Normalized defining polynomial
\( x^{17} + 9x - 9 \)
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: | \(42756627362882478930611573841\) \(\medspace = 3^{16}\cdot 89\cdot 421\cdot 811\cdot 827\cdot 39524364797\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(48.33\) | 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: | $3^{16/17}89^{1/2}421^{1/2}811^{1/2}827^{1/2}39524364797^{1/2}\approx 88631271802.77887$ | ||
Ramified primes: | \(3\), \(89\), \(421\), \(811\), \(827\), \(39524364797\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{99326\!\cdots\!28721}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}$, $\frac{1}{3}a^{10}$, $\frac{1}{3}a^{11}$, $\frac{1}{3}a^{12}$, $\frac{1}{3}a^{13}$, $\frac{1}{3}a^{14}$, $\frac{1}{3}a^{15}$, $\frac{1}{3}a^{16}$
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: | $a-1$, $\frac{1}{3}a^{11}+\frac{1}{3}a^{10}+a^{4}-a^{2}-a+1$, $\frac{1}{3}a^{15}+\frac{1}{3}a^{14}-\frac{1}{3}a^{12}-\frac{1}{3}a^{11}+\frac{1}{3}a^{10}+a^{9}+a^{8}-a^{6}-a^{5}+2a^{3}+2a^{2}-2$, $\frac{1}{3}a^{15}+\frac{1}{3}a^{14}+\frac{1}{3}a^{10}+a^{5}-a^{3}+a+1$, $a^{16}+\frac{4}{3}a^{15}+\frac{4}{3}a^{14}+a^{13}+a^{12}+\frac{4}{3}a^{11}+a^{10}-a^{8}-a^{7}-a^{5}-3a^{4}-3a^{3}-2a^{2}-a+7$, $a^{16}+\frac{4}{3}a^{15}+a^{14}-a^{12}-\frac{5}{3}a^{11}-2a^{10}-\frac{5}{3}a^{9}+2a^{7}+3a^{6}+3a^{5}+2a^{4}-a^{3}-4a^{2}-5a+4$, $\frac{1}{3}a^{14}+\frac{1}{3}a^{13}+a^{12}+\frac{1}{3}a^{11}+\frac{4}{3}a^{10}-\frac{1}{3}a^{9}+a^{8}-a^{7}-a^{5}-a^{4}-a^{2}+a-1$, $\frac{1}{3}a^{16}+a^{15}+\frac{5}{3}a^{14}-\frac{5}{3}a^{13}-\frac{7}{3}a^{12}+\frac{5}{3}a^{11}-3a^{10}-\frac{16}{3}a^{9}-a^{8}-6a^{6}-7a^{5}+8a^{4}-3a^{3}-11a^{2}+11a+11$ (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: | \( 46129308.1634 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{8}\cdot 46129308.1634 \cdot 1}{2\cdot\sqrt{42756627362882478930611573841}}\cr\approx \mathstrut & 0.541893781974 \end{aligned}\] (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 | ${\href{/padicField/2.12.0.1}{12} }{,}\,{\href{/padicField/2.3.0.1}{3} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ | R | ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | $17$ | ${\href{/padicField/11.9.0.1}{9} }{,}\,{\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ | $16{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.8.0.1}{8} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.9.0.1}{9} }{,}\,{\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ | $16{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.11.0.1}{11} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | ${\href{/padicField/53.13.0.1}{13} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.17.16.1 | $x^{17} + 3$ | $17$ | $1$ | $16$ | $F_{17}$ | $[\ ]_{17}^{16}$ |
\(89\) | 89.2.1.2 | $x^{2} + 267$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
89.3.0.1 | $x^{3} + 3 x + 86$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
89.3.0.1 | $x^{3} + 3 x + 86$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
89.9.0.1 | $x^{9} + 5 x^{3} + 12 x^{2} + 6 x + 86$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
\(421\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
\(811\) | $\Q_{811}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
\(827\) | $\Q_{827}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{827}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
\(39524364797\) | $\Q_{39524364797}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{39524364797}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ |