Normalized defining polynomial
\( x^{20} - 10 x^{19} + 103 x^{18} - 642 x^{17} + 4144 x^{16} - 19892 x^{15} + 98580 x^{14} - 386422 x^{13} + 1574718 x^{12} - 5165926 x^{11} + 17844628 x^{10} - 49158330 x^{9} + 145900146 x^{8} - 333136184 x^{7} + 852233649 x^{6} - 1553286060 x^{5} + 3412831600 x^{4} - 4549085736 x^{3} + 8494460005 x^{2} - 6434708372 x + 10169681941 \)
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: | \(4778429517306302115692167241186705145856=2^{20}\cdot 11^{18}\cdot 31^{10}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $96.37$ | 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: | $2, 11, 31$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1364=2^{2}\cdot 11\cdot 31\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1364}(1,·)$, $\chi_{1364}(993,·)$, $\chi_{1364}(371,·)$, $\chi_{1364}(1363,·)$, $\chi_{1364}(1301,·)$, $\chi_{1364}(1239,·)$, $\chi_{1364}(1241,·)$, $\chi_{1364}(1179,·)$, $\chi_{1364}(1055,·)$, $\chi_{1364}(929,·)$, $\chi_{1364}(931,·)$, $\chi_{1364}(743,·)$, $\chi_{1364}(621,·)$, $\chi_{1364}(433,·)$, $\chi_{1364}(435,·)$, $\chi_{1364}(309,·)$, $\chi_{1364}(185,·)$, $\chi_{1364}(123,·)$, $\chi_{1364}(125,·)$, $\chi_{1364}(63,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{23} a^{17} + \frac{3}{23} a^{16} - \frac{9}{23} a^{15} - \frac{4}{23} a^{14} - \frac{10}{23} a^{13} + \frac{1}{23} a^{12} + \frac{5}{23} a^{11} + \frac{2}{23} a^{10} - \frac{4}{23} a^{9} - \frac{6}{23} a^{8} + \frac{7}{23} a^{7} - \frac{10}{23} a^{6} - \frac{3}{23} a^{5} - \frac{7}{23} a^{4} + \frac{10}{23} a^{3} - \frac{11}{23} a^{2} + \frac{2}{23} a + \frac{5}{23}$, $\frac{1}{8192685223971436401620823221521} a^{18} - \frac{9}{8192685223971436401620823221521} a^{17} - \frac{4015936587399631582555431162028}{8192685223971436401620823221521} a^{16} - \frac{643248196688692946039843589656}{8192685223971436401620823221521} a^{15} - \frac{123272165555811094425223258648}{8192685223971436401620823221521} a^{14} + \frac{3927063376971367815053002407281}{8192685223971436401620823221521} a^{13} - \frac{2655657462534623776770572526335}{8192685223971436401620823221521} a^{12} + \frac{1155809717015923233088106308829}{8192685223971436401620823221521} a^{11} + \frac{1187521166912210111527465134036}{8192685223971436401620823221521} a^{10} + \frac{786101501879682637310918164073}{8192685223971436401620823221521} a^{9} - \frac{329228307228078430480561020380}{8192685223971436401620823221521} a^{8} - \frac{1196750044463526364439999186351}{8192685223971436401620823221521} a^{7} + \frac{1387310563223077548069688760125}{8192685223971436401620823221521} a^{6} - \frac{2620567915904063067037115460350}{8192685223971436401620823221521} a^{5} - \frac{2001146813991481395451022716845}{8192685223971436401620823221521} a^{4} - \frac{2671235987088230864955841458516}{8192685223971436401620823221521} a^{3} + \frac{1610003225690197452323063814561}{8192685223971436401620823221521} a^{2} - \frac{1989451294809755676837457431309}{8192685223971436401620823221521} a + \frac{638918269136401259329696512158}{8192685223971436401620823221521}$, $\frac{1}{33989132464339105713823136853118938557} a^{19} + \frac{2074349}{33989132464339105713823136853118938557} a^{18} + \frac{351078548781082124736883559596481471}{33989132464339105713823136853118938557} a^{17} + \frac{5679906371996936126505747817713029554}{33989132464339105713823136853118938557} a^{16} - \frac{2110555950937497062092257988108166413}{33989132464339105713823136853118938557} a^{15} + \frac{14265658330790961133303738618183508145}{33989132464339105713823136853118938557} a^{14} - \frac{16362389538326121220273261074590167048}{33989132464339105713823136853118938557} a^{13} + \frac{15932147309277756188665418059282185929}{33989132464339105713823136853118938557} a^{12} - \frac{14249440598690633396762474611737087298}{33989132464339105713823136853118938557} a^{11} + \frac{550870978625214521021957102597393282}{1477788368014743726687962471874736459} a^{10} + \frac{3881213312514390712360181812535931306}{33989132464339105713823136853118938557} a^{9} - \frac{553177086292681501523200282795996206}{1477788368014743726687962471874736459} a^{8} - \frac{11430453735173544572383465454906856888}{33989132464339105713823136853118938557} a^{7} - \frac{14837387868111183370000340497981556753}{33989132464339105713823136853118938557} a^{6} - \frac{9859372324561185155497709147020795826}{33989132464339105713823136853118938557} a^{5} + \frac{8243089788459796557034320960524232895}{33989132464339105713823136853118938557} a^{4} - \frac{414010675303253516709241379777605276}{1477788368014743726687962471874736459} a^{3} + \frac{3859457552115906531057617063905002902}{33989132464339105713823136853118938557} a^{2} - \frac{4958330921887595904102019448751646701}{33989132464339105713823136853118938557} a - \frac{83887823742419873565925558740859148}{33989132464339105713823136853118938557}$
Class group and class number
$C_{678342}$, which has order $678342$ (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: | \( 281202.490766 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times C_{10}$ (as 20T3):
| An abelian group of order 20 |
| The 20 conjugacy class representatives for $C_2\times C_{10}$ |
| Character table for $C_2\times C_{10}$ |
Intermediate fields
| \(\Q(\sqrt{-31}) \), \(\Q(\sqrt{-341}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{11}, \sqrt{-31})\), \(\Q(\zeta_{11})^+\), 10.0.6136912772340031.1, 10.0.69126185467638109184.1, \(\Q(\zeta_{44})^+\) |
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 | ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ | R | ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.10.10.11 | $x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$ | $2$ | $5$ | $10$ | $C_{10}$ | $[2]^{5}$ |
| 2.10.10.11 | $x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$ | $2$ | $5$ | $10$ | $C_{10}$ | $[2]^{5}$ | |
| 11 | Data not computed | ||||||
| 31 | Data not computed | ||||||