Normalized defining polynomial
\( x^{22} + 350979911981 x^{20} + \cdots + 28\!\cdots\!43 \)
Invariants
Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[4, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-898\!\cdots\!528\) \(\medspace = -\,2^{22}\cdot 151^{11}\cdot 2311^{11}\cdot 24910163^{11}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(5\,896\,668.68\) | 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: | \(2\), \(151\), \(2311\), \(24910163\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-8692675390643}$) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{8692675390643}a^{6}+\frac{350979911981}{8692675390643}a^{4}-\frac{1927973413914}{8692675390643}a^{2}$, $\frac{1}{8692675390643}a^{7}+\frac{350979911981}{8692675390643}a^{5}-\frac{1927973413914}{8692675390643}a^{3}$, $\frac{1}{75\!\cdots\!49}a^{8}+\frac{350979911981}{75\!\cdots\!49}a^{6}+\frac{31\!\cdots\!23}{75\!\cdots\!49}a^{4}-\frac{1845369017376}{8692675390643}a^{2}$, $\frac{1}{75\!\cdots\!49}a^{9}+\frac{350979911981}{75\!\cdots\!49}a^{7}+\frac{31\!\cdots\!23}{75\!\cdots\!49}a^{5}-\frac{1845369017376}{8692675390643}a^{3}$, $\frac{1}{65\!\cdots\!07}a^{10}+\frac{350979911981}{65\!\cdots\!07}a^{8}+\frac{31\!\cdots\!23}{65\!\cdots\!07}a^{6}-\frac{11\!\cdots\!80}{75\!\cdots\!49}a^{4}-\frac{547807863198}{8692675390643}a^{2}$, $\frac{1}{65\!\cdots\!07}a^{11}+\frac{350979911981}{65\!\cdots\!07}a^{9}+\frac{31\!\cdots\!23}{65\!\cdots\!07}a^{7}-\frac{11\!\cdots\!80}{75\!\cdots\!49}a^{5}-\frac{547807863198}{8692675390643}a^{3}$, $\frac{1}{57\!\cdots\!01}a^{12}+\frac{350979911981}{57\!\cdots\!01}a^{10}+\frac{31\!\cdots\!23}{57\!\cdots\!01}a^{8}-\frac{11\!\cdots\!80}{65\!\cdots\!07}a^{6}-\frac{42\!\cdots\!76}{75\!\cdots\!49}a^{4}-\frac{2770395461904}{8692675390643}a^{2}$, $\frac{1}{57\!\cdots\!01}a^{13}+\frac{350979911981}{57\!\cdots\!01}a^{11}+\frac{31\!\cdots\!23}{57\!\cdots\!01}a^{9}-\frac{11\!\cdots\!80}{65\!\cdots\!07}a^{7}-\frac{42\!\cdots\!76}{75\!\cdots\!49}a^{5}-\frac{2770395461904}{8692675390643}a^{3}$, $\frac{1}{49\!\cdots\!43}a^{14}+\frac{350979911981}{49\!\cdots\!43}a^{12}+\frac{31\!\cdots\!23}{49\!\cdots\!43}a^{10}-\frac{11\!\cdots\!80}{57\!\cdots\!01}a^{8}-\frac{42\!\cdots\!76}{65\!\cdots\!07}a^{6}-\frac{29\!\cdots\!64}{75\!\cdots\!49}a^{4}-\frac{2307218677902}{8692675390643}a^{2}$, $\frac{1}{49\!\cdots\!43}a^{15}+\frac{350979911981}{49\!\cdots\!43}a^{13}+\frac{31\!\cdots\!23}{49\!\cdots\!43}a^{11}-\frac{11\!\cdots\!80}{57\!\cdots\!01}a^{9}-\frac{42\!\cdots\!76}{65\!\cdots\!07}a^{7}-\frac{29\!\cdots\!64}{75\!\cdots\!49}a^{5}-\frac{2307218677902}{8692675390643}a^{3}$, $\frac{1}{43\!\cdots\!49}a^{16}+\frac{350979911981}{43\!\cdots\!49}a^{14}+\frac{31\!\cdots\!23}{43\!\cdots\!49}a^{12}-\frac{11\!\cdots\!80}{49\!\cdots\!43}a^{10}-\frac{42\!\cdots\!76}{57\!\cdots\!01}a^{8}-\frac{29\!\cdots\!64}{65\!\cdots\!07}a^{6}-\frac{71848621803046}{75\!\cdots\!49}a^{4}+\frac{108613817096}{8692675390643}a^{2}$, $\frac{1}{43\!\cdots\!49}a^{17}+\frac{350979911981}{43\!\cdots\!49}a^{15}+\frac{31\!\cdots\!23}{43\!\cdots\!49}a^{13}-\frac{11\!\cdots\!80}{49\!\cdots\!43}a^{11}-\frac{42\!\cdots\!76}{57\!\cdots\!01}a^{9}-\frac{29\!\cdots\!64}{65\!\cdots\!07}a^{7}-\frac{71848621803046}{75\!\cdots\!49}a^{5}+\frac{108613817096}{8692675390643}a^{3}$, $\frac{1}{37\!\cdots\!07}a^{18}+\frac{350979911981}{37\!\cdots\!07}a^{16}+\frac{31\!\cdots\!23}{37\!\cdots\!07}a^{14}-\frac{11\!\cdots\!80}{43\!\cdots\!49}a^{12}-\frac{42\!\cdots\!76}{49\!\cdots\!43}a^{10}-\frac{29\!\cdots\!64}{57\!\cdots\!01}a^{8}-\frac{71848621803046}{65\!\cdots\!07}a^{6}+\frac{108613817096}{75\!\cdots\!49}a^{4}+\frac{1047116611}{8692675390643}a^{2}$, $\frac{1}{37\!\cdots\!07}a^{19}+\frac{350979911981}{37\!\cdots\!07}a^{17}+\frac{31\!\cdots\!23}{37\!\cdots\!07}a^{15}-\frac{11\!\cdots\!80}{43\!\cdots\!49}a^{13}-\frac{42\!\cdots\!76}{49\!\cdots\!43}a^{11}-\frac{29\!\cdots\!64}{57\!\cdots\!01}a^{9}-\frac{71848621803046}{65\!\cdots\!07}a^{7}+\frac{108613817096}{75\!\cdots\!49}a^{5}+\frac{1047116611}{8692675390643}a^{3}$, $\frac{1}{22\!\cdots\!93}a^{20}-\frac{28\!\cdots\!83}{22\!\cdots\!93}a^{18}-\frac{86\!\cdots\!43}{22\!\cdots\!93}a^{16}+\frac{13\!\cdots\!29}{25\!\cdots\!51}a^{14}+\frac{64\!\cdots\!36}{29\!\cdots\!57}a^{12}-\frac{31\!\cdots\!06}{34\!\cdots\!99}a^{10}-\frac{28\!\cdots\!57}{39\!\cdots\!93}a^{8}-\frac{18\!\cdots\!30}{45\!\cdots\!51}a^{6}+\frac{12\!\cdots\!44}{52\!\cdots\!57}a^{4}-\frac{62\!\cdots\!47}{59\!\cdots\!99}a^{2}-\frac{13\!\cdots\!54}{68\!\cdots\!93}$, $\frac{1}{22\!\cdots\!93}a^{21}-\frac{28\!\cdots\!83}{22\!\cdots\!93}a^{19}-\frac{86\!\cdots\!43}{22\!\cdots\!93}a^{17}+\frac{13\!\cdots\!29}{25\!\cdots\!51}a^{15}+\frac{64\!\cdots\!36}{29\!\cdots\!57}a^{13}-\frac{31\!\cdots\!06}{34\!\cdots\!99}a^{11}-\frac{28\!\cdots\!57}{39\!\cdots\!93}a^{9}-\frac{18\!\cdots\!30}{45\!\cdots\!51}a^{7}+\frac{12\!\cdots\!44}{52\!\cdots\!57}a^{5}-\frac{62\!\cdots\!47}{59\!\cdots\!99}a^{3}-\frac{13\!\cdots\!54}{68\!\cdots\!93}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
not computed
Unit group
Rank: | $12$ | 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
$C_2^{10}.S_{11}$ (as 22T50):
A non-solvable group of order 40874803200 |
The 376 conjugacy class representatives for $C_2^{10}.S_{11}$ |
Character table for $C_2^{10}.S_{11}$ |
Intermediate fields
11.9.8692675390643.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 22 sibling: | data not computed |
Degree 44 sibling: | data not computed |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.11.0.1}{11} }^{2}$ | ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.5.0.1}{5} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.11.0.1}{11} }^{2}$ | ${\href{/padicField/13.11.0.1}{11} }^{2}$ | $18{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | $16{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | $16{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.11.0.1}{11} }^{2}$ | ${\href{/padicField/43.11.0.1}{11} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{6}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.10.0.1}{10} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.10.10.3 | $x^{10} + 10 x^{9} + 74 x^{8} + 320 x^{7} + 1104 x^{6} + 2752 x^{5} + 6176 x^{4} + 12096 x^{3} + 17712 x^{2} + 15968 x + 8416$ | $2$ | $5$ | $10$ | $C_2^4 : C_5$ | $[2, 2, 2, 2]^{5}$ |
2.12.12.6 | $x^{12} + 16 x^{11} + 120 x^{10} - 1052 x^{8} - 1904 x^{7} - 736 x^{6} + 3104 x^{5} + 15856 x^{4} + 18368 x^{3} + 28160 x^{2} + 14208 x + 11200$ | $2$ | $6$ | $12$ | 12T105 | $[2, 2, 2, 2]^{12}$ | |
\(151\) | 151.2.1.2 | $x^{2} + 151$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
151.2.1.1 | $x^{2} + 453$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
151.2.1.1 | $x^{2} + 453$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
151.16.8.1 | $x^{16} + 1208 x^{14} + 638446 x^{12} + 280 x^{11} + 192800064 x^{10} - 295874 x^{9} + 36384536085 x^{8} - 140711360 x^{7} + 4394339526520 x^{6} - 13467739022 x^{5} + 331762240073492 x^{4} + 736312937712 x^{3} + 14318349431500569 x^{2} + 109600247918124 x + 270475362208485995$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ | |
\(2311\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $2$ | $2$ | $2$ | ||||
Deg $14$ | $2$ | $7$ | $7$ | ||||
\(24910163\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $14$ | $2$ | $7$ | $7$ |