Normalized defining polynomial
\( x^{16} + 799x^{8} + 6561 \)
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: | \(229607785695641627262976\) \(\medspace = 2^{48}\cdot 13^{8}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(28.84\) | 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: | $2^{3}13^{1/2}\approx 28.844410203711913$ | ||
Ramified primes: | \(2\), \(13\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $16$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois and abelian over $\Q$. | |||
Conductor: | \(208=2^{4}\cdot 13\) | ||
Dirichlet character group: | $\lbrace$$\chi_{208}(1,·)$, $\chi_{208}(131,·)$, $\chi_{208}(129,·)$, $\chi_{208}(79,·)$, $\chi_{208}(77,·)$, $\chi_{208}(207,·)$, $\chi_{208}(25,·)$, $\chi_{208}(27,·)$, $\chi_{208}(157,·)$, $\chi_{208}(155,·)$, $\chi_{208}(103,·)$, $\chi_{208}(105,·)$, $\chi_{208}(51,·)$, $\chi_{208}(53,·)$, $\chi_{208}(183,·)$, $\chi_{208}(181,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{128}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{217}a^{8}+\frac{74}{217}$, $\frac{1}{651}a^{9}-\frac{143}{651}a$, $\frac{1}{1953}a^{10}-\frac{794}{1953}a^{2}$, $\frac{1}{5859}a^{11}-\frac{794}{5859}a^{3}$, $\frac{1}{17577}a^{12}-\frac{6653}{17577}a^{4}$, $\frac{1}{52731}a^{13}-\frac{24230}{52731}a^{5}$, $\frac{1}{158193}a^{14}+\frac{28501}{158193}a^{6}$, $\frac{1}{474579}a^{15}+\frac{186694}{474579}a^{7}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{15}$, which has order $15$ (assuming GRH)
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( \frac{97}{474579} a^{15} + \frac{75316}{474579} a^{7} \) (order $16$) | 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: | $\frac{40}{158193}a^{14}-\frac{1}{1953}a^{10}+\frac{32689}{158193}a^{6}-\frac{1159}{1953}a^{2}+1$, $\frac{1}{2511}a^{12}-\frac{4}{5859}a^{11}+\frac{1}{1953}a^{10}+\frac{880}{2511}a^{4}-\frac{2683}{5859}a^{3}+\frac{1159}{1953}a^{2}$, $\frac{19}{52731}a^{13}-\frac{4}{5859}a^{11}+\frac{14209}{52731}a^{5}-\frac{2683}{5859}a^{3}+1$, $\frac{1}{217}a^{8}+\frac{74}{217}$, $\frac{74}{474579}a^{15}+\frac{2}{52731}a^{13}+\frac{1}{651}a^{9}+\frac{52565}{474579}a^{7}+\frac{4271}{52731}a^{5}-\frac{143}{651}a$, $\frac{97}{474579}a^{15}-\frac{80}{158193}a^{14}+\frac{1}{2511}a^{13}-\frac{1}{1953}a^{11}+\frac{2}{1953}a^{10}-\frac{1}{651}a^{9}+\frac{1}{217}a^{8}+\frac{75316}{474579}a^{7}-\frac{65378}{158193}a^{6}+\frac{880}{2511}a^{5}-\frac{1159}{1953}a^{3}+\frac{2318}{1953}a^{2}-\frac{508}{651}a+\frac{291}{217}$, $\frac{1}{2187}a^{15}-\frac{40}{158193}a^{14}+\frac{1}{2511}a^{12}-\frac{1}{837}a^{11}+\frac{2}{1953}a^{10}+\frac{799}{2187}a^{7}-\frac{32689}{158193}a^{6}+\frac{880}{2511}a^{4}-\frac{880}{837}a^{3}+\frac{2318}{1953}a^{2}-a+2$ (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)]
| |
Regulator: | \( 108889.885552 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 108889.885552 \cdot 15}{16\cdot\sqrt{229607785695641627262976}}\cr\approx \mathstrut & 0.517492976016 \end{aligned}\] (assuming GRH)
Galois group
$C_2^2\times C_4$ (as 16T2):
An abelian group of order 16 |
The 16 conjugacy class representatives for $C_4\times C_2^2$ |
Character table for $C_4\times C_2^2$ |
Intermediate fields
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 | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | ${\href{/padicField/7.2.0.1}{2} }^{8}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/17.1.0.1}{1} }^{16}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.16.48.1 | $x^{16} - 8 x^{15} + 64 x^{14} + 8 x^{13} + 76 x^{12} + 48 x^{11} + 64 x^{10} + 256 x^{9} + 56 x^{8} + 144 x^{7} + 160 x^{6} + 432 x^{5} + 456 x^{4} + 256 x^{2} + 288 x + 516$ | $8$ | $2$ | $48$ | $C_4\times C_2^2$ | $[2, 3, 4]^{2}$ |
\(13\) | 13.8.4.1 | $x^{8} + 520 x^{7} + 101458 x^{6} + 8810644 x^{5} + 288610205 x^{4} + 142111548 x^{3} + 982314112 x^{2} + 3617879976 x + 920156436$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
13.8.4.1 | $x^{8} + 520 x^{7} + 101458 x^{6} + 8810644 x^{5} + 288610205 x^{4} + 142111548 x^{3} + 982314112 x^{2} + 3617879976 x + 920156436$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |