Normalized defining polynomial
\( x^{16} - 4 x^{14} - 4 x^{13} + 12 x^{12} + 8 x^{11} - 14 x^{10} - 12 x^{9} + 6 x^{8} + 12 x^{7} + \cdots + 1 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[2, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-401647199580061696\) \(\medspace = -\,2^{44}\cdot 17^{2}\cdot 79\) | 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: | \(12.60\) | 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: | not computed | ||
Ramified primes: | \(2\), \(17\), \(79\) | 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: | \(\Q(\sqrt{-79}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{11}-\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}+\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{15}-\frac{1}{4}a^{11}-\frac{1}{4}a^{8}+\frac{1}{4}a^{7}-\frac{1}{2}a^{6}+\frac{1}{4}a^{5}-\frac{1}{4}a^{2}+\frac{1}{4}a-\frac{1}{4}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $8$ | 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: | $\frac{1}{4}a^{15}-\frac{3}{4}a^{14}-\frac{7}{4}a^{13}+a^{12}+8a^{11}-\frac{3}{2}a^{10}-\frac{49}{4}a^{9}-3a^{8}+11a^{7}+\frac{33}{4}a^{6}-\frac{11}{4}a^{5}-8a^{4}-4a^{3}+\frac{5}{4}a^{2}+\frac{9}{2}a+\frac{11}{4}$, $a$, $\frac{3}{4}a^{15}-\frac{1}{2}a^{14}-\frac{13}{4}a^{13}-\frac{5}{4}a^{12}+\frac{47}{4}a^{11}+\frac{7}{4}a^{10}-\frac{31}{2}a^{9}-5a^{8}+\frac{21}{2}a^{7}+\frac{33}{4}a^{6}-\frac{1}{2}a^{5}-6a^{4}-\frac{11}{2}a^{3}-\frac{3}{4}a^{2}+\frac{17}{4}a+3$, $\frac{1}{2}a^{15}+a^{14}-\frac{1}{2}a^{13}-\frac{9}{2}a^{12}-2a^{11}+\frac{13}{2}a^{10}+8a^{9}-4a^{8}-11a^{7}-\frac{5}{2}a^{6}+\frac{11}{2}a^{5}+7a^{4}+\frac{5}{2}a^{3}-\frac{7}{2}a^{2}-\frac{9}{2}a-2$, $2a^{15}+\frac{3}{2}a^{14}-\frac{21}{4}a^{13}-\frac{23}{2}a^{12}+11a^{11}+\frac{69}{4}a^{10}-\frac{13}{4}a^{9}-19a^{8}-\frac{31}{4}a^{7}+8a^{6}+\frac{41}{4}a^{5}+\frac{27}{4}a^{4}-\frac{11}{4}a^{3}-\frac{31}{4}a^{2}-\frac{21}{4}a$, $\frac{3}{4}a^{15}+a^{14}-2a^{13}-\frac{11}{2}a^{12}+\frac{11}{4}a^{11}+\frac{21}{2}a^{10}-\frac{1}{2}a^{9}-\frac{39}{4}a^{8}-\frac{21}{4}a^{7}+4a^{6}+\frac{21}{4}a^{5}+3a^{4}-a^{3}-\frac{13}{4}a^{2}-\frac{7}{4}a-\frac{3}{4}$, $\frac{3}{4}a^{15}-\frac{3}{4}a^{14}-\frac{13}{4}a^{13}-\frac{3}{4}a^{12}+\frac{25}{2}a^{11}-\frac{31}{2}a^{9}-\frac{19}{4}a^{8}+\frac{23}{2}a^{7}+9a^{6}-\frac{11}{4}a^{5}-\frac{23}{4}a^{4}-\frac{17}{4}a^{3}-\frac{3}{2}a^{2}+\frac{15}{4}a+3$, $\frac{3}{2}a^{15}+\frac{1}{4}a^{14}-5a^{13}-\frac{23}{4}a^{12}+\frac{57}{4}a^{11}+\frac{33}{4}a^{10}-\frac{59}{4}a^{9}-9a^{8}+3a^{7}+\frac{19}{2}a^{6}+\frac{17}{4}a^{5}-\frac{3}{2}a^{4}-\frac{11}{2}a^{3}-4a^{2}+\frac{5}{4}a+\frac{9}{4}$ | 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: | \( 159.18386616 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{7}\cdot 159.18386616 \cdot 1}{2\cdot\sqrt{401647199580061696}}\cr\approx \mathstrut & 0.19420738964 \end{aligned}\]
Galois group
$C_4^4.C_2\wr C_4$ (as 16T1771):
A solvable group of order 16384 |
The 190 conjugacy class representatives for $C_4^4.C_2\wr C_4$ |
Character table for $C_4^4.C_2\wr C_4$ |
Intermediate fields
\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.4.71303168.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 16 siblings: | data not computed |
Degree 32 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 | $16$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | $16$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | $16$ | ${\href{/padicField/41.4.0.1}{4} }^{3}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 $16$ | $4$ | $4$ | $44$ | |||
\(17\) | $\Q_{17}$ | $x + 14$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{17}$ | $x + 14$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
17.4.0.1 | $x^{4} + 7 x^{2} + 10 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
17.4.0.1 | $x^{4} + 7 x^{2} + 10 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(79\) | $\Q_{79}$ | $x + 76$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{79}$ | $x + 76$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{79}$ | $x + 76$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{79}$ | $x + 76$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{79}$ | $x + 76$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{79}$ | $x + 76$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
79.2.1.1 | $x^{2} + 237$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
79.2.0.1 | $x^{2} + 78 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
79.2.0.1 | $x^{2} + 78 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
79.4.0.1 | $x^{4} + 2 x^{2} + 66 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |