Normalized defining polynomial
\( x^{20} - 2 x^{19} - 58 x^{18} + 86 x^{17} + 1411 x^{16} - 1312 x^{15} - 18748 x^{14} + 7872 x^{13} + 146699 x^{12} - 2200 x^{11} - 686190 x^{10} - 175274 x^{9} + 1918052 x^{8} + 779496 x^{7} - 3153426 x^{6} - 1467848 x^{5} + 2874345 x^{4} + 1298692 x^{3} - 1234282 x^{2} - 447034 x + 150173 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[20, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(12569066605710630176407970532818944=2^{30}\cdot 11^{18}\cdot 1451^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $50.70$ | 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: | $2, 11, 1451$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{4622416655471866688418555035869727} a^{19} + \frac{715226105381047814600563905988922}{4622416655471866688418555035869727} a^{18} - \frac{382101336734451242298892680177065}{4622416655471866688418555035869727} a^{17} - \frac{904792875790838676058933073850766}{4622416655471866688418555035869727} a^{16} + \frac{1626704058945221007980696671117255}{4622416655471866688418555035869727} a^{15} - \frac{1369190748129232772996338953374327}{4622416655471866688418555035869727} a^{14} - \frac{401928427049152667380516951673949}{4622416655471866688418555035869727} a^{13} - \frac{941296096519193674939854150206531}{4622416655471866688418555035869727} a^{12} + \frac{1203755110460084861707900121735175}{4622416655471866688418555035869727} a^{11} - \frac{1836430902841214773808223829248759}{4622416655471866688418555035869727} a^{10} + \frac{796282771628945847114033360607117}{4622416655471866688418555035869727} a^{9} + \frac{560592088050110084127198045347733}{4622416655471866688418555035869727} a^{8} + \frac{426890152107454257234874950468003}{4622416655471866688418555035869727} a^{7} + \frac{130188259647188165801537281414963}{4622416655471866688418555035869727} a^{6} + \frac{2240349909429889252556257805747450}{4622416655471866688418555035869727} a^{5} + \frac{1796251247115401662917746871451971}{4622416655471866688418555035869727} a^{4} + \frac{1013950202409313293514137816660113}{4622416655471866688418555035869727} a^{3} - \frac{215187587116804696098364945086740}{4622416655471866688418555035869727} a^{2} - \frac{1775403257173958847136222622089466}{4622416655471866688418555035869727} a + \frac{2050941573431462777435776294768313}{4622416655471866688418555035869727}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $19$ | 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: | \( 43178494373.9 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 2560 |
| The 40 conjugacy class representatives for t20n262 |
| Character table for t20n262 is not computed |
Intermediate fields
| \(\Q(\sqrt{11}) \), \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{44})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 20 siblings: | data not computed |
| Degree 32 siblings: | 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 | ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{12}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$ |
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 | $x^{10} - 11$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ | |
| 1451 | Data not computed | ||||||