Normalized defining polynomial
\( x^{44} - 3 x^{22} + 3 \)
Invariants
| Degree: | $44$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
| |
| Signature: | $[0, 22]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
| |
| Discriminant: | \(178\!\cdots\!352\)\(\medspace = 2^{44}\cdot 3^{43}\cdot 11^{36}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
| Root discriminant: | $41.63$ | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
| |
| Ramified primes: | $2, 3, 11$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
| |
| $|\Aut(K/\Q)|$: | $2$ | ||
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $\frac{1}{11} a^{40} + \frac{3}{11} a^{38} - \frac{5}{11} a^{36} - \frac{2}{11} a^{34} - \frac{2}{11} a^{32} - \frac{5}{11} a^{28} - \frac{4}{11} a^{26} + \frac{3}{11} a^{24} - \frac{1}{11} a^{22} - \frac{1}{11} a^{20} - \frac{3}{11} a^{18} + \frac{5}{11} a^{16} + \frac{2}{11} a^{14} + \frac{2}{11} a^{12} + \frac{5}{11} a^{8} + \frac{4}{11} a^{6} - \frac{3}{11} a^{4} + \frac{1}{11} a^{2} + \frac{1}{11}$, $\frac{1}{11} a^{41} + \frac{3}{11} a^{39} - \frac{5}{11} a^{37} - \frac{2}{11} a^{35} - \frac{2}{11} a^{33} - \frac{5}{11} a^{29} - \frac{4}{11} a^{27} + \frac{3}{11} a^{25} - \frac{1}{11} a^{23} - \frac{1}{11} a^{21} - \frac{3}{11} a^{19} + \frac{5}{11} a^{17} + \frac{2}{11} a^{15} + \frac{2}{11} a^{13} + \frac{5}{11} a^{9} + \frac{4}{11} a^{7} - \frac{3}{11} a^{5} + \frac{1}{11} a^{3} + \frac{1}{11} a$, $\frac{1}{11} a^{42} - \frac{3}{11} a^{38} + \frac{2}{11} a^{36} + \frac{4}{11} a^{34} - \frac{5}{11} a^{32} - \frac{5}{11} a^{30} + \frac{4}{11} a^{26} + \frac{1}{11} a^{24} + \frac{2}{11} a^{22} + \frac{3}{11} a^{18} - \frac{2}{11} a^{16} - \frac{4}{11} a^{14} + \frac{5}{11} a^{12} + \frac{5}{11} a^{10} - \frac{4}{11} a^{6} - \frac{1}{11} a^{4} - \frac{2}{11} a^{2} - \frac{3}{11}$, $\frac{1}{11} a^{43} - \frac{3}{11} a^{39} + \frac{2}{11} a^{37} + \frac{4}{11} a^{35} - \frac{5}{11} a^{33} - \frac{5}{11} a^{31} + \frac{4}{11} a^{27} + \frac{1}{11} a^{25} + \frac{2}{11} a^{23} + \frac{3}{11} a^{19} - \frac{2}{11} a^{17} - \frac{4}{11} a^{15} + \frac{5}{11} a^{13} + \frac{5}{11} a^{11} - \frac{4}{11} a^{7} - \frac{1}{11} a^{5} - \frac{2}{11} a^{3} - \frac{3}{11} a$
Class group and class number
not computed
Unit group
| Rank: | $21$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
| |
| Torsion generator: | \( a^{22} - 1 \) (order $6$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
| |
| Fundamental units: | not computed | sage: UK.fundamental_units()
gp: K.fu
magma: [K!f(g): g in Generators(UK)];
| |
| Regulator: | not computed | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
|
Class number formula
Galois group
| A solvable group of order 880 |
| The 55 conjugacy class representatives for t44n32 are not computed |
| Character table for t44n32 is not computed |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 4.0.432.1, 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.
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 | $20^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ | ${\href{/LocalNumberField/7.10.0.1}{10} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ | R | ${\href{/LocalNumberField/13.10.0.1}{10} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ | $20^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ | ${\href{/LocalNumberField/19.10.0.1}{10} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | $22^{2}$ | $20^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }$ | ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/37.5.0.1}{5} }^{8}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ | $20^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{21}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ | $20^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ | ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$ |
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 | Data not computed | ||||||
| 3 | Data not computed | ||||||
| 11 | Data not computed | ||||||