Normalized defining polynomial
\( x^{44} - 6144x^{22} + 12582912 \)
Invariants
Degree: | $44$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 22]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(748738788633404023046417283703695162922995794376200461450129397733922342699008\) \(\medspace = 2^{66}\cdot 3^{43}\cdot 11^{36}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(58.87\) | 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: | $2^{2}3^{43/44}11^{9/10}\approx 101.29569716579633$ | ||
Ramified primes: | \(2\), \(3\), \(11\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{3}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{4}a^{5}$, $\frac{1}{8}a^{6}$, $\frac{1}{8}a^{7}$, $\frac{1}{16}a^{8}$, $\frac{1}{16}a^{9}$, $\frac{1}{32}a^{10}$, $\frac{1}{32}a^{11}$, $\frac{1}{64}a^{12}$, $\frac{1}{64}a^{13}$, $\frac{1}{128}a^{14}$, $\frac{1}{128}a^{15}$, $\frac{1}{256}a^{16}$, $\frac{1}{256}a^{17}$, $\frac{1}{512}a^{18}$, $\frac{1}{512}a^{19}$, $\frac{1}{1024}a^{20}$, $\frac{1}{1024}a^{21}$, $\frac{1}{2048}a^{22}$, $\frac{1}{2048}a^{23}$, $\frac{1}{4096}a^{24}$, $\frac{1}{4096}a^{25}$, $\frac{1}{8192}a^{26}$, $\frac{1}{8192}a^{27}$, $\frac{1}{16384}a^{28}$, $\frac{1}{16384}a^{29}$, $\frac{1}{32768}a^{30}$, $\frac{1}{32768}a^{31}$, $\frac{1}{65536}a^{32}$, $\frac{1}{65536}a^{33}$, $\frac{1}{131072}a^{34}$, $\frac{1}{131072}a^{35}$, $\frac{1}{262144}a^{36}$, $\frac{1}{262144}a^{37}$, $\frac{1}{524288}a^{38}$, $\frac{1}{524288}a^{39}$, $\frac{1}{11534336}a^{40}+\frac{3}{5767168}a^{38}-\frac{5}{2883584}a^{36}-\frac{1}{720896}a^{34}-\frac{1}{360448}a^{32}-\frac{5}{180224}a^{28}-\frac{1}{22528}a^{26}+\frac{3}{45056}a^{24}-\frac{1}{22528}a^{22}-\frac{1}{11264}a^{20}-\frac{3}{5632}a^{18}+\frac{5}{2816}a^{16}+\frac{1}{704}a^{14}+\frac{1}{352}a^{12}+\frac{5}{176}a^{8}+\frac{1}{22}a^{6}-\frac{3}{44}a^{4}+\frac{1}{22}a^{2}+\frac{1}{11}$, $\frac{1}{11534336}a^{41}+\frac{3}{5767168}a^{39}-\frac{5}{2883584}a^{37}-\frac{1}{720896}a^{35}-\frac{1}{360448}a^{33}-\frac{5}{180224}a^{29}-\frac{1}{22528}a^{27}+\frac{3}{45056}a^{25}-\frac{1}{22528}a^{23}-\frac{1}{11264}a^{21}-\frac{3}{5632}a^{19}+\frac{5}{2816}a^{17}+\frac{1}{704}a^{15}+\frac{1}{352}a^{13}+\frac{5}{176}a^{9}+\frac{1}{22}a^{7}-\frac{3}{44}a^{5}+\frac{1}{22}a^{3}+\frac{1}{11}a$, $\frac{1}{23068672}a^{42}-\frac{3}{5767168}a^{38}+\frac{1}{1441792}a^{36}+\frac{1}{360448}a^{34}-\frac{5}{720896}a^{32}-\frac{5}{360448}a^{30}+\frac{1}{22528}a^{26}+\frac{1}{45056}a^{24}+\frac{1}{11264}a^{22}+\frac{3}{5632}a^{18}-\frac{1}{1408}a^{16}-\frac{1}{352}a^{14}+\frac{5}{704}a^{12}+\frac{5}{352}a^{10}-\frac{1}{22}a^{6}-\frac{1}{44}a^{4}-\frac{1}{11}a^{2}-\frac{3}{11}$, $\frac{1}{23068672}a^{43}-\frac{3}{5767168}a^{39}+\frac{1}{1441792}a^{37}+\frac{1}{360448}a^{35}-\frac{5}{720896}a^{33}-\frac{5}{360448}a^{31}+\frac{1}{22528}a^{27}+\frac{1}{45056}a^{25}+\frac{1}{11264}a^{23}+\frac{3}{5632}a^{19}-\frac{1}{1408}a^{17}-\frac{1}{352}a^{15}+\frac{5}{704}a^{13}+\frac{5}{352}a^{11}-\frac{1}{22}a^{7}-\frac{1}{44}a^{5}-\frac{1}{11}a^{3}-\frac{3}{11}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
not computed
Unit group
Rank: | $21$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -\frac{1}{2048} a^{22} + 2 \) (order $6$) | 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
$D_4\times F_{11}$ (as 44T32):
A solvable group of order 880 |
The 55 conjugacy class representatives for $D_4\times F_{11}$ |
Character table for $D_4\times F_{11}$ |
Intermediate fields
\(\Q(\sqrt{-3}) \), 4.0.1728.1, 11.1.139234453205859.1, 22.0.58158698878603618687895783643.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 44 siblings: | 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 | R | $20^{2}{,}\,{\href{/padicField/5.4.0.1}{4} }$ | ${\href{/padicField/7.10.0.1}{10} }^{4}{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/13.10.0.1}{10} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ | $20^{2}{,}\,{\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.10.0.1}{10} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | $22^{2}$ | $20^{2}{,}\,{\href{/padicField/29.4.0.1}{4} }$ | ${\href{/padicField/31.10.0.1}{10} }^{2}{,}\,{\href{/padicField/31.5.0.1}{5} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.10.0.1}{10} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | $20^{2}{,}\,{\href{/padicField/41.4.0.1}{4} }$ | ${\href{/padicField/43.2.0.1}{2} }^{21}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.10.0.1}{10} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | $20^{2}{,}\,{\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.10.0.1}{10} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.6.6 | $x^{4} - 4 x^{3} + 28 x^{2} - 24 x + 36$ | $2$ | $2$ | $6$ | $D_{4}$ | $[2, 3]^{2}$ |
Deg $20$ | $2$ | $10$ | $30$ | ||||
Deg $20$ | $2$ | $10$ | $30$ | ||||
\(3\) | Deg $44$ | $44$ | $1$ | $43$ | |||
\(11\) | 11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
11.20.18.1 | $x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241675 x^{10} + 2355135790 x^{9} + 1953262935 x^{8} + 1157863560 x^{7} + 500734950 x^{6} + 191763012 x^{5} + 243790230 x^{4} + 806750280 x^{3} + 2014356815 x^{2} + 2999040310 x + 2009802620$ | $10$ | $2$ | $18$ | 20T3 | $[\ ]_{10}^{2}$ | |
11.20.18.1 | $x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241675 x^{10} + 2355135790 x^{9} + 1953262935 x^{8} + 1157863560 x^{7} + 500734950 x^{6} + 191763012 x^{5} + 243790230 x^{4} + 806750280 x^{3} + 2014356815 x^{2} + 2999040310 x + 2009802620$ | $10$ | $2$ | $18$ | 20T3 | $[\ ]_{10}^{2}$ |