Normalized defining polynomial
\( x^{20} - 4 x^{19} + 4 x^{18} - 12 x^{17} + 55 x^{16} - 98 x^{15} + 194 x^{14} - 348 x^{13} + 615 x^{12} - 1188 x^{11} + 1970 x^{10} - 3806 x^{9} + 3640 x^{8} - 4066 x^{7} + 7168 x^{6} - 12880 x^{5} + 11693 x^{4} - 4082 x^{3} + 70 x^{2} + 84 x + 1 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[4, 8]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(5969915757478328440239161344=2^{30}\cdot 11^{18}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $24.48$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $2, 11$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{6962752769028589082674710482473951639} a^{19} - \frac{633299696217553160985820047035250017}{6962752769028589082674710482473951639} a^{18} + \frac{2186707637641797311980248638153158171}{6962752769028589082674710482473951639} a^{17} - \frac{841121900165807412502359231827313522}{6962752769028589082674710482473951639} a^{16} + \frac{342980618625530536413568394511004011}{6962752769028589082674710482473951639} a^{15} + \frac{572614298010121590134823617373603200}{6962752769028589082674710482473951639} a^{14} - \frac{3436236637124549683482769029008993580}{6962752769028589082674710482473951639} a^{13} + \frac{875857796147042382967577118310529751}{6962752769028589082674710482473951639} a^{12} - \frac{1419476925826206387270864288356380823}{6962752769028589082674710482473951639} a^{11} + \frac{22498683628474946113244430796402582}{6962752769028589082674710482473951639} a^{10} - \frac{841672971878138617992077493038430521}{6962752769028589082674710482473951639} a^{9} - \frac{2998838678859522667052716957099639201}{6962752769028589082674710482473951639} a^{8} + \frac{1786611585777620395436804395271762738}{6962752769028589082674710482473951639} a^{7} - \frac{2364325103068202063327398720724738523}{6962752769028589082674710482473951639} a^{6} + \frac{3131421128380120139521034295478021579}{6962752769028589082674710482473951639} a^{5} - \frac{2340853484723469556236298003773359600}{6962752769028589082674710482473951639} a^{4} - \frac{150343949508140616392733358590719636}{6962752769028589082674710482473951639} a^{3} + \frac{203440770154286209414741327134503695}{6962752769028589082674710482473951639} a^{2} + \frac{3325463855149896526763388190167403880}{6962752769028589082674710482473951639} a + \frac{2647311576530998225728400166199313770}{6962752769028589082674710482473951639}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $11$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 1092701.97171 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times C_2^4:C_5$ (as 20T46):
| A solvable group of order 160 |
| The 16 conjugacy class representatives for $C_2\times C_2^4:C_5$ |
| Character table for $C_2\times C_2^4:C_5$ |
Intermediate fields
| \(\Q(\zeta_{11})^+\), 10.6.2414538435584.2, 10.6.2414538435584.1, 10.2.219503494144.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 10 siblings: | data not computed |
| Degree 20 siblings: | data not computed |
| Degree 32 sibling: | data not computed |
| Degree 40 siblings: | data not computed |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ | R | ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$ |
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 | ||||||
| $11$ | 11.10.9.1 | $x^{10} - 11$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ |
| 11.10.9.1 | $x^{10} - 11$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ | |