Normalized defining polynomial
\( x^{20} - 10 x^{19} + 125 x^{18} - 840 x^{17} + 6350 x^{16} - 33050 x^{15} + 188095 x^{14} - 798615 x^{13} + 3689530 x^{12} - 13040835 x^{11} + 50563722 x^{10} - 149057585 x^{9} + 492237900 x^{8} - 1191982015 x^{7} + 3364246225 x^{6} - 6431451625 x^{5} + 15431468120 x^{4} - 21255032915 x^{3} + 42774415730 x^{2} - 32795293410 x + 54172578001 \)
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: | \(4738339971047593862749636173248291015625=3^{10}\cdot 5^{34}\cdot 13^{10}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $96.33$ | 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: | $3, 5, 13$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(975=3\cdot 5^{2}\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{975}(896,·)$, $\chi_{975}(1,·)$, $\chi_{975}(194,·)$, $\chi_{975}(196,·)$, $\chi_{975}(389,·)$, $\chi_{975}(391,·)$, $\chi_{975}(584,·)$, $\chi_{975}(586,·)$, $\chi_{975}(779,·)$, $\chi_{975}(781,·)$, $\chi_{975}(974,·)$, $\chi_{975}(79,·)$, $\chi_{975}(274,·)$, $\chi_{975}(469,·)$, $\chi_{975}(664,·)$, $\chi_{975}(859,·)$, $\chi_{975}(116,·)$, $\chi_{975}(311,·)$, $\chi_{975}(506,·)$, $\chi_{975}(701,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{7} a^{17} + \frac{3}{7} a^{16} - \frac{1}{7} a^{15} + \frac{2}{7} a^{14} - \frac{3}{7} a^{12} - \frac{1}{7} a^{11} - \frac{2}{7} a^{9} - \frac{3}{7} a^{6} + \frac{3}{7} a^{5} + \frac{2}{7} a^{4} - \frac{1}{7} a^{2} - \frac{3}{7} a + \frac{3}{7}$, $\frac{1}{7} a^{18} - \frac{3}{7} a^{16} - \frac{2}{7} a^{15} + \frac{1}{7} a^{14} - \frac{3}{7} a^{13} + \frac{1}{7} a^{12} + \frac{3}{7} a^{11} - \frac{2}{7} a^{10} - \frac{1}{7} a^{9} - \frac{3}{7} a^{7} - \frac{2}{7} a^{6} + \frac{1}{7} a^{4} - \frac{1}{7} a^{3} - \frac{2}{7} a - \frac{2}{7}$, $\frac{1}{827130801190832676485232167291977778997429054734458694282615537251280501} a^{19} - \frac{10951781458127625874042825778557237107979037232535909283202973481471031}{827130801190832676485232167291977778997429054734458694282615537251280501} a^{18} + \frac{21943947810045859398018573023852751170734410241866605738727285522701526}{827130801190832676485232167291977778997429054734458694282615537251280501} a^{17} + \frac{141192277604825623560468682978328090206337153602253608904270660937873045}{827130801190832676485232167291977778997429054734458694282615537251280501} a^{16} + \frac{220429830929283597369632701829089413511656529577108913917348354658768789}{827130801190832676485232167291977778997429054734458694282615537251280501} a^{15} + \frac{312171441525189627203953975031332307918685203251542405103197007748675072}{827130801190832676485232167291977778997429054734458694282615537251280501} a^{14} + \frac{4550817097416151318812485724937800495211700685662934617972545534189007}{118161543027261810926461738184568254142489864962065527754659362464468643} a^{13} - \frac{318433584379304477127144608179530160075707567803110744332155813233570747}{827130801190832676485232167291977778997429054734458694282615537251280501} a^{12} - \frac{176924432472028796090639235924970378132164846195337892946925757594564732}{827130801190832676485232167291977778997429054734458694282615537251280501} a^{11} - \frac{383014613701822041846003994925716885072354841543410569976991316124632494}{827130801190832676485232167291977778997429054734458694282615537251280501} a^{10} + \frac{327827050279797119066864504887765961646575932653525151933053456084290028}{827130801190832676485232167291977778997429054734458694282615537251280501} a^{9} + \frac{354068980995582957845570879592201619628671411571166993553648330403539715}{827130801190832676485232167291977778997429054734458694282615537251280501} a^{8} - \frac{6937627906680654896214657956928960852186301264353460962894549736852002}{827130801190832676485232167291977778997429054734458694282615537251280501} a^{7} + \frac{65366122044962153322207599934834699045818320107819996595915647614543549}{827130801190832676485232167291977778997429054734458694282615537251280501} a^{6} - \frac{177662466586828469462293667159212928864233310776698443614155210974420753}{827130801190832676485232167291977778997429054734458694282615537251280501} a^{5} + \frac{732569655090920911388751104168519774176292959889664333859456306918279}{1842162140736821105757755383723781244983138206535542748959054648666549} a^{4} - \frac{258215594294748884563090853469874685777017545369082101115378529911838555}{827130801190832676485232167291977778997429054734458694282615537251280501} a^{3} - \frac{211007082025843230509837680030077851115005717599674853651376540651144305}{827130801190832676485232167291977778997429054734458694282615537251280501} a^{2} + \frac{50319046187937131015858389710276689898170911850013731627269684675756855}{827130801190832676485232167291977778997429054734458694282615537251280501} a + \frac{386303898095254328744280790919506506878593340505978828421448133039759}{1376257572696892972521184970535736737100547512037368875678228847339901}$
Class group and class number
$C_{2018840}$, which has order $2018840$ (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: | \( 161406.837641 \) (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
| \(\Q(\sqrt{-195}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-39}) \), \(\Q(\sqrt{5}, \sqrt{-39})\), 5.5.390625.1, 10.0.68835601043701171875.3, \(\Q(\zeta_{25})^+\), 10.0.13767120208740234375.2 |
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 | ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ | R | R | ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ | R | ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| 3 | Data not computed | ||||||
| $5$ | 5.10.17.1 | $x^{10} - 5 x^{8} + 5$ | $10$ | $1$ | $17$ | $C_{10}$ | $[2]_{2}$ |
| 5.10.17.1 | $x^{10} - 5 x^{8} + 5$ | $10$ | $1$ | $17$ | $C_{10}$ | $[2]_{2}$ | |
| 13 | Data not computed | ||||||