Normalized defining polynomial
\( x^{17} - 6 \)
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: | \(2333735698654040712778226966003712\) \(\medspace = 2^{16}\cdot 3^{16}\cdot 17^{17}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(91.80\) | 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: | $2^{16/17}3^{16/17}17^{287/272}\approx 107.32038150548227$ | ||
Ramified primes: | \(2\), \(3\), \(17\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{17}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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: | $3a^{16}+7a^{15}+4a^{14}-5a^{13}-10a^{12}-3a^{11}+8a^{10}+13a^{9}+4a^{8}-15a^{7}-17a^{6}+3a^{5}+19a^{4}+19a^{3}-3a^{2}-32a-23$, $4a^{16}+3a^{15}+3a^{14}+6a^{13}+3a^{12}+7a^{11}+4a^{10}+7a^{9}+9a^{8}+5a^{7}+12a^{6}+7a^{5}+14a^{4}+12a^{3}+9a^{2}+20a+13$, $11a^{16}-6a^{15}+13a^{14}+8a^{13}+7a^{12}+2a^{11}-31a^{10}+4a^{9}-33a^{8}+41a^{7}-34a^{6}+75a^{5}-33a^{4}+73a^{3}-101a^{2}+25a-97$, $13a^{16}+13a^{15}+16a^{14}+19a^{13}+19a^{12}+22a^{11}+23a^{10}+23a^{9}+29a^{8}+31a^{7}+32a^{6}+38a^{5}+38a^{4}+43a^{3}+56a^{2}+59a+67$, $3a^{16}+36a^{15}+54a^{14}+44a^{13}+8a^{12}-46a^{11}-78a^{10}-82a^{9}-25a^{8}+44a^{7}+122a^{6}+125a^{5}+77a^{4}-52a^{3}-156a^{2}-216a-139$, $6a^{16}+13a^{15}+22a^{14}+30a^{13}+35a^{12}+34a^{11}+23a^{10}+4a^{9}-17a^{8}-36a^{7}-49a^{6}-50a^{5}-41a^{4}-30a^{3}-18a^{2}-3a+13$, $28a^{16}-a^{15}-40a^{14}+64a^{13}-52a^{12}+10a^{11}+24a^{10}-16a^{9}-46a^{8}+121a^{7}-147a^{6}+96a^{5}+10a^{4}-86a^{3}+67a^{2}+53a-167$, $2a^{16}+a^{15}-14a^{14}+11a^{13}+2a^{11}-5a^{9}-8a^{8}+7a^{7}+27a^{6}-35a^{5}+2a^{4}+8a^{3}+30a^{2}-70a+49$ (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: | \( 19074539557.9 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{8}\cdot 19074539557.9 \cdot 1}{2\cdot\sqrt{2333735698654040712778226966003712}}\cr\approx \mathstrut & 0.959107253255 \end{aligned}\] (assuming GRH)
Galois group
A solvable group of order 272 |
The 17 conjugacy class representatives for $F_{17}$ |
Character table for $F_{17}$ |
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 | R | $16{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | R | ${\href{/padicField/19.8.0.1}{8} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.8.0.1}{8} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.8.0.1}{8} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.8.0.1}{8} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 | $x^{17} + 2$ | $17$ | $1$ | $16$ | $C_{17}:C_{8}$ | $[\ ]_{17}^{8}$ |
\(3\) | 3.17.16.1 | $x^{17} + 3$ | $17$ | $1$ | $16$ | $F_{17}$ | $[\ ]_{17}^{16}$ |
\(17\) | 17.17.17.1 | $x^{17} + 17 x + 17$ | $17$ | $1$ | $17$ | $F_{17}$ | $[17/16]_{16}$ |