Normalized defining polynomial
\( x^{17} - 4x - 8 \)
Invariants
Degree: | $17$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[1, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(3469692417024426965689434112\) \(\medspace = 2^{22}\cdot 83\cdot 9966736578149758091\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(41.69\) | 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: | not computed | ||
Ramified primes: | \(2\), \(83\), \(9966736578149758091\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{82723\!\cdots\!21553}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}$, $\frac{1}{2}a^{9}$, $\frac{1}{2}a^{10}$, $\frac{1}{2}a^{11}$, $\frac{1}{2}a^{12}$, $\frac{1}{4}a^{13}-\frac{1}{2}a^{5}$, $\frac{1}{4}a^{14}-\frac{1}{2}a^{6}$, $\frac{1}{4}a^{15}-\frac{1}{2}a^{7}$, $\frac{1}{4}a^{16}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $8$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
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)
| |
Fundamental units: | $\frac{1}{2}a^{9}+a+1$, $\frac{1}{4}a^{13}-\frac{1}{2}a^{9}+\frac{1}{2}a^{5}-a-1$, $\frac{1}{4}a^{15}-\frac{1}{2}a^{12}+\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-a^{6}+a^{4}+a^{3}-a^{2}-2a-1$, $\frac{1}{4}a^{16}+\frac{1}{4}a^{15}-\frac{1}{2}a^{14}+\frac{3}{4}a^{13}-\frac{1}{2}a^{11}+a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}+\frac{3}{2}a^{7}-a^{6}+\frac{1}{2}a^{5}+a^{4}-2a^{3}+2a^{2}+2a-3$, $\frac{1}{4}a^{16}-\frac{1}{2}a^{15}-\frac{3}{4}a^{14}+a^{12}+a^{11}-\frac{1}{2}a^{10}-2a^{9}-\frac{3}{2}a^{8}+a^{7}+\frac{5}{2}a^{6}+a^{5}-2a^{4}-3a^{3}+5a+5$, $\frac{1}{2}a^{16}-\frac{3}{4}a^{14}+\frac{3}{4}a^{13}-\frac{1}{2}a^{11}+\frac{1}{2}a^{10}-\frac{1}{2}a^{9}+a^{7}-\frac{3}{2}a^{6}+\frac{1}{2}a^{5}+a^{4}-a^{3}+a^{2}-a-3$, $\frac{3}{4}a^{16}-a^{15}+\frac{1}{4}a^{14}-a^{13}+\frac{3}{2}a^{10}+\frac{3}{2}a^{8}-a^{7}-\frac{3}{2}a^{6}-a^{5}-2a^{4}+a^{3}+2a^{2}+3a-1$, $\frac{1}{4}a^{16}-\frac{1}{4}a^{15}+\frac{1}{4}a^{14}+\frac{1}{4}a^{13}+\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}+\frac{3}{2}a^{6}-\frac{1}{2}a^{5}+a^{4}-a^{3}-3$ (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: | \( 29494487.3135 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{8}\cdot 29494487.3135 \cdot 1}{2\cdot\sqrt{3469692417024426965689434112}}\cr\approx \mathstrut & 1.21628208480 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 355687428096000 |
The 297 conjugacy class representatives for $S_{17}$ |
Character table for $S_{17}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | $17$ | ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.5.0.1}{5} }^{2}$ | ${\href{/padicField/11.13.0.1}{13} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.4.0.1}{4} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.4.0.1}{4} }$ | ${\href{/padicField/37.13.0.1}{13} }{,}\,{\href{/padicField/37.4.0.1}{4} }$ | $15{,}\,{\href{/padicField/41.2.0.1}{2} }$ | $15{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | $15{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $16$ | $16$ | $1$ | $22$ | ||||
\(83\) | 83.2.1.1 | $x^{2} + 166$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
83.4.0.1 | $x^{4} + 4 x^{2} + 42 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
83.5.0.1 | $x^{5} + 9 x + 81$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
83.6.0.1 | $x^{6} + x^{4} + 76 x^{3} + 32 x^{2} + 17 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
\(9966736578149758091\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |