Normalized defining polynomial
\( x^{39} - x - 4 \)
Invariants
Degree: | $39$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[1, 19]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-30949918466425163032455347262376548002357192825523150128829529791279710855\) \(\medspace = -\,5\cdot 7\cdot 41\cdot 659\cdot 2251\cdot 31793\cdot 2678539\cdot 118932731307271627\cdot 14\!\cdots\!53\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(76.63\) | 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: | $5^{1/2}7^{1/2}41^{1/2}659^{1/2}2251^{1/2}31793^{1/2}2678539^{1/2}118932731307271627^{1/2}1435541730552723797020749364164204653^{1/2}\approx 5.563265090432521e+36$ | ||
Ramified primes: | \(5\), \(7\), \(41\), \(659\), \(2251\), \(31793\), \(2678539\), \(118932731307271627\), \(14355\!\cdots\!04653\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-30949\!\cdots\!10855}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{2}a^{20}-\frac{1}{2}a$, $\frac{1}{2}a^{21}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{22}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{23}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{24}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{25}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{26}-\frac{1}{2}a^{7}$, $\frac{1}{2}a^{27}-\frac{1}{2}a^{8}$, $\frac{1}{2}a^{28}-\frac{1}{2}a^{9}$, $\frac{1}{2}a^{29}-\frac{1}{2}a^{10}$, $\frac{1}{2}a^{30}-\frac{1}{2}a^{11}$, $\frac{1}{2}a^{31}-\frac{1}{2}a^{12}$, $\frac{1}{2}a^{32}-\frac{1}{2}a^{13}$, $\frac{1}{2}a^{33}-\frac{1}{2}a^{14}$, $\frac{1}{2}a^{34}-\frac{1}{2}a^{15}$, $\frac{1}{2}a^{35}-\frac{1}{2}a^{16}$, $\frac{1}{2}a^{36}-\frac{1}{2}a^{17}$, $\frac{1}{2}a^{37}-\frac{1}{2}a^{18}$, $\frac{1}{2}a^{38}-\frac{1}{2}a^{19}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
not computed
Unit group
Rank: | $19$ | 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: | not computed | 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: | not computed | 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 $
Galois group
A non-solvable group of order 20397882081197443358640281739902897356800000000 |
The 31185 conjugacy class representatives for $S_{39}$ are not computed |
Character table for $S_{39}$ 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 | $36{,}\,{\href{/padicField/2.2.0.1}{2} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | $27{,}\,{\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | R | R | $34{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | $39$ | $18{,}\,17{,}\,{\href{/padicField/17.4.0.1}{4} }$ | $20{,}\,19$ | $22{,}\,{\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | $22{,}\,17$ | $26{,}\,{\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ | $29{,}\,{\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ | R | $39$ | $15{,}\,{\href{/padicField/47.9.0.1}{9} }{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/53.13.0.1}{13} }{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $19^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.3.0.1 | $x^{3} + 3 x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
5.13.0.1 | $x^{13} + 4 x^{2} + 3 x + 3$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | |
5.21.0.1 | $x^{21} + 4 x^{7} + 2 x^{6} + 2 x^{5} + 2 x^{4} + x^{3} + 2 x^{2} + 2 x + 3$ | $1$ | $21$ | $0$ | $C_{21}$ | $[\ ]^{21}$ | |
\(7\) | 7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
7.5.0.1 | $x^{5} + x + 4$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
Deg $32$ | $1$ | $32$ | $0$ | 32T33 | $[\ ]^{32}$ | ||
\(41\) | $\Q_{41}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
41.2.1.2 | $x^{2} + 123$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
41.4.0.1 | $x^{4} + 23 x + 6$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
41.16.0.1 | $x^{16} - 8 x + 11$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ | |
41.16.0.1 | $x^{16} - 8 x + 11$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ | |
\(659\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
Deg $16$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ | ||
\(2251\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $32$ | $1$ | $32$ | $0$ | 32T33 | $[\ ]^{32}$ | ||
\(31793\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $37$ | $1$ | $37$ | $0$ | $C_{37}$ | $[\ ]^{37}$ | ||
\(2678539\) | $\Q_{2678539}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $34$ | $1$ | $34$ | $0$ | $C_{34}$ | $[\ ]^{34}$ | ||
\(118932731307271627\) | $\Q_{118932731307271627}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $36$ | $1$ | $36$ | $0$ | $C_{36}$ | $[\ ]^{36}$ | ||
\(143\!\cdots\!653\) | $\Q_{14\!\cdots\!53}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ |