Normalized defining polynomial
\( x^{17} - 7x - 2 \)
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)
| |
Discriminant: | \(-1072818875260874969805419538268160\) \(\medspace = -\,2^{14}\cdot 5\cdot 3251\cdot 9473\cdot 425237890924282842001\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(87.69\) | 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: | $2^{14/15}5^{1/2}3251^{1/2}9473^{1/2}425237890924282842001^{1/2}\approx 488668744925295.3$ | ||
Ramified primes: | \(2\), \(5\), \(3251\), \(9473\), \(425237890924282842001\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-65479\!\cdots\!13615}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
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: | $a^{16}+2a^{15}+2a^{14}+a^{13}-2a^{12}-3a^{11}-4a^{10}-a^{9}+2a^{8}+6a^{7}+7a^{6}+2a^{5}-4a^{4}-11a^{3}-9a^{2}-5a-1$, $2a^{16}-3a^{15}-2a^{14}-2a^{13}-2a^{12}-a^{11}+3a^{9}+4a^{8}+5a^{7}+6a^{6}+9a^{5}+5a^{4}+2a^{3}-a^{2}-3a-27$, $a^{16}-a^{15}+a^{14}-a^{13}+a^{12}-a^{11}+a^{10}-a^{9}-2a^{8}+2a^{7}-2a^{6}+2a^{5}-2a^{4}+2a^{3}-2a^{2}+2a-1$, $4a^{16}-5a^{14}+4a^{13}+3a^{12}-8a^{11}+3a^{10}+7a^{9}-9a^{8}-2a^{7}+13a^{6}-9a^{5}-10a^{4}+20a^{3}-5a^{2}-20a-7$, $2a^{16}-4a^{14}+a^{13}+4a^{12}+a^{11}-2a^{10}+8a^{8}+3a^{7}-8a^{6}-a^{5}+6a^{4}-14a^{2}-14a-1$, $24a^{16}-9a^{14}+6a^{13}+6a^{12}-14a^{11}+15a^{10}-10a^{9}-7a^{8}+25a^{7}-23a^{6}+9a^{5}+10a^{4}-37a^{3}+43a^{2}-10a-193$, $2a^{16}-a^{14}+4a^{12}-10a^{11}+16a^{10}-19a^{9}+18a^{8}-13a^{7}+7a^{6}-3a^{5}+4a^{4}-8a^{3}+12a^{2}-11a-9$, $2a^{16}+6a^{15}+5a^{14}-2a^{13}-7a^{12}-3a^{11}+7a^{10}+10a^{9}+a^{8}-10a^{7}-6a^{6}+13a^{5}+29a^{4}+22a^{3}-a^{2}-12a-3$, $30a^{16}-38a^{15}+32a^{14}-13a^{13}-7a^{12}+13a^{11}+5a^{10}-41a^{9}+76a^{8}-87a^{7}+65a^{6}-20a^{5}-20a^{4}+27a^{3}+17a^{2}-99a-29$ (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: | \( 35856250594.5 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{7}\cdot 35856250594.5 \cdot 1}{2\cdot\sqrt{1072818875260874969805419538268160}}\cr\approx \mathstrut & 1.69285882275 \end{aligned}\] (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 | $17$ | R | $16{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $17$ | ${\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | $16{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | $17$ | ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ | $16{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.9.0.1}{9} }{,}\,{\href{/padicField/53.8.0.1}{8} }$ | ${\href{/padicField/59.13.0.1}{13} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
2.15.14.1 | $x^{15} + 2$ | $15$ | $1$ | $14$ | $C_{15} : C_4$ | $[\ ]_{15}^{4}$ | |
\(5\) | 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.3.0.1 | $x^{3} + 3 x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
5.12.0.1 | $x^{12} + x^{7} + x^{6} + 4 x^{4} + 4 x^{3} + 3 x^{2} + 2 x + 2$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
\(3251\) | $\Q_{3251}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{3251}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
\(9473\) | 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 $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
\(425\!\cdots\!001\) | $\Q_{42\!\cdots\!01}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ |