Normalized defining polynomial
\( x^{17} - x^{16} - 4 x^{15} + 4 x^{14} + 6 x^{13} - 6 x^{12} - 8 x^{11} + 8 x^{10} + 9 x^{9} - 7 x^{8} + \cdots - 1 \)
Invariants
Degree: | $17$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[5, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(37475477362914554710500388\) \(\medspace = 2^{2}\cdot 9368869340728638677625097\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(31.94\) | 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: | $2^{2/3}9368869340728638677625097^{1/2}\approx 4858813784474.296$ | ||
Ramified primes: | \(2\), \(9368869340728638677625097\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{93688\!\cdots\!25097}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{29}a^{16}+\frac{3}{29}a^{15}+\frac{8}{29}a^{14}+\frac{7}{29}a^{13}+\frac{5}{29}a^{12}+\frac{14}{29}a^{11}-\frac{10}{29}a^{10}-\frac{3}{29}a^{9}-\frac{3}{29}a^{8}+\frac{10}{29}a^{7}+\frac{2}{29}a^{6}+\frac{11}{29}a^{5}-\frac{5}{29}a^{4}+\frac{7}{29}a^{3}-\frac{7}{29}a^{2}+\frac{5}{29}a-\frac{7}{29}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $10$ | 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: | $a$, $\frac{12}{29}a^{16}+\frac{7}{29}a^{15}-\frac{49}{29}a^{14}-\frac{32}{29}a^{13}+\frac{60}{29}a^{12}+\frac{52}{29}a^{11}-\frac{33}{29}a^{10}-\frac{36}{29}a^{9}+\frac{22}{29}a^{8}+\frac{4}{29}a^{7}-\frac{34}{29}a^{6}-\frac{13}{29}a^{5}+\frac{56}{29}a^{4}+\frac{55}{29}a^{3}-\frac{26}{29}a^{2}+\frac{2}{29}a+\frac{32}{29}$, $a^{16}-4a^{14}+6a^{12}-8a^{10}+9a^{8}+2a^{7}-7a^{6}-4a^{5}+5a^{4}+3a^{3}-3a^{2}+a+2$, $\frac{17}{29}a^{16}-\frac{36}{29}a^{15}-\frac{38}{29}a^{14}+\frac{119}{29}a^{13}-\frac{2}{29}a^{12}-\frac{110}{29}a^{11}-\frac{25}{29}a^{10}+\frac{123}{29}a^{9}+\frac{36}{29}a^{8}-\frac{91}{29}a^{7}-\frac{111}{29}a^{6}+\frac{42}{29}a^{5}+\frac{205}{29}a^{4}-\frac{113}{29}a^{3}-\frac{90}{29}a^{2}+\frac{143}{29}a-\frac{32}{29}$, $\frac{16}{29}a^{16}-\frac{39}{29}a^{15}-\frac{46}{29}a^{14}+\frac{141}{29}a^{13}+\frac{22}{29}a^{12}-\frac{182}{29}a^{11}-\frac{15}{29}a^{10}+\frac{242}{29}a^{9}-\frac{19}{29}a^{8}-\frac{217}{29}a^{7}+\frac{3}{29}a^{6}+\frac{147}{29}a^{5}+\frac{94}{29}a^{4}-\frac{149}{29}a^{3}-\frac{25}{29}a^{2}+\frac{138}{29}a-\frac{54}{29}$, $\frac{16}{29}a^{16}-\frac{39}{29}a^{15}-\frac{46}{29}a^{14}+\frac{112}{29}a^{13}+\frac{51}{29}a^{12}-\frac{95}{29}a^{11}-\frac{102}{29}a^{10}+\frac{155}{29}a^{9}+\frac{68}{29}a^{8}-\frac{72}{29}a^{7}-\frac{142}{29}a^{6}+\frac{31}{29}a^{5}+\frac{123}{29}a^{4}-\frac{4}{29}a^{3}+\frac{4}{29}a^{2}+\frac{22}{29}a-\frac{25}{29}$, $\frac{39}{29}a^{16}-\frac{28}{29}a^{15}-\frac{152}{29}a^{14}+\frac{99}{29}a^{13}+\frac{224}{29}a^{12}-\frac{121}{29}a^{11}-\frac{303}{29}a^{10}+\frac{173}{29}a^{9}+\frac{318}{29}a^{8}-\frac{132}{29}a^{7}-\frac{299}{29}a^{6}+\frac{23}{29}a^{5}+\frac{298}{29}a^{4}-\frac{17}{29}a^{3}-\frac{157}{29}a^{2}+\frac{79}{29}a+\frac{75}{29}$, $\frac{8}{29}a^{16}-\frac{5}{29}a^{15}-\frac{23}{29}a^{14}-\frac{2}{29}a^{13}+\frac{40}{29}a^{12}+\frac{25}{29}a^{11}-\frac{80}{29}a^{10}+\frac{5}{29}a^{9}+\frac{34}{29}a^{8}+\frac{51}{29}a^{7}-\frac{71}{29}a^{6}+\frac{1}{29}a^{5}+\frac{18}{29}a^{4}+\frac{27}{29}a^{3}-\frac{27}{29}a^{2}+\frac{11}{29}a+\frac{31}{29}$, $\frac{25}{29}a^{16}-\frac{41}{29}a^{15}-\frac{61}{29}a^{14}+\frac{146}{29}a^{13}+\frac{9}{29}a^{12}-\frac{172}{29}a^{11}-\frac{18}{29}a^{10}+\frac{215}{29}a^{9}-\frac{17}{29}a^{8}-\frac{185}{29}a^{7}-\frac{8}{29}a^{6}+\frac{130}{29}a^{5}+\frac{78}{29}a^{4}-\frac{144}{29}a^{3}-\frac{1}{29}a^{2}+\frac{154}{29}a-\frac{59}{29}$, $\frac{14}{29}a^{16}-\frac{16}{29}a^{15}-\frac{62}{29}a^{14}+\frac{69}{29}a^{13}+\frac{128}{29}a^{12}-\frac{94}{29}a^{11}-\frac{227}{29}a^{10}+\frac{74}{29}a^{9}+\frac{306}{29}a^{8}-\frac{34}{29}a^{7}-\frac{378}{29}a^{6}-\frac{49}{29}a^{5}+\frac{394}{29}a^{4}+\frac{98}{29}a^{3}-\frac{272}{29}a^{2}-\frac{75}{29}a+\frac{47}{29}$ (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: | \( 3367432.32884 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{6}\cdot 3367432.32884 \cdot 1}{2\cdot\sqrt{37475477362914554710500388}}\cr\approx \mathstrut & 0.541532446550 \end{aligned}\] (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 | R | ${\href{/padicField/3.13.0.1}{13} }{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ | ${\href{/padicField/5.14.0.1}{14} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.7.0.1}{7} }$ | $16{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.5.0.1}{5} }$ | ${\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ | ${\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{5}$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.9.0.1}{9} }{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | $17$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
2.14.0.1 | $x^{14} + x^{7} + x^{5} + x^{3} + 1$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | |
\(936\!\cdots\!097\) | $\Q_{93\!\cdots\!97}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |