Normalized defining polynomial
\( x^{17} + 3x - 3 \)
Invariants
Degree: | $17$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[1, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(1519687851595448316396690681\) \(\medspace = 3^{16}\cdot 35388253\cdot 997597133837\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(39.71\) | 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: | $3^{16/17}35388253^{1/2}997597133837^{1/2}\approx 16709461602.49277$ | ||
Ramified primes: | \(3\), \(35388253\), \(997597133837\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{35303\!\cdots\!16761}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{5}a^{16}+\frac{2}{5}a^{15}-\frac{1}{5}a^{14}-\frac{2}{5}a^{13}+\frac{1}{5}a^{12}+\frac{2}{5}a^{11}-\frac{1}{5}a^{10}-\frac{2}{5}a^{9}+\frac{1}{5}a^{8}+\frac{2}{5}a^{7}-\frac{1}{5}a^{6}-\frac{2}{5}a^{5}+\frac{1}{5}a^{4}+\frac{2}{5}a^{3}-\frac{1}{5}a^{2}-\frac{2}{5}a-\frac{1}{5}$
Monogenic: | Not computed | |
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)
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Fundamental units: | $a-1$, $\frac{7}{5}a^{16}+\frac{9}{5}a^{15}+\frac{8}{5}a^{14}+\frac{6}{5}a^{13}+\frac{7}{5}a^{12}+\frac{4}{5}a^{11}+\frac{3}{5}a^{10}+\frac{6}{5}a^{9}+\frac{7}{5}a^{8}+\frac{4}{5}a^{7}+\frac{3}{5}a^{6}+\frac{1}{5}a^{5}+\frac{2}{5}a^{4}+\frac{4}{5}a^{3}+\frac{3}{5}a^{2}+\frac{6}{5}a+\frac{23}{5}$, $a^{10}+a^{9}-a^{7}-a^{6}+a^{5}+a^{4}-a^{2}-a+1$, $\frac{1}{5}a^{16}+\frac{2}{5}a^{15}+\frac{4}{5}a^{14}+\frac{3}{5}a^{13}+\frac{1}{5}a^{12}+\frac{2}{5}a^{11}-\frac{1}{5}a^{10}-\frac{2}{5}a^{9}+\frac{1}{5}a^{8}-\frac{3}{5}a^{7}-\frac{1}{5}a^{6}-\frac{2}{5}a^{5}-\frac{9}{5}a^{4}-\frac{3}{5}a^{3}-\frac{1}{5}a^{2}-\frac{2}{5}a+\frac{4}{5}$, $\frac{1}{5}a^{16}+\frac{2}{5}a^{15}-\frac{1}{5}a^{14}-\frac{2}{5}a^{13}-\frac{4}{5}a^{12}-\frac{3}{5}a^{11}-\frac{1}{5}a^{10}-\frac{2}{5}a^{9}-\frac{4}{5}a^{8}-\frac{3}{5}a^{7}-\frac{1}{5}a^{6}-\frac{2}{5}a^{5}+\frac{1}{5}a^{4}-\frac{3}{5}a^{3}+\frac{4}{5}a^{2}+\frac{3}{5}a+\frac{4}{5}$, $a^{16}+a^{15}-a^{10}-a^{9}+a^{8}+a^{7}+a^{6}+a^{3}-a^{2}-a+1$, $\frac{2}{5}a^{16}-\frac{1}{5}a^{15}-\frac{2}{5}a^{14}+\frac{1}{5}a^{13}+\frac{7}{5}a^{12}+\frac{9}{5}a^{11}+\frac{3}{5}a^{10}-\frac{4}{5}a^{9}-\frac{8}{5}a^{8}-\frac{6}{5}a^{7}+\frac{3}{5}a^{6}+\frac{6}{5}a^{5}+\frac{2}{5}a^{4}-\frac{6}{5}a^{3}-\frac{12}{5}a^{2}-\frac{4}{5}a+\frac{13}{5}$, $\frac{1}{5}a^{16}+\frac{2}{5}a^{15}-\frac{1}{5}a^{14}-\frac{2}{5}a^{13}+\frac{1}{5}a^{12}+\frac{2}{5}a^{11}-\frac{6}{5}a^{10}-\frac{2}{5}a^{9}+\frac{6}{5}a^{8}-\frac{3}{5}a^{7}-\frac{6}{5}a^{6}+\frac{3}{5}a^{5}+\frac{6}{5}a^{4}-\frac{3}{5}a^{3}-\frac{6}{5}a^{2}+\frac{13}{5}a-\frac{1}{5}$ (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: | \( 10106858.0011 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{8}\cdot 10106858.0011 \cdot 1}{2\cdot\sqrt{1519687851595448316396690681}}\cr\approx \mathstrut & 0.629764121997 \end{aligned}\] (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 | ${\href{/padicField/2.12.0.1}{12} }{,}\,{\href{/padicField/2.3.0.1}{3} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ | R | ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.9.0.1}{9} }{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | $17$ | ${\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.8.0.1}{8} }$ | $16{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.14.0.1}{14} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.13.0.1}{13} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.9.0.1}{9} }{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | $16{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.17.16.1 | $x^{17} + 3$ | $17$ | $1$ | $16$ | $F_{17}$ | $[\ ]_{17}^{16}$ |
\(35388253\) | $\Q_{35388253}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
\(997597133837\) | $\Q_{997597133837}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{997597133837}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |