Normalized defining polynomial
\( x^{17} + 7x - 4 \)
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: | \(65533882524540008146301681664\) \(\medspace = 2^{16}\cdot 3\cdot 31\cdot 10752340751490440474693\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(49.55\) | 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: | \(2\), \(3\), \(31\), \(10752340751490440474693\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{99996\!\cdots\!46449}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{7}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{8}$
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)
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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: | $\frac{1}{2}a^{16}+\frac{1}{2}a^{14}-a^{10}-\frac{3}{2}a^{8}-a^{7}-\frac{1}{2}a^{6}-2a^{5}-2a^{3}-a+1$, $\frac{1}{2}a^{15}+a^{14}+a^{13}+\frac{1}{2}a^{12}+\frac{1}{2}a^{11}+\frac{3}{2}a^{10}+\frac{5}{2}a^{9}+2a^{8}+\frac{1}{2}a^{7}+a^{5}+\frac{3}{2}a^{4}+\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-1$, $\frac{1}{2}a^{15}+\frac{1}{2}a^{14}-\frac{1}{2}a^{9}-a^{8}-\frac{1}{2}a^{7}+\frac{1}{2}a^{6}+a^{5}+a^{3}+a^{2}+\frac{1}{2}a-1$, $\frac{1}{2}a^{9}-a^{7}+a^{6}-a^{4}+a^{3}+a^{2}-\frac{5}{2}a+1$, $a^{15}-\frac{1}{2}a^{14}-a^{13}+a^{12}+\frac{1}{2}a^{11}-2a^{10}+3a^{8}-a^{7}-\frac{7}{2}a^{6}+3a^{5}+3a^{4}-\frac{11}{2}a^{3}-a^{2}+7a-3$, $\frac{1}{2}a^{16}+a^{15}+a^{14}+a^{13}+a^{12}+\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{3}{2}a^{8}-2a^{7}-2a^{6}-3a^{5}-3a^{4}-\frac{7}{2}a^{3}-5a^{2}-\frac{9}{2}a-1$, $\frac{1}{2}a^{16}+\frac{1}{2}a^{15}+\frac{1}{2}a^{14}-\frac{1}{2}a^{12}+\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}+\frac{1}{2}a^{6}+a^{5}-\frac{1}{2}a^{4}-a^{3}+\frac{1}{2}a^{2}+a+5$, $\frac{1}{2}a^{16}+2a^{15}-\frac{1}{2}a^{14}+2a^{13}-a^{12}+2a^{11}-2a^{10}+\frac{1}{2}a^{9}-\frac{5}{2}a^{8}+a^{7}-\frac{1}{2}a^{6}+2a^{4}-a^{3}+6a^{2}-\frac{11}{2}a+9$ (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: | \( 129563914.275 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{8}\cdot 129563914.275 \cdot 1}{2\cdot\sqrt{65533882524540008146301681664}}\cr\approx \mathstrut & 1.22939108020 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 355687428096000 |
The 297 conjugacy class representatives for $S_{17}$ are not computed |
Character table for $S_{17}$ is not computed |
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 | R | ${\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.13.0.1}{13} }{,}\,{\href{/padicField/11.4.0.1}{4} }$ | $16{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.13.0.1}{13} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.9.0.1}{9} }{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/37.11.0.1}{11} }{,}\,{\href{/padicField/37.6.0.1}{6} }$ | ${\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.11.0.1}{11} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.13.0.1}{13} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $16$ | $8$ | $2$ | $16$ | ||||
\(3\) | 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.3.0.1 | $x^{3} + 2 x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
3.5.0.1 | $x^{5} + 2 x + 1$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
3.7.0.1 | $x^{7} + 2 x^{2} + 1$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
\(31\) | $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
31.2.1.2 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
31.14.0.1 | $x^{14} + 10 x^{7} + 5 x^{6} + x^{5} + x^{4} + 18 x^{3} + 18 x^{2} + 6 x + 3$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | |
\(107\!\cdots\!693\) | $\Q_{10\!\cdots\!93}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |