Normalized defining polynomial
\( x^{20} - 2 x^{19} + 43 x^{18} - 152 x^{17} + 1879 x^{16} - 2726 x^{15} + 61371 x^{14} - 17956 x^{13} + 1522777 x^{12} + 1027682 x^{11} + 30527583 x^{10} + 39950494 x^{9} + 479660823 x^{8} + 786037928 x^{7} + 5655758447 x^{6} + 9090696442 x^{5} + 45429641276 x^{4} + 61368607614 x^{3} + 225869813356 x^{2} + 196984421920 x + 487148467849 \)
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: | \(432835576015865621834099207027388579840000000000=2^{30}\cdot 3^{10}\cdot 5^{10}\cdot 31^{18}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $240.89$ | 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: | $2, 3, 5, 31$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(3720=2^{3}\cdot 3\cdot 5\cdot 31\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{3720}(1,·)$, $\chi_{3720}(3139,·)$, $\chi_{3720}(2441,·)$, $\chi_{3720}(1379,·)$, $\chi_{3720}(401,·)$, $\chi_{3720}(721,·)$, $\chi_{3720}(2899,·)$, $\chi_{3720}(3259,·)$, $\chi_{3720}(3161,·)$, $\chi_{3720}(1859,·)$, $\chi_{3720}(481,·)$, $\chi_{3720}(1139,·)$, $\chi_{3720}(2321,·)$, $\chi_{3720}(1459,·)$, $\chi_{3720}(841,·)$, $\chi_{3720}(2419,·)$, $\chi_{3720}(2761,·)$, $\chi_{3720}(2819,·)$, $\chi_{3720}(2681,·)$, $\chi_{3720}(1019,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{23249} a^{18} + \frac{8751}{23249} a^{17} + \frac{1317}{23249} a^{16} - \frac{11006}{23249} a^{15} + \frac{1632}{23249} a^{14} - \frac{5689}{23249} a^{13} + \frac{9894}{23249} a^{12} - \frac{8771}{23249} a^{11} - \frac{9552}{23249} a^{10} + \frac{7997}{23249} a^{9} + \frac{6855}{23249} a^{8} - \frac{7146}{23249} a^{7} + \frac{8526}{23249} a^{6} - \frac{6806}{23249} a^{5} - \frac{2005}{23249} a^{4} - \frac{4670}{23249} a^{3} + \frac{5879}{23249} a^{2} - \frac{6935}{23249} a - \frac{9824}{23249}$, $\frac{1}{1882733889241675745957371706350023089883402899398817842769466280656899053568679077} a^{19} + \frac{29051276523914845221536710076353481622853052604503651603823282099925918447156}{1882733889241675745957371706350023089883402899398817842769466280656899053568679077} a^{18} + \frac{261156382315871713572991326446204153497011770979294198784233547284246301860773609}{1882733889241675745957371706350023089883402899398817842769466280656899053568679077} a^{17} + \frac{498377033266802676331216027216818117382706485571787030554179984070942435511131151}{1882733889241675745957371706350023089883402899398817842769466280656899053568679077} a^{16} - \frac{82618379129683009058855371990264509410696202248370317599069621156753153464133674}{1882733889241675745957371706350023089883402899398817842769466280656899053568679077} a^{15} + \frac{799869214047192747109047829330708165206369671022461497000165424450712033251933374}{1882733889241675745957371706350023089883402899398817842769466280656899053568679077} a^{14} - \frac{787762077466580683629332113143089155736218508296053429818292420173400616524819213}{1882733889241675745957371706350023089883402899398817842769466280656899053568679077} a^{13} - \frac{342411502220153374405219328867573430195323134106359050089316471113832308357431080}{1882733889241675745957371706350023089883402899398817842769466280656899053568679077} a^{12} + \frac{266954593591770794802533241078592061360459231788776337106667050908017756066824658}{1882733889241675745957371706350023089883402899398817842769466280656899053568679077} a^{11} + \frac{365239613318872866457473862032044372772230401297265818020812263986349491699389137}{1882733889241675745957371706350023089883402899398817842769466280656899053568679077} a^{10} + \frac{780340746916735562743060979677207443805186145641104062427375023249563876228257999}{1882733889241675745957371706350023089883402899398817842769466280656899053568679077} a^{9} - \frac{96317979930801341733971573986857031776163480900361302139212129105754763780347201}{1882733889241675745957371706350023089883402899398817842769466280656899053568679077} a^{8} + \frac{38927601600879140901389484255986722219622567413248268660569999050665152248361226}{1882733889241675745957371706350023089883402899398817842769466280656899053568679077} a^{7} + \frac{734204969091383892791943398486883339665385020347312303260949566233772170910317682}{1882733889241675745957371706350023089883402899398817842769466280656899053568679077} a^{6} + \frac{38561002480315391983380884397656822910719431239335204346364174097606644948047617}{1882733889241675745957371706350023089883402899398817842769466280656899053568679077} a^{5} - \frac{599053089436272429110029787311345166925611103431336110928565735858244784438408232}{1882733889241675745957371706350023089883402899398817842769466280656899053568679077} a^{4} + \frac{388201787222590347637879272724077986764581270950755150678328094219274750770853656}{1882733889241675745957371706350023089883402899398817842769466280656899053568679077} a^{3} + \frac{269193906112049284741074659174625529487601724191156423641123737942935546181271}{5425746078506270161260437194092285561623639479535498105963879771345530413742591} a^{2} - \frac{632857785195299133888900921506336663751139945654154739506988924321166613795025244}{1882733889241675745957371706350023089883402899398817842769466280656899053568679077} a - \frac{514398479304384535798273053076772166991667682508122899366871242129470251053641914}{1882733889241675745957371706350023089883402899398817842769466280656899053568679077}$
Class group and class number
$C_{2}\times C_{4}\times C_{4}\times C_{4}\times C_{4}\times C_{330132}$, which has order $169027584$ (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: | \( 12967416.87984626 \) (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
| \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{93}) \), \(\Q(\sqrt{-930}) \), \(\Q(\sqrt{-10}, \sqrt{93})\), 5.5.923521.1, 10.0.87336042233958400000.1, 10.10.6424828185043053.1, 10.0.657902406148408627200000.1 |
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 | R | R | ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$ |
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 | ||||||
| 3 | Data not computed | ||||||
| $5$ | 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 31 | Data not computed | ||||||