Normalized defining polynomial
\( x^{17} - 4x - 4 \)
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: |
\(17578609656292078845952\)
\(\medspace = 2^{16}\cdot 63535033\cdot 4221738529\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(20.35\) | 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: | $2^{16/17}63535033^{1/2}4221738529^{1/2}\approx 994430978.0542748$ | ||
Ramified primes: |
\(2\), \(63535033\), \(4221738529\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{26822\!\cdots\!86457}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}$, $\frac{1}{2}a^{10}$, $\frac{1}{2}a^{11}$, $\frac{1}{2}a^{12}$, $\frac{1}{2}a^{13}$, $\frac{1}{2}a^{14}$, $\frac{1}{2}a^{15}$, $\frac{1}{106}a^{16}-\frac{11}{106}a^{15}+\frac{15}{106}a^{14}-\frac{3}{53}a^{13}+\frac{13}{106}a^{12}+\frac{8}{53}a^{11}-\frac{17}{106}a^{10}-\frac{25}{106}a^{9}+\frac{5}{53}a^{8}-\frac{2}{53}a^{7}+\frac{22}{53}a^{6}+\frac{23}{53}a^{5}+\frac{12}{53}a^{4}-\frac{26}{53}a^{3}+\frac{21}{53}a^{2}-\frac{19}{53}a-\frac{5}{53}$
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)
| |
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: |
$\frac{5}{106}a^{16}-\frac{1}{53}a^{15}+\frac{11}{53}a^{14}-\frac{15}{53}a^{13}+\frac{6}{53}a^{12}-\frac{13}{53}a^{11}+\frac{21}{106}a^{10}-\frac{19}{106}a^{9}+\frac{25}{53}a^{8}-\frac{10}{53}a^{7}+\frac{4}{53}a^{6}+\frac{9}{53}a^{5}+\frac{7}{53}a^{4}-\frac{24}{53}a^{3}-\frac{1}{53}a^{2}+\frac{11}{53}a-\frac{25}{53}$, $\frac{11}{53}a^{16}-\frac{15}{53}a^{15}+\frac{6}{53}a^{14}-\frac{13}{53}a^{13}+\frac{21}{106}a^{12}-\frac{19}{106}a^{11}-\frac{3}{106}a^{10}-\frac{10}{53}a^{9}+\frac{4}{53}a^{8}+\frac{9}{53}a^{7}+\frac{7}{53}a^{6}+\frac{29}{53}a^{5}-\frac{1}{53}a^{4}+\frac{11}{53}a^{3}-\frac{15}{53}a^{2}+\frac{6}{53}a-\frac{57}{53}$, $\frac{11}{53}a^{16}-\frac{15}{53}a^{15}+\frac{6}{53}a^{14}-\frac{13}{53}a^{13}+\frac{21}{106}a^{12}-\frac{19}{106}a^{11}-\frac{3}{106}a^{10}+\frac{33}{106}a^{9}+\frac{4}{53}a^{8}+\frac{9}{53}a^{7}+\frac{7}{53}a^{6}+\frac{29}{53}a^{5}-\frac{1}{53}a^{4}+\frac{11}{53}a^{3}-\frac{15}{53}a^{2}+\frac{6}{53}a-\frac{57}{53}$, $\frac{1}{106}a^{16}-\frac{11}{106}a^{15}+\frac{15}{106}a^{14}-\frac{3}{53}a^{13}+\frac{13}{106}a^{12}+\frac{8}{53}a^{11}-\frac{17}{106}a^{10}+\frac{14}{53}a^{9}+\frac{5}{53}a^{8}-\frac{2}{53}a^{7}+\frac{22}{53}a^{6}-\frac{30}{53}a^{5}+\frac{12}{53}a^{4}-\frac{26}{53}a^{3}+\frac{21}{53}a^{2}-\frac{19}{53}a-\frac{5}{53}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{15}+\frac{1}{2}a^{14}-\frac{1}{2}a^{13}+\frac{1}{2}a^{12}-\frac{1}{2}a^{11}+\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-3$, $\frac{37}{106}a^{16}-\frac{18}{53}a^{15}+\frac{25}{106}a^{14}-\frac{5}{53}a^{13}+\frac{2}{53}a^{12}+\frac{9}{106}a^{11}+\frac{7}{106}a^{10}-\frac{12}{53}a^{9}+\frac{26}{53}a^{8}-\frac{21}{53}a^{7}+\frac{19}{53}a^{6}+\frac{3}{53}a^{5}+\frac{20}{53}a^{4}-\frac{8}{53}a^{3}+\frac{35}{53}a^{2}-\frac{14}{53}a-\frac{79}{53}$, $\frac{23}{106}a^{16}-\frac{41}{106}a^{15}+\frac{27}{106}a^{14}-\frac{16}{53}a^{13}+\frac{17}{53}a^{12}-\frac{3}{106}a^{11}+\frac{33}{106}a^{10}-\frac{45}{106}a^{9}+\frac{9}{53}a^{8}+\frac{7}{53}a^{7}+\frac{29}{53}a^{6}-\frac{1}{53}a^{5}+\frac{11}{53}a^{4}-\frac{15}{53}a^{3}+\frac{6}{53}a^{2}+\frac{40}{53}a-\frac{9}{53}$, $\frac{6}{53}a^{16}-\frac{13}{53}a^{15}+\frac{21}{106}a^{14}-\frac{19}{106}a^{13}-\frac{3}{106}a^{12}+\frac{33}{106}a^{11}+\frac{4}{53}a^{10}+\frac{9}{53}a^{9}+\frac{7}{53}a^{8}-\frac{24}{53}a^{7}-\frac{1}{53}a^{6}+\frac{11}{53}a^{5}-\frac{15}{53}a^{4}+\frac{6}{53}a^{3}-\frac{13}{53}a^{2}-\frac{16}{53}a-\frac{7}{53}$
| 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: | \( 38635.7554956 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{8}\cdot 38635.7554956 \cdot 1}{2\cdot\sqrt{17578609656292078845952}}\cr\approx \mathstrut & 0.707841619545 \end{aligned}\]
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 | $17$ | ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.9.0.1}{9} }{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.13.0.1}{13} }{,}\,{\href{/padicField/11.4.0.1}{4} }$ | $16{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.4.0.1}{4} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $17$ | ${\href{/padicField/23.13.0.1}{13} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.5.0.1}{5} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.13.0.1}{13} }{,}\,{\href{/padicField/47.4.0.1}{4} }$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\)
| 2.17.16.1 | $x^{17} + 2$ | $17$ | $1$ | $16$ | $C_{17}:C_{8}$ | $[\ ]_{17}^{8}$ |
\(63535033\)
| $\Q_{63535033}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{63535033}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
\(4221738529\)
| $\Q_{4221738529}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ |