Normalized defining polynomial
\( x^{17} - 7 \)
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: | \(27491618187647178051040055821983377\) \(\medspace = 7^{16}\cdot 17^{17}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(106.13\) | 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: | $7^{16/17}17^{287/272}\approx 124.07690477638815$ | ||
Ramified primes: | \(7\), \(17\) | 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: | $a^{5}-a^{3}-a^{2}+a+1$, $7a^{16}+4a^{15}-2a^{14}-3a^{13}+5a^{12}+14a^{11}+13a^{10}-12a^{8}-8a^{7}+14a^{6}+33a^{5}+28a^{4}-20a^{2}-7a+29$, $11a^{16}+2a^{15}-4a^{14}+13a^{13}+22a^{12}+4a^{11}+22a^{9}+25a^{8}+7a^{7}+5a^{6}+13a^{5}+18a^{4}+25a^{3}+9a^{2}-14a+29$, $7a^{16}+14a^{15}-5a^{14}-14a^{13}-2a^{12}+18a^{11}+9a^{10}-30a^{9}-11a^{8}+39a^{7}+15a^{6}-33a^{5}-27a^{4}+22a^{3}+52a^{2}-32a-83$, $95a^{16}-29a^{15}-101a^{14}-13a^{13}+173a^{12}+25a^{11}-190a^{10}-118a^{9}+228a^{8}+202a^{7}-222a^{6}-277a^{5}+154a^{4}+408a^{3}-194a^{2}-394a+62$, $9a^{16}-79a^{15}-94a^{14}+10a^{13}+124a^{12}+111a^{11}-36a^{10}-182a^{9}-134a^{8}+101a^{7}+254a^{6}+132a^{5}-184a^{4}-365a^{3}-114a^{2}+349a+477$, $5a^{16}-121a^{15}+44a^{14}-39a^{13}-153a^{12}+86a^{11}-108a^{10}-163a^{9}+128a^{8}-230a^{7}-153a^{6}+163a^{5}-390a^{4}-78a^{3}+142a^{2}-625a+69$, $68a^{16}+72a^{15}-34a^{14}-153a^{13}-90a^{12}+143a^{11}+240a^{10}+7a^{9}-284a^{8}-222a^{7}+138a^{6}+330a^{5}+135a^{4}-176a^{3}-300a^{2}-204a+85$ (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: | \( 70086534327.5 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{8}\cdot 70086534327.5 \cdot 1}{2\cdot\sqrt{27491618187647178051040055821983377}}\cr\approx \mathstrut & 1.02677049486 \end{aligned}\] (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 | ${\href{/padicField/2.8.0.1}{8} }^{2}{,}\,{\href{/padicField/2.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/3.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/5.1.0.1}{1} }$ | R | $16{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | 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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.17.16.1 | $x^{17} + 7$ | $17$ | $1$ | $16$ | $F_{17}$ | $[\ ]_{17}^{16}$ |
\(17\) | 17.17.17.1 | $x^{17} + 17 x + 17$ | $17$ | $1$ | $17$ | $F_{17}$ | $[17/16]_{16}$ |