016}a^{12}+\frac{4355}{2616}a^{11}-\frac{757}{5668}a^{10}-\frac{42073}{7848}a^{9}+\frac{2692}{4251}a^{8}+\frac{12725}{1308}a^{7}-\frac{2403}{1417}a^{6}-\frac{26221}{3924}a^{5}+\frac{17971}{8502}a^{4}-\frac{3025}{1962}a^{3}-\frac{185}{4251}a^{2}-\frac{571}{654}a+\frac{3267}{2834}$, $\frac{3499}{204048}a^{15}+\frac{685}{34008}a^{14}-\frac{46843}{204048}a^{13}-\frac{1052}{4251}a^{12}+\frac{42905}{34008}a^{11}+\frac{13929}{11336}a^{10}-\frac{100327}{25506}a^{9}-\frac{59479}{17004}a^{8}+\frac{114155}{17004}a^{7}+\frac{29541}{5668}a^{6}-\frac{94655}{25506}a^{5}-\frac{18049}{8502}a^{4}-\frac{24665}{12753}a^{3}+\frac{1345}{8502}a^{2}-\frac{4159}{8502}a-\frac{1237}{1417}$ (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: | \( 79816.8724552 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 79816.8724552 \cdot 1}{2\cdot\sqrt{2599167103947239325696}}\cr\approx \mathstrut & 0.312325215696 \end{aligned}\] (assuming GRH)
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
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 | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{8}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 $16$ | $8$ | $2$ | $36$ | |||
\(3\) | 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(7\) | 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 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |