Normalized defining polynomial
\( x^{45} + x - 3 \)
Invariants
Degree: | $45$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[1, 22]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(461\!\cdots\!549\) \(\medspace = 41\cdot 281\cdot 2689\cdot 3061\cdot 9790127\cdot 49\!\cdots\!43\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(114.61\) | 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: | $41^{1/2}281^{1/2}2689^{1/2}3061^{1/2}9790127^{1/2}497404856720178834753693302562002159288580122855457096966807286973147607143^{1/2}\approx 2.1489253433498218e+46$ | ||
Ramified primes: | \(41\), \(281\), \(2689\), \(3061\), \(9790127\), \(49740\!\cdots\!07143\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{46178\!\cdots\!95549}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$, $a^{42}$, $a^{43}$, $\frac{1}{23}a^{44}-\frac{10}{23}a^{43}+\frac{8}{23}a^{42}-\frac{11}{23}a^{41}-\frac{5}{23}a^{40}+\frac{4}{23}a^{39}+\frac{6}{23}a^{38}+\frac{9}{23}a^{37}+\frac{2}{23}a^{36}+\frac{3}{23}a^{35}-\frac{7}{23}a^{34}+\frac{1}{23}a^{33}-\frac{10}{23}a^{32}+\frac{8}{23}a^{31}-\frac{11}{23}a^{30}-\frac{5}{23}a^{29}+\frac{4}{23}a^{28}+\frac{6}{23}a^{27}+\frac{9}{23}a^{26}+\frac{2}{23}a^{25}+\frac{3}{23}a^{24}-\frac{7}{23}a^{23}+\frac{1}{23}a^{22}-\frac{10}{23}a^{21}+\frac{8}{23}a^{20}-\frac{11}{23}a^{19}-\frac{5}{23}a^{18}+\frac{4}{23}a^{17}+\frac{6}{23}a^{16}+\frac{9}{23}a^{15}+\frac{2}{23}a^{14}+\frac{3}{23}a^{13}-\frac{7}{23}a^{12}+\frac{1}{23}a^{11}-\frac{10}{23}a^{10}+\frac{8}{23}a^{9}-\frac{11}{23}a^{8}-\frac{5}{23}a^{7}+\frac{4}{23}a^{6}+\frac{6}{23}a^{5}+\frac{9}{23}a^{4}+\frac{2}{23}a^{3}+\frac{3}{23}a^{2}-\frac{7}{23}a+\frac{2}{23}$
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 119622220865480194561963161495657715064383733760000000000 |
The 89134 conjugacy class representatives for $S_{45}$ are not computed |
Character table for $S_{45}$ |
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 | $22{,}\,15{,}\,{\href{/padicField/2.5.0.1}{5} }{,}\,{\href{/padicField/2.3.0.1}{3} }$ | ${\href{/padicField/3.10.0.1}{10} }^{4}{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }$ | $37{,}\,{\href{/padicField/5.7.0.1}{7} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $41{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ | $37{,}\,{\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | $40{,}\,{\href{/padicField/17.5.0.1}{5} }$ | $23{,}\,{\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.4.0.1}{4} }$ | $43{,}\,{\href{/padicField/23.2.0.1}{2} }$ | $19{,}\,{\href{/padicField/29.14.0.1}{14} }{,}\,{\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.5.0.1}{5} }$ | $18{,}\,{\href{/padicField/31.11.0.1}{11} }{,}\,{\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.11.0.1}{11} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | R | $36{,}\,{\href{/padicField/43.9.0.1}{9} }$ | $45$ | $26{,}\,{\href{/padicField/53.14.0.1}{14} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.14.0.1}{14} }{,}\,{\href{/padicField/59.10.0.1}{10} }^{2}{,}\,{\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(41\) | $\Q_{41}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
41.2.1.2 | $x^{2} + 123$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
41.4.0.1 | $x^{4} + 23 x + 6$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
Deg $36$ | $1$ | $36$ | $0$ | $C_{36}$ | $[\ ]^{36}$ | ||
\(281\) | 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 $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $27$ | $1$ | $27$ | $0$ | $C_{27}$ | $[\ ]^{27}$ | ||
\(2689\) | $\Q_{2689}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{2689}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $32$ | $1$ | $32$ | $0$ | 32T33 | $[\ ]^{32}$ | ||
\(3061\) | $\Q_{3061}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
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 $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $27$ | $1$ | $27$ | $0$ | $C_{27}$ | $[\ ]^{27}$ | ||
\(9790127\) | $\Q_{9790127}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{9790127}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | ||
\(497\!\cdots\!143\) | $\Q_{49\!\cdots\!43}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{49\!\cdots\!43}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{49\!\cdots\!43}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | ||
Deg $24$ | $1$ | $24$ | $0$ | $C_{24}$ | $[\ ]^{24}$ |