Normalized defining polynomial
\( x^{17} + 2x - 2 \)
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: | \(56631869442212224526516224\) \(\medspace = 2^{16}\cdot 23\cdot 37\cdot 67\cdot 15155721101316377\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(32.73\) | 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: | $2^{16/17}23^{1/2}37^{1/2}67^{1/2}15155721101316377^{1/2}\approx 56443356590.24881$ | ||
Ramified primes: | \(2\), \(23\), \(37\), \(67\), \(15155721101316377\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{86413\!\cdots\!67409}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$
Monogenic: | Yes | |
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: | $a-1$, $a^{2}-a+1$, $a^{6}-a+1$, $a^{13}+a^{12}-a^{8}-a^{7}+a^{3}+a^{2}-1$, $a^{13}-a^{9}+a-1$, $a^{16}+a^{14}+a^{13}+a^{10}+a^{9}-a^{8}+a^{7}+a^{6}-a^{5}+a^{3}+a^{2}-2a+3$, $a^{15}+a^{14}+a^{11}+a^{10}+a^{5}+a^{4}-a^{3}-a^{2}+2a+1$, $a^{16}+a^{15}+a^{14}+a^{13}+a^{12}+a^{11}+a^{10}+a^{9}-a^{5}-a^{2}-a+1$ (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: | \( 1744681.36197 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{8}\cdot 1744681.36197 \cdot 1}{2\cdot\sqrt{56631869442212224526516224}}\cr\approx \mathstrut & 0.563150688530 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 355687428096000 |
The 297 conjugacy class representatives for $S_{17}$ are not computed |
Character table for $S_{17}$ is not computed |
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 | $17$ | ${\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $17$ | ${\href{/padicField/11.13.0.1}{13} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | $15{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $17$ | R | $17$ | ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.5.0.1}{5} }$ | R | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.5.0.1}{5} }$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.7.0.1}{7} }$ | ${\href{/padicField/47.9.0.1}{9} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.17.16.1 | $x^{17} + 2$ | $17$ | $1$ | $16$ | $C_{17}:C_{8}$ | $[\ ]_{17}^{8}$ |
\(23\) | $\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
23.5.0.1 | $x^{5} + 3 x + 18$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
23.7.0.1 | $x^{7} + 21 x + 18$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
\(37\) | $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
37.2.1.2 | $x^{2} + 74$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
37.14.0.1 | $x^{14} + 4 x^{8} + 35 x^{7} + 35 x^{6} + x^{5} + 32 x^{4} + 16 x^{3} + x^{2} + 9 x + 2$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | |
\(67\) | 67.2.1.2 | $x^{2} + 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
67.15.0.1 | $x^{15} + x^{7} + x^{6} + 52 x^{5} + 41 x^{4} + 20 x^{3} + 21 x^{2} + 46 x + 65$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | |
\(15155721101316377\) | $\Q_{15155721101316377}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{15155721101316377}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{15155721101316377}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ |