Normalized defining polynomial
\( x^{17} - 2 x^{16} - 2 x^{15} + 9 x^{14} - 8 x^{13} - 6 x^{12} + 22 x^{11} - 20 x^{10} - 4 x^{9} + \cdots + 1 \)
Invariants
Degree: | $17$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[5, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(47814524918055414168841988\) \(\medspace = 2^{2}\cdot 71\cdot 113\cdot 398023\cdot 3743302227733793\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(32.40\) | 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))
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Galois root discriminant: | $2^{2/3}71^{1/2}113^{1/2}398023^{1/2}3743302227733793^{1/2}\approx 5488284177909.384$ | ||
Ramified primes: | \(2\), \(71\), \(113\), \(398023\), \(3743302227733793\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | $\Q(\sqrt{11953\!\cdots\!10497}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{17}a^{16}+\frac{3}{17}a^{15}-\frac{4}{17}a^{14}+\frac{6}{17}a^{13}+\frac{5}{17}a^{12}+\frac{2}{17}a^{11}-\frac{2}{17}a^{10}+\frac{4}{17}a^{9}-\frac{1}{17}a^{8}+\frac{1}{17}a^{7}+\frac{2}{17}a^{6}+\frac{1}{17}a^{4}-\frac{6}{17}a^{3}-\frac{2}{17}a^{2}+\frac{6}{17}a-\frac{7}{17}$
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)
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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)
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Fundamental units: | $a$, $\frac{1}{17}a^{16}-\frac{14}{17}a^{15}+\frac{13}{17}a^{14}+\frac{40}{17}a^{13}-\frac{80}{17}a^{12}+\frac{36}{17}a^{11}+\frac{83}{17}a^{10}-\frac{183}{17}a^{9}+\frac{118}{17}a^{8}+\frac{86}{17}a^{7}-\frac{168}{17}a^{6}+7a^{5}-\frac{33}{17}a^{4}-\frac{23}{17}a^{3}+\frac{117}{17}a^{2}-\frac{11}{17}a-\frac{24}{17}$, $\frac{40}{17}a^{16}-\frac{67}{17}a^{15}-\frac{92}{17}a^{14}+\frac{308}{17}a^{13}-\frac{225}{17}a^{12}-\frac{243}{17}a^{11}+\frac{702}{17}a^{10}-\frac{554}{17}a^{9}-\frac{210}{17}a^{8}+\frac{652}{17}a^{7}-\frac{498}{17}a^{6}+11a^{5}-\frac{28}{17}a^{4}-\frac{342}{17}a^{3}+\frac{260}{17}a^{2}+\frac{36}{17}a-\frac{59}{17}$, $\frac{9}{17}a^{16}-\frac{7}{17}a^{15}-\frac{36}{17}a^{14}+\frac{54}{17}a^{13}+\frac{11}{17}a^{12}-\frac{101}{17}a^{11}+\frac{118}{17}a^{10}-\frac{15}{17}a^{9}-\frac{128}{17}a^{8}+\frac{128}{17}a^{7}-\frac{67}{17}a^{6}+2a^{5}+\frac{43}{17}a^{4}-\frac{207}{17}a^{3}+\frac{67}{17}a^{2}+\frac{88}{17}a-\frac{46}{17}$, $\frac{33}{17}a^{16}-\frac{71}{17}a^{15}-\frac{30}{17}a^{14}+\frac{249}{17}a^{13}-\frac{328}{17}a^{12}+\frac{32}{17}a^{11}+\frac{495}{17}a^{10}-\frac{718}{17}a^{9}+\frac{324}{17}a^{8}+\frac{237}{17}a^{7}-\frac{512}{17}a^{6}+27a^{5}-\frac{341}{17}a^{4}-\frac{45}{17}a^{3}+\frac{172}{17}a^{2}-\frac{74}{17}a+\frac{7}{17}$, $\frac{25}{17}a^{16}-\frac{44}{17}a^{15}-\frac{49}{17}a^{14}+\frac{184}{17}a^{13}-\frac{164}{17}a^{12}-\frac{86}{17}a^{11}+\frac{392}{17}a^{10}-\frac{393}{17}a^{9}+\frac{26}{17}a^{8}+\frac{263}{17}a^{7}-\frac{324}{17}a^{6}+14a^{5}-\frac{145}{17}a^{4}-\frac{116}{17}a^{3}+\frac{154}{17}a^{2}-\frac{20}{17}a-\frac{22}{17}$, $\frac{3}{17}a^{16}-\frac{8}{17}a^{15}-\frac{12}{17}a^{14}+\frac{52}{17}a^{13}-\frac{19}{17}a^{12}-\frac{113}{17}a^{11}+\frac{181}{17}a^{10}-\frac{5}{17}a^{9}-\frac{309}{17}a^{8}+\frac{360}{17}a^{7}+\frac{23}{17}a^{6}-23a^{5}+\frac{309}{17}a^{4}+\frac{16}{17}a^{3}-\frac{125}{17}a^{2}+\frac{86}{17}a-\frac{21}{17}$, $\frac{24}{17}a^{16}-\frac{30}{17}a^{15}-\frac{79}{17}a^{14}+\frac{178}{17}a^{13}-\frac{50}{17}a^{12}-\frac{258}{17}a^{11}+\frac{428}{17}a^{10}-\frac{159}{17}a^{9}-\frac{381}{17}a^{8}+\frac{483}{17}a^{7}-\frac{190}{17}a^{6}-5a^{5}+\frac{143}{17}a^{4}-\frac{280}{17}a^{3}+\frac{139}{17}a^{2}+\frac{110}{17}a-\frac{66}{17}$, $\frac{52}{17}a^{16}-\frac{184}{17}a^{15}+\frac{64}{17}a^{14}+\frac{550}{17}a^{13}-\frac{1015}{17}a^{12}+\frac{461}{17}a^{11}+\frac{1069}{17}a^{10}-\frac{2223}{17}a^{9}+\frac{1529}{17}a^{8}+\frac{375}{17}a^{7}-\frac{1511}{17}a^{6}+91a^{5}-\frac{1138}{17}a^{4}+\frac{351}{17}a^{3}+\frac{559}{17}a^{2}-\frac{334}{17}a+\frac{27}{17}$, $\frac{55}{17}a^{16}-\frac{90}{17}a^{15}-\frac{152}{17}a^{14}+\frac{449}{17}a^{13}-\frac{252}{17}a^{12}-\frac{468}{17}a^{11}+\frac{1063}{17}a^{10}-\frac{664}{17}a^{9}-\frac{565}{17}a^{8}+\frac{1109}{17}a^{7}-\frac{638}{17}a^{6}+5a^{5}+\frac{123}{17}a^{4}-\frac{568}{17}a^{3}+\frac{383}{17}a^{2}+\frac{109}{17}a-\frac{130}{17}$ (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)]
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Regulator: | \( 4697339.96204 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{6}\cdot 4697339.96204 \cdot 1}{2\cdot\sqrt{47814524918055414168841988}}\cr\approx \mathstrut & 0.668761499602 \end{aligned}\] (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 | ${\href{/padicField/3.14.0.1}{14} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ | ${\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.7.0.1}{7} }$ | ${\href{/padicField/7.11.0.1}{11} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $17$ | ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | $15{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.9.0.1}{9} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | $17$ | ${\href{/padicField/41.9.0.1}{9} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.5.0.1}{5} }$ | $16{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.3.0.1 | $x^{3} + x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
2.4.0.1 | $x^{4} + x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
2.7.0.1 | $x^{7} + x + 1$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
\(71\) | 71.2.0.1 | $x^{2} + 69 x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
71.2.1.1 | $x^{2} + 497$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
71.6.0.1 | $x^{6} + x^{4} + 10 x^{3} + 13 x^{2} + 29 x + 7$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
71.7.0.1 | $x^{7} + 2 x + 64$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
\(113\) | $\Q_{113}$ | $x + 110$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
113.2.0.1 | $x^{2} + 101 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
113.2.0.1 | $x^{2} + 101 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
113.2.1.2 | $x^{2} + 339$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
113.3.0.1 | $x^{3} + 8 x + 110$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
113.3.0.1 | $x^{3} + 8 x + 110$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
113.4.0.1 | $x^{4} + 62 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(398023\) | $\Q_{398023}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{398023}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{398023}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{398023}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
\(3743302227733793\) | $\Q_{3743302227733793}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |