Normalized defining polynomial
\( x^{20} - 8 x^{19} + 28 x^{18} - 68 x^{17} + 170 x^{16} - 454 x^{15} + 1146 x^{14} - 2830 x^{13} + 6881 x^{12} - 15568 x^{11} + 31984 x^{10} - 58918 x^{9} + 96846 x^{8} - 143610 x^{7} + 189640 x^{6} - 213758 x^{5} + 197354 x^{4} - 143246 x^{3} + 77376 x^{2} - 28442 x + 5543 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 10]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(154181708612560135336755200000=2^{30}\cdot 5^{5}\cdot 11^{16}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $28.80$ | 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: | $2, 5, 11$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{1226694146133331287024282107009802317} a^{19} - \frac{321112107269335247248934612800389151}{1226694146133331287024282107009802317} a^{18} + \frac{232355305651391392169955787591932594}{1226694146133331287024282107009802317} a^{17} + \frac{152532438364942777771486983557382588}{1226694146133331287024282107009802317} a^{16} + \frac{566724241920778516784270093253339616}{1226694146133331287024282107009802317} a^{15} - \frac{63991792526558158286285186517043100}{1226694146133331287024282107009802317} a^{14} - \frac{29713302789356755623783053348695683}{1226694146133331287024282107009802317} a^{13} - \frac{13543438496435725729142854977192}{11254074735168176945176900064310113} a^{12} - \frac{163462199105851118820015642718666902}{1226694146133331287024282107009802317} a^{11} - \frac{394564390329606067532796737256785764}{1226694146133331287024282107009802317} a^{10} + \frac{134618317104391388890543335893913662}{1226694146133331287024282107009802317} a^{9} + \frac{405138591731025666203985851768331469}{1226694146133331287024282107009802317} a^{8} - \frac{229550144890778924997014693373605330}{1226694146133331287024282107009802317} a^{7} + \frac{460012085807591434031312334048208878}{1226694146133331287024282107009802317} a^{6} + \frac{473192486610002169365098940279152151}{1226694146133331287024282107009802317} a^{5} + \frac{507242652419155398399207603822542294}{1226694146133331287024282107009802317} a^{4} + \frac{455015677364019725027676257875131055}{1226694146133331287024282107009802317} a^{3} + \frac{10155913275865130635176974512577842}{1226694146133331287024282107009802317} a^{2} + \frac{205885263184743381166870640349493419}{1226694146133331287024282107009802317} a + \frac{62303921835440200551272376769974523}{1226694146133331287024282107009802317}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $9$ | 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: | \( 1827536.7207 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 10240 |
| The 136 conjugacy class representatives for t20n427 are not computed |
| Character table for t20n427 is not computed |
Intermediate fields
| \(\Q(\zeta_{11})^+\), 10.4.219503494144.1 |
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 | $20$ | R | $20$ | R | ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ | ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ | $20$ | ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$ |
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 | ||||||
| $5$ | 5.10.5.2 | $x^{10} - 625 x^{2} + 6250$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
| 5.10.0.1 | $x^{10} + x^{2} - x + 3$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
| $11$ | 11.10.8.5 | $x^{10} - 2321 x^{5} + 2033647$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ |
| 11.10.8.5 | $x^{10} - 2321 x^{5} + 2033647$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ | |