Normalized defining polynomial
\( x^{20} - x^{19} + 90 x^{18} - 91 x^{17} + 3360 x^{16} - 3451 x^{15} + 68499 x^{14} - 71950 x^{13} + 849729 x^{12} - 921679 x^{11} + 6805360 x^{10} - 7799638 x^{9} + 36820719 x^{8} - 45394658 x^{7} + 141833529 x^{6} - 176867903 x^{5} + 410339451 x^{4} - 382542201 x^{3} + 863744092 x^{2} - 219225038 x + 1102456301 \)
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: | \(47929047471561558446078911407470703125=5^{15}\cdot 7^{10}\cdot 11^{18}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $76.56$ | 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: | $5, 7, 11$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(385=5\cdot 7\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{385}(64,·)$, $\chi_{385}(1,·)$, $\chi_{385}(246,·)$, $\chi_{385}(71,·)$, $\chi_{385}(13,·)$, $\chi_{385}(141,·)$, $\chi_{385}(272,·)$, $\chi_{385}(83,·)$, $\chi_{385}(344,·)$, $\chi_{385}(153,·)$, $\chi_{385}(36,·)$, $\chi_{385}(293,·)$, $\chi_{385}(167,·)$, $\chi_{385}(169,·)$, $\chi_{385}(237,·)$, $\chi_{385}(307,·)$, $\chi_{385}(309,·)$, $\chi_{385}(118,·)$, $\chi_{385}(379,·)$, $\chi_{385}(62,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{80415941} a^{16} - \frac{3881556}{80415941} a^{15} - \frac{1150415}{80415941} a^{14} + \frac{32061802}{80415941} a^{13} - \frac{7407639}{80415941} a^{12} - \frac{9441280}{80415941} a^{11} + \frac{15119286}{80415941} a^{10} + \frac{36515030}{80415941} a^{9} + \frac{31133217}{80415941} a^{8} - \frac{38607937}{80415941} a^{7} - \frac{2334298}{80415941} a^{6} - \frac{8848246}{80415941} a^{5} - \frac{21030612}{80415941} a^{4} + \frac{15887651}{80415941} a^{3} + \frac{31004591}{80415941} a^{2} - \frac{24024538}{80415941} a + \frac{3380100}{80415941}$, $\frac{1}{80415941} a^{17} + \frac{11326386}{80415941} a^{15} - \frac{31812090}{80415941} a^{14} - \frac{27376802}{80415941} a^{13} + \frac{27152632}{80415941} a^{12} - \frac{10943638}{80415941} a^{11} - \frac{36114580}{80415941} a^{10} - \frac{21907069}{80415941} a^{9} - \frac{9947858}{80415941} a^{8} + \frac{13758457}{80415941} a^{7} - \frac{11935641}{80415941} a^{6} + \frac{21692184}{80415941} a^{5} + \frac{26059535}{80415941} a^{4} + \frac{24147054}{80415941} a^{3} + \frac{38191154}{80415941} a^{2} - \frac{9411080}{80415941} a + \frac{25829568}{80415941}$, $\frac{1}{1424246731051} a^{18} - \frac{3720}{1424246731051} a^{17} + \frac{1081}{1424246731051} a^{16} + \frac{181821774900}{1424246731051} a^{15} + \frac{614772440369}{1424246731051} a^{14} - \frac{492675199597}{1424246731051} a^{13} + \frac{320925096276}{1424246731051} a^{12} - \frac{511222408317}{1424246731051} a^{11} + \frac{198197002106}{1424246731051} a^{10} + \frac{34390738147}{1424246731051} a^{9} - \frac{121958285787}{1424246731051} a^{8} + \frac{118102699376}{1424246731051} a^{7} + \frac{48660944658}{1424246731051} a^{6} - \frac{111904223058}{1424246731051} a^{5} + \frac{546641405130}{1424246731051} a^{4} + \frac{318185464555}{1424246731051} a^{3} + \frac{410857655424}{1424246731051} a^{2} + \frac{328608422815}{1424246731051} a + \frac{694622166446}{1424246731051}$, $\frac{1}{802865673019937542851616029310267250061272283715699} a^{19} + \frac{41848868381596174041553749059905887460}{802865673019937542851616029310267250061272283715699} a^{18} - \frac{1399471687905814457588137370951162633232486}{802865673019937542851616029310267250061272283715699} a^{17} + \frac{2491972828647615605066524702781345069770962}{802865673019937542851616029310267250061272283715699} a^{16} + \frac{351438573883829816677850358620236252139701805081259}{802865673019937542851616029310267250061272283715699} a^{15} - \frac{202737830852709192726088172070868599799797932395209}{802865673019937542851616029310267250061272283715699} a^{14} - \frac{188684354762757306523225178451085511491489415707430}{802865673019937542851616029310267250061272283715699} a^{13} - \frac{333303850783530446025523772427257717234095535679295}{802865673019937542851616029310267250061272283715699} a^{12} - \frac{391961873646828078748301632920496429430215844867797}{802865673019937542851616029310267250061272283715699} a^{11} - \frac{323779374294761560234600098975276341314624916357603}{802865673019937542851616029310267250061272283715699} a^{10} + \frac{20399199367718093167688301919121327732448850498593}{802865673019937542851616029310267250061272283715699} a^{9} - \frac{107735113730785182481113852805266819049473119992257}{802865673019937542851616029310267250061272283715699} a^{8} + \frac{231561993833226690479712879251702837160615233334245}{802865673019937542851616029310267250061272283715699} a^{7} - \frac{351584427275895957771261829323551581034727344444008}{802865673019937542851616029310267250061272283715699} a^{6} - \frac{3612533936192138689333677606611158175563788513922}{7365740119448968283042348892754745413406167740511} a^{5} - \frac{251504192940433923568274323181796171213275740943278}{802865673019937542851616029310267250061272283715699} a^{4} - \frac{229472466973029786870476303365871493070263012925063}{802865673019937542851616029310267250061272283715699} a^{3} - \frac{89602768388333058514025427909863717045378857643022}{802865673019937542851616029310267250061272283715699} a^{2} + \frac{322401643710807556548400273754656672961264483645511}{802865673019937542851616029310267250061272283715699} a + \frac{342068344074588418745718803817825340455094}{728251697860210735782819957151541782571999}$
Class group and class number
$C_{2}\times C_{2}\times C_{64564}$, which has order $258256$ (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: | \( 140644.599182 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 20 |
| The 20 conjugacy class representatives for $C_{20}$ |
| Character table for $C_{20}$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.0.741125.2, \(\Q(\zeta_{11})^+\), 10.10.669871503125.1 |
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 | $20$ | $20$ | R | R | R | $20$ | $20$ | ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ | $20$ | ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ | $20$ | $20$ | ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$ |
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 | ||||||
| 7 | Data not computed | ||||||
| $11$ | 11.10.9.7 | $x^{10} + 2673$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ |
| 11.10.9.7 | $x^{10} + 2673$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ | |