Normalized defining polynomial
\( x^{47} + 2x - 2 \)
Invariants
Degree: | $47$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[1, 23]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-277\!\cdots\!320\) \(\medspace = -\,2^{46}\cdot 5\cdot 13\cdot 17\cdot 89\cdot 361179289\cdot 582997712425903553\cdot 19\!\cdots\!87\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(92.65\) | 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: | $2^{46/47}5^{1/2}13^{1/2}17^{1/2}89^{1/2}361179289^{1/2}582997712425903553^{1/2}190228429934148238189141498229911082130769295887^{1/2}\approx 3.911414135180125e+39$ | ||
Ramified primes: | \(2\), \(5\), \(13\), \(17\), \(89\), \(361179289\), \(582997712425903553\), \(19022\!\cdots\!95887\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-39392\!\cdots\!26255}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$, $a^{42}$, $a^{43}$, $a^{44}$, $a^{45}$, $a^{46}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
not computed
Unit group
Rank: | $23$ | 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 258623241511168180642964355153611979969197632389120000000000 |
The 124754 conjugacy class representatives for $S_{47}$ are not computed |
Character table for $S_{47}$ |
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 | $22{,}\,16{,}\,{\href{/padicField/3.7.0.1}{7} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | R | $21{,}\,19{,}\,{\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | $43{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | R | R | $27{,}\,18{,}\,{\href{/padicField/19.2.0.1}{2} }$ | $20{,}\,{\href{/padicField/23.13.0.1}{13} }{,}\,{\href{/padicField/23.5.0.1}{5} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $26{,}\,{\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.9.0.1}{9} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | $36{,}\,{\href{/padicField/31.11.0.1}{11} }$ | $24{,}\,15{,}\,{\href{/padicField/37.8.0.1}{8} }$ | $31{,}\,{\href{/padicField/41.13.0.1}{13} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ | $32{,}\,{\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $46{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $30{,}\,{\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ | $31{,}\,15{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | Deg $47$ | $47$ | $1$ | $46$ | |||
\(5\) | $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.10.0.1 | $x^{10} + 3 x^{5} + 3 x^{4} + 2 x^{3} + 4 x^{2} + x + 2$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
5.15.0.1 | $x^{15} + 2 x^{5} + 3 x^{3} + 3 x^{2} + 4 x + 3$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | |
5.19.0.1 | $x^{19} + x^{3} + 2 x + 3$ | $1$ | $19$ | $0$ | $C_{19}$ | $[\ ]^{19}$ | |
\(13\) | $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
13.2.1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $42$ | $1$ | $42$ | $0$ | $C_{42}$ | $[\ ]^{42}$ | ||
\(17\) | $\Q_{17}$ | $x + 14$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
17.5.0.1 | $x^{5} + x + 14$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
17.16.0.1 | $x^{16} + x^{8} + 13 x^{7} + 5 x^{6} + 2 x^{5} + 12 x^{4} + 13 x^{3} + 12 x^{2} + x + 3$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ | |
17.23.0.1 | $x^{23} + 15 x^{2} + 16 x + 14$ | $1$ | $23$ | $0$ | $C_{23}$ | $[\ ]^{23}$ | |
\(89\) | 89.2.1.1 | $x^{2} + 89$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
89.4.0.1 | $x^{4} + 4 x^{2} + 72 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
Deg $41$ | $1$ | $41$ | $0$ | $C_{41}$ | $[\ ]^{41}$ | ||
\(361179289\) | $\Q_{361179289}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
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 $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
Deg $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | ||
\(582997712425903553\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
Deg $17$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ | ||
\(190\!\cdots\!887\) | $\Q_{19\!\cdots\!87}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{19\!\cdots\!87}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
Deg $28$ | $1$ | $28$ | $0$ | $C_{28}$ | $[\ ]^{28}$ |