Normalized defining polynomial
\( x^{16} - 80 x^{14} + 2600 x^{12} - 44000 x^{10} + 412500 x^{8} - 2100000 x^{6} + 5250000 x^{4} + \cdots + 781250 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[16, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(236118324143482260684800000000\) \(\medspace = 2^{79}\cdot 5^{8}\) | 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: | \(68.52\) | 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: | $2^{79/16}5^{1/2}\approx 68.52051299580233$ | ||
Ramified primes: | \(2\), \(5\) | 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: | \(\Q(\sqrt{2}) \) | ||
$\card{ \Gal(K/\Q) }$: | $16$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois and abelian over $\Q$. | |||
Conductor: | \(320=2^{6}\cdot 5\) | ||
Dirichlet character group: | $\lbrace$$\chi_{320}(1,·)$, $\chi_{320}(69,·)$, $\chi_{320}(201,·)$, $\chi_{320}(269,·)$, $\chi_{320}(81,·)$, $\chi_{320}(149,·)$, $\chi_{320}(281,·)$, $\chi_{320}(29,·)$, $\chi_{320}(161,·)$, $\chi_{320}(229,·)$, $\chi_{320}(41,·)$, $\chi_{320}(109,·)$, $\chi_{320}(241,·)$, $\chi_{320}(309,·)$, $\chi_{320}(121,·)$, $\chi_{320}(189,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{5}a^{2}$, $\frac{1}{5}a^{3}$, $\frac{1}{25}a^{4}$, $\frac{1}{25}a^{5}$, $\frac{1}{125}a^{6}$, $\frac{1}{125}a^{7}$, $\frac{1}{625}a^{8}$, $\frac{1}{625}a^{9}$, $\frac{1}{3125}a^{10}$, $\frac{1}{3125}a^{11}$, $\frac{1}{15625}a^{12}$, $\frac{1}{15625}a^{13}$, $\frac{1}{78125}a^{14}$, $\frac{1}{78125}a^{15}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$ (assuming GRH)
Unit group
Rank: | $15$ | 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: | $\frac{1}{625}a^{8}-\frac{8}{125}a^{6}+\frac{4}{5}a^{4}-\frac{16}{5}a^{2}+1$, $\frac{1}{25}a^{4}-\frac{4}{5}a^{2}+1$, $\frac{1}{15625}a^{12}-\frac{12}{3125}a^{10}+\frac{54}{625}a^{8}-\frac{112}{125}a^{6}+\frac{21}{5}a^{4}-\frac{36}{5}a^{2}+3$, $\frac{1}{78125}a^{14}-\frac{14}{15625}a^{12}+\frac{77}{3125}a^{10}-\frac{42}{125}a^{8}+\frac{294}{125}a^{6}-\frac{196}{25}a^{4}+\frac{49}{5}a^{2}-3$, $\frac{1}{3125}a^{10}-\frac{2}{125}a^{8}+\frac{7}{25}a^{6}-2a^{4}+5a^{2}-3$, $\frac{1}{125}a^{6}-\frac{6}{25}a^{4}+\frac{9}{5}a^{2}-3$, $\frac{1}{78125}a^{14}-\frac{3}{3125}a^{12}+\frac{89}{3125}a^{10}-\frac{264}{625}a^{8}+\frac{81}{25}a^{6}-\frac{59}{5}a^{4}+\frac{76}{5}a^{2}-3$, $\frac{1}{3125}a^{10}-\frac{2}{125}a^{8}+\frac{7}{25}a^{6}-\frac{1}{25}a^{5}-2a^{4}+a^{3}+5a^{2}-5a+1$, $\frac{1}{78125}a^{15}-\frac{3}{3125}a^{13}+\frac{18}{625}a^{11}-\frac{11}{25}a^{9}+\frac{18}{5}a^{7}-\frac{378}{25}a^{5}+28a^{3}+\frac{1}{5}a^{2}-15a-5$, $\frac{1}{15625}a^{13}-\frac{13}{3125}a^{11}+\frac{13}{125}a^{9}-\frac{156}{125}a^{7}+\frac{1}{125}a^{6}+\frac{182}{25}a^{5}-\frac{6}{25}a^{4}-\frac{91}{5}a^{3}+\frac{9}{5}a^{2}+13a-5$, $\frac{1}{78125}a^{14}-\frac{14}{15625}a^{12}+\frac{77}{3125}a^{10}-\frac{42}{125}a^{8}+\frac{1}{125}a^{7}+\frac{294}{125}a^{6}-\frac{7}{25}a^{5}-\frac{196}{25}a^{4}+\frac{14}{5}a^{3}+\frac{49}{5}a^{2}-7a+1$, $\frac{1}{3125}a^{11}-\frac{1}{3125}a^{10}-\frac{11}{625}a^{9}+\frac{2}{125}a^{8}+\frac{44}{125}a^{7}-\frac{7}{25}a^{6}-\frac{77}{25}a^{5}+2a^{4}+11a^{3}-5a^{2}-11a+5$, $\frac{1}{5}a^{2}+a+1$, $\frac{1}{125}a^{6}-\frac{6}{25}a^{4}+\frac{1}{5}a^{3}+\frac{9}{5}a^{2}-3a+1$, $\frac{1}{78125}a^{14}-\frac{14}{15625}a^{12}+\frac{77}{3125}a^{10}+\frac{1}{625}a^{9}-\frac{42}{125}a^{8}-\frac{9}{125}a^{7}+\frac{294}{125}a^{6}+\frac{27}{25}a^{5}-\frac{196}{25}a^{4}-6a^{3}+\frac{49}{5}a^{2}+9a-5$ (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: | \( 2057965528.1974995 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{16}\cdot(2\pi)^{0}\cdot 2057965528.1974995 \cdot 2}{2\cdot\sqrt{236118324143482260684800000000}}\cr\approx \mathstrut & 0.277557636344263 \end{aligned}\] (assuming GRH)
Galois group
A cyclic group of order 16 |
The 16 conjugacy class representatives for $C_{16}$ |
Character table for $C_{16}$ |
Intermediate fields
\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), \(\Q(\zeta_{32})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | $16$ | R | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | $16$ | $16$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | $16$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | $16$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | $16$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | $16$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | $16$ | $16$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.16.79.4 | $x^{16} + 32 x^{15} + 16 x^{14} + 56 x^{12} + 48 x^{10} + 4 x^{8} + 32 x^{5} + 32 x^{4} + 32 x^{2} + 66$ | $16$ | $1$ | $79$ | $C_{16}$ | $[3, 4, 5, 6]$ |
\(5\) | 5.16.8.2 | $x^{16} + 625 x^{8} + 46875 x^{4} - 312500 x^{2} + 781250$ | $2$ | $8$ | $8$ | $C_{16}$ | $[\ ]_{2}^{8}$ |