Normalized defining polynomial
\( x^{47} - 4x - 4 \)
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: | \(-264\!\cdots\!456\) \(\medspace = -\,2^{46}\cdot 1630520977\cdot 36906152269379\cdot 62\!\cdots\!13\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(92.56\) | 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: | $2^{46/47}1630520977^{1/2}36906152269379^{1/2}62403398979076007530508335002787240525799903609762789813^{1/2}\approx 3.818930938814277e+39$ | ||
Ramified primes: | \(2\), \(1630520977\), \(36906152269379\), \(62403\!\cdots\!89813\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-37552\!\cdots\!36079}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{2}a^{24}$, $\frac{1}{2}a^{25}$, $\frac{1}{2}a^{26}$, $\frac{1}{2}a^{27}$, $\frac{1}{2}a^{28}$, $\frac{1}{2}a^{29}$, $\frac{1}{2}a^{30}$, $\frac{1}{2}a^{31}$, $\frac{1}{2}a^{32}$, $\frac{1}{2}a^{33}$, $\frac{1}{2}a^{34}$, $\frac{1}{2}a^{35}$, $\frac{1}{2}a^{36}$, $\frac{1}{2}a^{37}$, $\frac{1}{2}a^{38}$, $\frac{1}{2}a^{39}$, $\frac{1}{2}a^{40}$, $\frac{1}{2}a^{41}$, $\frac{1}{2}a^{42}$, $\frac{1}{2}a^{43}$, $\frac{1}{2}a^{44}$, $\frac{1}{2}a^{45}$, $\frac{1}{2}a^{46}$
Monogenic: | Not computed | |
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 | $22{,}\,16{,}\,{\href{/padicField/3.7.0.1}{7} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | $17{,}\,{\href{/padicField/5.14.0.1}{14} }{,}\,{\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | $17{,}\,{\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | $35{,}\,{\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | $18{,}\,{\href{/padicField/13.11.0.1}{11} }{,}\,{\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | $28{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | $29{,}\,{\href{/padicField/19.11.0.1}{11} }{,}\,{\href{/padicField/19.7.0.1}{7} }$ | $33{,}\,{\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | $25{,}\,22$ | $25{,}\,{\href{/padicField/31.14.0.1}{14} }{,}\,{\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | $38{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $21{,}\,17{,}\,{\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $47$ | $23^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $44{,}\,{\href{/padicField/53.3.0.1}{3} }$ | $47$ |
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 $47$ | $47$ | $1$ | $46$ | |||
\(1630520977\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
Deg $30$ | $1$ | $30$ | $0$ | $C_{30}$ | $[\ ]^{30}$ | ||
\(36906152269379\) | $\Q_{36906152269379}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $42$ | $1$ | $42$ | $0$ | $C_{42}$ | $[\ ]^{42}$ | ||
\(624\!\cdots\!813\) | 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 $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ |