Normalized defining polynomial
\( x^{17} - 9x - 4 \)
Invariants
Degree: | $17$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[3, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-2538753443209963749199249408\) \(\medspace = -\,2^{16}\cdot 38738303271636409747303\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(40.93\) | 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: | not computed | ||
Ramified primes: | \(2\), \(38738303271636409747303\) | 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: | $\Q(\sqrt{-38738\!\cdots\!47303}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{7}$, $\frac{1}{86}a^{16}-\frac{8}{43}a^{15}-\frac{1}{43}a^{14}-\frac{11}{86}a^{13}+\frac{2}{43}a^{12}-\frac{21}{86}a^{11}-\frac{4}{43}a^{10}-\frac{1}{86}a^{9}-\frac{27}{86}a^{8}+\frac{1}{43}a^{7}-\frac{16}{43}a^{6}+\frac{39}{86}a^{5}-\frac{11}{43}a^{4}-\frac{35}{86}a^{3}-\frac{21}{43}a^{2}+\frac{27}{86}a+\frac{16}{43}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $9$ | 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)
| |
Fundamental units: | $\frac{10}{43}a^{16}-\frac{19}{86}a^{15}+\frac{3}{86}a^{14}-\frac{5}{86}a^{13}-\frac{3}{43}a^{12}+\frac{5}{43}a^{11}-\frac{31}{86}a^{10}+\frac{23}{86}a^{9}-\frac{12}{43}a^{8}-\frac{3}{86}a^{7}+\frac{5}{86}a^{6}-\frac{37}{86}a^{5}-\frac{5}{43}a^{4}-\frac{6}{43}a^{3}-\frac{23}{86}a^{2}-\frac{19}{86}a-\frac{67}{43}$, $\frac{3}{86}a^{16}-\frac{5}{86}a^{15}-\frac{3}{43}a^{14}+\frac{5}{43}a^{13}+\frac{6}{43}a^{12}-\frac{10}{43}a^{11}-\frac{12}{43}a^{10}+\frac{20}{43}a^{9}+\frac{5}{86}a^{8}-\frac{37}{86}a^{7}-\frac{5}{43}a^{6}+\frac{37}{43}a^{5}+\frac{10}{43}a^{4}-\frac{31}{43}a^{3}+\frac{23}{43}a^{2}+\frac{19}{43}a+\frac{5}{43}$, $\frac{13}{86}a^{16}+\frac{7}{86}a^{15}+\frac{17}{86}a^{14}-\frac{7}{43}a^{13}+\frac{9}{86}a^{12}-\frac{15}{86}a^{11}+\frac{25}{86}a^{10}-\frac{13}{86}a^{9}-\frac{7}{86}a^{8}-\frac{17}{86}a^{7}-\frac{29}{86}a^{6}+\frac{17}{43}a^{5}+\frac{15}{86}a^{4}+\frac{61}{86}a^{3}+\frac{13}{86}a^{2}+\frac{7}{86}a-\frac{7}{43}$, $\frac{17}{86}a^{16}-\frac{7}{43}a^{15}+\frac{9}{86}a^{14}+\frac{14}{43}a^{13}+\frac{25}{86}a^{12}-\frac{13}{86}a^{11}-\frac{7}{86}a^{10}+\frac{13}{43}a^{9}+\frac{57}{86}a^{8}+\frac{17}{43}a^{7}+\frac{15}{86}a^{6}+\frac{9}{43}a^{5}+\frac{13}{86}a^{4}+\frac{7}{86}a^{3}+\frac{103}{86}a^{2}+\frac{79}{43}a-\frac{29}{43}$, $\frac{106}{43}a^{16}-\frac{81}{86}a^{15}+\frac{49}{86}a^{14}-\frac{5}{43}a^{13}-\frac{6}{43}a^{12}-\frac{23}{86}a^{11}-\frac{19}{86}a^{10}+\frac{3}{86}a^{9}+\frac{19}{43}a^{8}+\frac{37}{86}a^{7}+\frac{53}{86}a^{6}+\frac{6}{43}a^{5}-\frac{10}{43}a^{4}-\frac{67}{86}a^{3}-\frac{89}{86}a^{2}-\frac{81}{86}a-\frac{951}{43}$, $\frac{61}{43}a^{16}-\frac{30}{43}a^{15}+\frac{7}{43}a^{14}-\frac{9}{86}a^{13}+\frac{15}{86}a^{12}+\frac{9}{43}a^{11}-\frac{15}{43}a^{10}+\frac{7}{86}a^{9}+\frac{30}{43}a^{8}-\frac{7}{43}a^{7}-\frac{17}{43}a^{6}-\frac{15}{86}a^{5}+\frac{25}{86}a^{4}+\frac{15}{43}a^{3}-\frac{68}{43}a^{2}-\frac{103}{86}a-\frac{499}{43}$, $\frac{17}{86}a^{16}-\frac{7}{43}a^{15}+\frac{9}{86}a^{14}-\frac{15}{86}a^{13}+\frac{25}{86}a^{12}-\frac{13}{86}a^{11}-\frac{7}{86}a^{10}-\frac{17}{86}a^{9}+\frac{57}{86}a^{8}-\frac{26}{43}a^{7}+\frac{15}{86}a^{6}-\frac{25}{86}a^{5}+\frac{99}{86}a^{4}-\frac{79}{86}a^{3}+\frac{17}{86}a^{2}-\frac{57}{86}a-\frac{29}{43}$, $\frac{21}{86}a^{16}-\frac{35}{86}a^{15}+\frac{22}{43}a^{14}-\frac{8}{43}a^{13}-\frac{1}{43}a^{12}+\frac{16}{43}a^{11}-\frac{39}{86}a^{10}-\frac{21}{86}a^{9}+\frac{35}{86}a^{8}-\frac{87}{86}a^{7}+\frac{51}{43}a^{6}+\frac{1}{43}a^{5}-\frac{16}{43}a^{4}+\frac{41}{43}a^{3}-\frac{65}{86}a^{2}-\frac{121}{86}a-\frac{51}{43}$, $\frac{109}{43}a^{16}-\frac{91}{86}a^{15}+\frac{40}{43}a^{14}-\frac{33}{86}a^{13}+\frac{6}{43}a^{12}-\frac{10}{43}a^{11}+\frac{19}{86}a^{10}-\frac{23}{43}a^{9}-\frac{19}{43}a^{8}+\frac{49}{86}a^{7}+\frac{38}{43}a^{6}+\frac{31}{86}a^{5}-\frac{33}{43}a^{4}+\frac{12}{43}a^{3}+\frac{3}{86}a^{2}-\frac{67}{43}a-\frac{1027}{43}$ (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: | \( 35818073.0947 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{7}\cdot 35818073.0947 \cdot 1}{2\cdot\sqrt{2538753443209963749199249408}}\cr\approx \mathstrut & 1.09928683146 \end{aligned}\] (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 | $16{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.11.0.1}{11} }{,}\,{\href{/padicField/7.6.0.1}{6} }$ | ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.11.0.1}{11} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.8.0.1}{8} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.14.0.1}{14} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.11.0.1}{11} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}$ | ${\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | $15{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $16$ | $8$ | $2$ | $16$ | ||||
\(387\!\cdots\!303\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ |