Normalized defining polynomial
\( x^{32} - x - 1 \)
Invariants
Degree: | $32$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 15]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-1478570811461626154162295475745342334411977277407\) \(\medspace = -\,191\cdot 19329543076986451\cdot 400485847292917407445603765627\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(32.01\) | 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: | $191^{1/2}19329543076986451^{1/2}400485847292917407445603765627^{1/2}\approx 1.2159649713135762e+24$ | ||
Ramified primes: | \(191\), \(19329543076986451\), \(400485847292917407445603765627\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-14785\!\cdots\!77407}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $16$ | 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)
| |
Fundamental units: | $a$, $a^{16}+1$, $a^{9}+a$, $a^{7}+a^{3}$, $a^{10}+a^{8}$, $a^{31}-a^{30}+a^{29}-a^{28}+a^{27}-a^{26}-a^{5}-1$, $a^{31}-a^{30}-a^{8}-1$, $a^{11}-a^{4}$, $a^{31}+a^{27}-a^{26}+a^{25}-1$, $a^{27}-a^{26}+a^{25}-a^{24}+a^{23}-a^{22}-a^{14}+a^{8}-a^{2}$, $a^{31}-a^{30}+a^{29}-a^{28}-a^{23}+a^{22}-a^{21}+a^{20}-a^{19}-a^{14}-a^{10}-a^{5}-a-1$, $a^{28}-a^{15}-a^{9}+a^{2}$, $3a^{31}-3a^{30}+3a^{29}-3a^{28}+3a^{27}-3a^{26}+2a^{25}-a^{24}+a^{23}-a^{22}+a^{21}-2a^{20}+2a^{19}-a^{18}+a^{17}-a^{16}+a^{15}-a^{14}+a^{13}-a^{12}-a^{8}+a^{7}-a^{6}+a^{5}-4$, $2a^{31}-a^{30}+2a^{29}-a^{28}+2a^{27}-2a^{26}+2a^{25}-a^{24}+a^{23}-a^{22}+a^{21}-a^{20}-a^{18}-a^{16}-a^{14}-a^{12}-a^{10}+a^{5}+a^{3}+a-1$, $a^{31}-2a^{30}+a^{29}-2a^{28}+2a^{27}-a^{26}+a^{25}-a^{24}+a^{23}-a^{22}-a^{20}+a^{17}+a^{16}-a^{15}-a^{13}+a^{10}-a^{9}+a^{8}-a^{5}-a^{3}+a^{2}+a-1$, $2a^{31}-2a^{30}+2a^{29}-2a^{28}+a^{27}-a^{26}+a^{25}-a^{24}+a^{23}-2a^{22}+a^{21}-a^{20}+a^{19}-a^{17}-a^{15}+a^{14}-a^{11}+a^{8}-a^{7}-a^{5}+a^{4}-a-2$ (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: | \( 189619653779.73328 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{15}\cdot 189619653779.73328 \cdot 1}{2\cdot\sqrt{1478570811461626154162295475745342334411977277407}}\cr\approx \mathstrut & 0.292880398006785 \end{aligned}\] (assuming GRH)
Galois group
$S_{32}$ (as 32T2801324):
A non-solvable group of order 263130836933693530167218012160000000 |
The 8349 conjugacy class representatives for $S_{32}$ are not computed |
Character table for $S_{32}$ |
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 | ${\href{/padicField/2.10.0.1}{10} }^{3}{,}\,{\href{/padicField/2.2.0.1}{2} }$ | $28{,}\,{\href{/padicField/3.4.0.1}{4} }$ | $17{,}\,{\href{/padicField/5.13.0.1}{13} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | $23{,}\,{\href{/padicField/7.9.0.1}{9} }$ | $26{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | $26{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | $29{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.13.0.1}{13} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.5.0.1}{5} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }$ | $20{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $30{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | $32$ | $15{,}\,{\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | $19{,}\,{\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.5.0.1}{5} }$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.14.0.1}{14} }{,}\,{\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | $21{,}\,{\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(191\) | 191.2.1.1 | $x^{2} + 1337$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
191.4.0.1 | $x^{4} + 7 x^{2} + 100 x + 19$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
191.12.0.1 | $x^{12} + 79 x^{7} + 168 x^{6} + 25 x^{5} + 49 x^{4} + 90 x^{3} + 7 x^{2} + 151 x + 19$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
191.14.0.1 | $x^{14} - x + 28$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | |
\(19329543076986451\) | $\Q_{19329543076986451}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $27$ | $1$ | $27$ | $0$ | $C_{27}$ | $[\ ]^{27}$ | ||
\(400\!\cdots\!627\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $19$ | $1$ | $19$ | $0$ | $C_{19}$ | $[\ ]^{19}$ |