Normalized defining polynomial
\( x^{46} - 4x + 5 \)
Invariants
Degree: | $46$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 23]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-177\!\cdots\!000\) \(\medspace = -\,2^{46}\cdot 5^{45}\cdot 7\cdot 101\cdot 1583\cdot 603448043502369483310514077\cdot 13174785488719670283249479881\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(204.07\) | 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: | not computed | ||
Ramified primes: | \(2\), \(5\), \(7\), \(101\), \(1583\), \(603448043502369483310514077\), \(13174785488719670283249479881\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-44489\!\cdots\!92485}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$, $a^{42}$, $a^{43}$, $a^{44}$, $\frac{1}{7}a^{45}-\frac{3}{7}a^{44}+\frac{2}{7}a^{43}+\frac{1}{7}a^{42}-\frac{3}{7}a^{41}+\frac{2}{7}a^{40}+\frac{1}{7}a^{39}-\frac{3}{7}a^{38}+\frac{2}{7}a^{37}+\frac{1}{7}a^{36}-\frac{3}{7}a^{35}+\frac{2}{7}a^{34}+\frac{1}{7}a^{33}-\frac{3}{7}a^{32}+\frac{2}{7}a^{31}+\frac{1}{7}a^{30}-\frac{3}{7}a^{29}+\frac{2}{7}a^{28}+\frac{1}{7}a^{27}-\frac{3}{7}a^{26}+\frac{2}{7}a^{25}+\frac{1}{7}a^{24}-\frac{3}{7}a^{23}+\frac{2}{7}a^{22}+\frac{1}{7}a^{21}-\frac{3}{7}a^{20}+\frac{2}{7}a^{19}+\frac{1}{7}a^{18}-\frac{3}{7}a^{17}+\frac{2}{7}a^{16}+\frac{1}{7}a^{15}-\frac{3}{7}a^{14}+\frac{2}{7}a^{13}+\frac{1}{7}a^{12}-\frac{3}{7}a^{11}+\frac{2}{7}a^{10}+\frac{1}{7}a^{9}-\frac{3}{7}a^{8}+\frac{2}{7}a^{7}+\frac{1}{7}a^{6}-\frac{3}{7}a^{5}+\frac{2}{7}a^{4}+\frac{1}{7}a^{3}-\frac{3}{7}a^{2}+\frac{2}{7}a-\frac{3}{7}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
not computed
Unit group
Rank: | $22$ | 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 5502622159812088949850305428800254892961651752960000000000 |
The 105558 conjugacy class representatives for $S_{46}$ are not computed |
Character table for $S_{46}$ |
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 | $36{,}\,{\href{/padicField/3.10.0.1}{10} }$ | R | R | ${\href{/padicField/11.11.0.1}{11} }{,}\,{\href{/padicField/11.9.0.1}{9} }{,}\,{\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.7.0.1}{7} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $42{,}\,{\href{/padicField/13.4.0.1}{4} }$ | $26{,}\,16{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $38{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | $23{,}\,20{,}\,{\href{/padicField/23.3.0.1}{3} }$ | $23{,}\,{\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $44{,}\,{\href{/padicField/31.2.0.1}{2} }$ | $34{,}\,{\href{/padicField/37.12.0.1}{12} }$ | $30{,}\,{\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $46$ | $23{,}\,{\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $18^{2}{,}\,{\href{/padicField/53.10.0.1}{10} }$ | $34{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
Deg $22$ | $2$ | $11$ | $22$ | ||||
Deg $22$ | $2$ | $11$ | $22$ | ||||
\(5\) | $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
5.5.5.4 | $x^{5} + 10 x + 5$ | $5$ | $1$ | $5$ | $F_5$ | $[5/4]_{4}$ | |
5.10.10.11 | $x^{10} - 20 x^{6} + 10 x^{5} - 100 x^{2} - 100 x + 25$ | $5$ | $2$ | $10$ | $(C_5^2 : C_4) : C_2$ | $[5/4, 5/4]_{4}^{2}$ | |
Deg $30$ | $5$ | $6$ | $30$ | ||||
\(7\) | 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $42$ | $1$ | $42$ | $0$ | $C_{42}$ | $[\ ]^{42}$ | ||
\(101\) | 101.2.1.2 | $x^{2} + 202$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
101.2.0.1 | $x^{2} + 97 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $42$ | $1$ | $42$ | $0$ | $C_{42}$ | $[\ ]^{42}$ | ||
\(1583\) | $\Q_{1583}$ | $x$ | $1$ | $1$ | $0$ | 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 $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | ||
\(603\!\cdots\!077\) | $\Q_{60\!\cdots\!77}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
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 $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | ||
Deg $19$ | $1$ | $19$ | $0$ | $C_{19}$ | $[\ ]^{19}$ | ||
\(131\!\cdots\!881\) | $\Q_{13\!\cdots\!81}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{13\!\cdots\!81}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
Deg $22$ | $1$ | $22$ | $0$ | 22T1 | $[\ ]^{22}$ |