Normalized defining polynomial
\( x^{20} - 10 x^{19} + 145 x^{18} - 1020 x^{17} + 8770 x^{16} - 48330 x^{15} + 309415 x^{14} - 1394755 x^{13} + 7198300 x^{12} - 26988435 x^{11} + 116371456 x^{10} - 362840905 x^{9} + 1328806310 x^{8} - 3388327335 x^{7} + 10592693335 x^{6} - 21196340165 x^{5} + 56364337770 x^{4} - 80639732115 x^{3} + 180301322200 x^{2} - 142233499330 x + 262135950049 \)
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: | \(30616724535258282558061182498931884765625=5^{34}\cdot 47^{10}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $105.75$ | 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: | $5, 47$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1175=5^{2}\cdot 47\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1175}(704,·)$, $\chi_{1175}(1,·)$, $\chi_{1175}(706,·)$, $\chi_{1175}(516,·)$, $\chi_{1175}(659,·)$, $\chi_{1175}(469,·)$, $\chi_{1175}(1174,·)$, $\chi_{1175}(471,·)$, $\chi_{1175}(281,·)$, $\chi_{1175}(986,·)$, $\chi_{1175}(424,·)$, $\chi_{1175}(1129,·)$, $\chi_{1175}(234,·)$, $\chi_{1175}(939,·)$, $\chi_{1175}(236,·)$, $\chi_{1175}(941,·)$, $\chi_{1175}(46,·)$, $\chi_{1175}(751,·)$, $\chi_{1175}(189,·)$, $\chi_{1175}(894,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{101} a^{18} + \frac{29}{101} a^{17} - \frac{25}{101} a^{16} - \frac{31}{101} a^{15} - \frac{11}{101} a^{14} - \frac{45}{101} a^{13} - \frac{17}{101} a^{12} - \frac{37}{101} a^{11} - \frac{1}{101} a^{10} - \frac{1}{101} a^{9} + \frac{13}{101} a^{8} - \frac{31}{101} a^{7} - \frac{31}{101} a^{6} - \frac{5}{101} a^{5} + \frac{22}{101} a^{4} - \frac{21}{101} a^{3} - \frac{30}{101} a^{2} - \frac{34}{101} a + \frac{25}{101}$, $\frac{1}{860220666930842793564040142651579185042177765527197302967334815909625521057} a^{19} + \frac{20547302750823567907022427638232337274672448818961454692683695228469400}{5773293066649951634657987534574356946591797084075149684344528965836412893} a^{18} + \frac{88735904361597363824451170880279444580142069068979036352683376477185064829}{860220666930842793564040142651579185042177765527197302967334815909625521057} a^{17} - \frac{170182527023751460409908958103668765146057171098079526902088719515031556211}{860220666930842793564040142651579185042177765527197302967334815909625521057} a^{16} - \frac{70911875396830540543754884266447318343970368363071964703803170283490709590}{860220666930842793564040142651579185042177765527197302967334815909625521057} a^{15} - \frac{152803069411903818688203600698031849670116466428307567022554558047024425361}{860220666930842793564040142651579185042177765527197302967334815909625521057} a^{14} - \frac{253672782839464467568829289961604845291725701454554352705302846730937910117}{860220666930842793564040142651579185042177765527197302967334815909625521057} a^{13} + \frac{250709284748607617694361840457684755050195372134347460600551909704553294659}{860220666930842793564040142651579185042177765527197302967334815909625521057} a^{12} + \frac{100175679412704949297726359402422253017412434698225742884711932269028009366}{860220666930842793564040142651579185042177765527197302967334815909625521057} a^{11} - \frac{266061347167372247982387307353717891591960937525379099128506859196093621622}{860220666930842793564040142651579185042177765527197302967334815909625521057} a^{10} + \frac{46705612336295519366089697337193068855621900852076502302651762341626179370}{860220666930842793564040142651579185042177765527197302967334815909625521057} a^{9} + \frac{106529042861696687476983436617987700654243696240640743194763147433281204763}{860220666930842793564040142651579185042177765527197302967334815909625521057} a^{8} - \frac{119771121830805298098355386653858570124382258718824598016213969485173766551}{860220666930842793564040142651579185042177765527197302967334815909625521057} a^{7} - \frac{5666034114174624069635937033842204924964148184160497528144145466005708793}{860220666930842793564040142651579185042177765527197302967334815909625521057} a^{6} + \frac{88783175570306565556403835217666246376936034649971512518090510853781408543}{860220666930842793564040142651579185042177765527197302967334815909625521057} a^{5} + \frac{284940885176690586709431026440969265110994265778808290726624959141180180771}{860220666930842793564040142651579185042177765527197302967334815909625521057} a^{4} + \frac{187091608759546262084527934377260067978249622052604898430579890051852992161}{860220666930842793564040142651579185042177765527197302967334815909625521057} a^{3} + \frac{68232395501348001488233414658632831881252458262283339329102689270871343322}{860220666930842793564040142651579185042177765527197302967334815909625521057} a^{2} - \frac{300294907442246909018361680995484638488854570677689665627044736284403052726}{860220666930842793564040142651579185042177765527197302967334815909625521057} a + \frac{136618415492866100081272075943916645221696796862487331456064258736809958121}{860220666930842793564040142651579185042177765527197302967334815909625521057}$
Class group and class number
$C_{3905275}$, which has order $3905275$ (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: | \( 161406.8376411007 \) (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
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-235}) \), \(\Q(\sqrt{-47}) \), \(\Q(\sqrt{5}, \sqrt{-47})\), 5.5.390625.1, \(\Q(\zeta_{25})^+\), 10.0.174976354217529296875.3, 10.0.34995270843505859375.3 |
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 | ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ | R | ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| 5 | Data not computed | ||||||
| 47 | Data not computed | ||||||