Normalized defining polynomial
\( x^{44} - 3 \)
Invariants
Degree: | $44$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 21]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-314\!\cdots\!432\) \(\medspace = -\,2^{88}\cdot 3^{43}\cdot 11^{36}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(83.25\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $\frac{1}{11}a^{40}+\frac{3}{11}a^{36}-\frac{2}{11}a^{32}+\frac{5}{11}a^{28}+\frac{4}{11}a^{24}+\frac{1}{11}a^{20}+\frac{3}{11}a^{16}-\frac{2}{11}a^{12}+\frac{5}{11}a^{8}+\frac{4}{11}a^{4}+\frac{1}{11}$, $\frac{1}{11}a^{41}+\frac{3}{11}a^{37}-\frac{2}{11}a^{33}+\frac{5}{11}a^{29}+\frac{4}{11}a^{25}+\frac{1}{11}a^{21}+\frac{3}{11}a^{17}-\frac{2}{11}a^{13}+\frac{5}{11}a^{9}+\frac{4}{11}a^{5}+\frac{1}{11}a$, $\frac{1}{11}a^{42}+\frac{3}{11}a^{38}-\frac{2}{11}a^{34}+\frac{5}{11}a^{30}+\frac{4}{11}a^{26}+\frac{1}{11}a^{22}+\frac{3}{11}a^{18}-\frac{2}{11}a^{14}+\frac{5}{11}a^{10}+\frac{4}{11}a^{6}+\frac{1}{11}a^{2}$, $\frac{1}{11}a^{43}+\frac{3}{11}a^{39}-\frac{2}{11}a^{35}+\frac{5}{11}a^{31}+\frac{4}{11}a^{27}+\frac{1}{11}a^{23}+\frac{3}{11}a^{19}-\frac{2}{11}a^{15}+\frac{5}{11}a^{11}+\frac{4}{11}a^{7}+\frac{1}{11}a^{3}$
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
$D_4\times F_{11}$ (as 44T32):
A solvable group of order 880 |
The 55 conjugacy class representatives for $D_4\times F_{11}$ are not computed |
Character table for $D_4\times F_{11}$ is not computed |
Intermediate fields
\(\Q(\sqrt{3}) \), 4.2.6912.1, 11.1.139234453205859.1, 22.2.243935263341322672277116036916969472.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
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 | $20^{2}{,}\,{\href{/padicField/5.4.0.1}{4} }$ | ${\href{/padicField/7.10.0.1}{10} }^{4}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/13.10.0.1}{10} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ | $20^{2}{,}\,{\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.10.0.1}{10} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | $22{,}\,{\href{/padicField/23.11.0.1}{11} }^{2}$ | $20^{2}{,}\,{\href{/padicField/29.4.0.1}{4} }$ | ${\href{/padicField/31.10.0.1}{10} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.10.0.1}{10} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | $20^{2}{,}\,{\href{/padicField/41.4.0.1}{4} }$ | ${\href{/padicField/43.2.0.1}{2} }^{22}$ | ${\href{/padicField/47.10.0.1}{10} }^{2}{,}\,{\href{/padicField/47.5.0.1}{5} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | $20^{2}{,}\,{\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.10.0.1}{10} }^{2}{,}\,{\href{/padicField/59.5.0.1}{5} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.8.5 | $x^{4} + 2 x^{2} + 4 x + 6$ | $4$ | $1$ | $8$ | $D_{4}$ | $[2, 3]^{2}$ |
Deg $40$ | $4$ | $10$ | $80$ | ||||
\(3\) | Deg $44$ | $44$ | $1$ | $43$ | |||
\(11\) | $\Q_{11}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{11}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
11.10.9.7 | $x^{10} + 11$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ | |
11.10.9.7 | $x^{10} + 11$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ | |
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 | $[\ ]_{10}^{2}$ |