Normalized defining polynomial
\( x^{20} - x^{19} + 5 x^{18} - 9 x^{17} + 29 x^{16} - 65 x^{15} + 181 x^{14} - 441 x^{13} + 1165 x^{12} - 2929 x^{11} + 7589 x^{10} + 11716 x^{9} + 18640 x^{8} + 28224 x^{7} + 46336 x^{6} + 66560 x^{5} + 118784 x^{4} + 147456 x^{3} + 327680 x^{2} + 262144 x + 1048576 \)
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: | \(11208759391001129236841977834969=11^{18}\cdot 17^{10}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $35.68$ | 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: | $11, 17$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(187=11\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{187}(1,·)$, $\chi_{187}(67,·)$, $\chi_{187}(69,·)$, $\chi_{187}(135,·)$, $\chi_{187}(137,·)$, $\chi_{187}(16,·)$, $\chi_{187}(18,·)$, $\chi_{187}(84,·)$, $\chi_{187}(86,·)$, $\chi_{187}(152,·)$, $\chi_{187}(35,·)$, $\chi_{187}(101,·)$, $\chi_{187}(103,·)$, $\chi_{187}(169,·)$, $\chi_{187}(171,·)$, $\chi_{187}(50,·)$, $\chi_{187}(52,·)$, $\chi_{187}(118,·)$, $\chi_{187}(120,·)$, $\chi_{187}(186,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{30356} a^{11} - \frac{1}{4} a^{10} + \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{2929}{7589}$, $\frac{1}{121424} a^{12} - \frac{1}{121424} a^{11} + \frac{1}{16} a^{10} - \frac{5}{16} a^{9} - \frac{7}{16} a^{8} + \frac{3}{16} a^{7} + \frac{1}{16} a^{6} - \frac{5}{16} a^{5} - \frac{7}{16} a^{4} + \frac{3}{16} a^{3} + \frac{1}{16} a^{2} + \frac{2929}{30356} a + \frac{1165}{7589}$, $\frac{1}{485696} a^{13} - \frac{1}{485696} a^{12} + \frac{5}{485696} a^{11} - \frac{21}{64} a^{10} + \frac{25}{64} a^{9} + \frac{19}{64} a^{8} + \frac{17}{64} a^{7} - \frac{5}{64} a^{6} + \frac{9}{64} a^{5} - \frac{29}{64} a^{4} + \frac{1}{64} a^{3} + \frac{2929}{121424} a^{2} + \frac{1165}{30356} a + \frac{441}{7589}$, $\frac{1}{1942784} a^{14} - \frac{1}{1942784} a^{13} + \frac{5}{1942784} a^{12} - \frac{9}{1942784} a^{11} + \frac{25}{256} a^{10} - \frac{109}{256} a^{9} - \frac{47}{256} a^{8} + \frac{123}{256} a^{7} - \frac{55}{256} a^{6} + \frac{35}{256} a^{5} + \frac{1}{256} a^{4} + \frac{2929}{485696} a^{3} + \frac{1165}{121424} a^{2} + \frac{441}{30356} a + \frac{181}{7589}$, $\frac{1}{7771136} a^{15} - \frac{1}{7771136} a^{14} + \frac{5}{7771136} a^{13} - \frac{9}{7771136} a^{12} + \frac{29}{7771136} a^{11} - \frac{365}{1024} a^{10} + \frac{465}{1024} a^{9} + \frac{123}{1024} a^{8} - \frac{311}{1024} a^{7} - \frac{221}{1024} a^{6} + \frac{1}{1024} a^{5} + \frac{2929}{1942784} a^{4} + \frac{1165}{485696} a^{3} + \frac{441}{121424} a^{2} + \frac{181}{30356} a + \frac{65}{7589}$, $\frac{1}{31084544} a^{16} - \frac{1}{31084544} a^{15} + \frac{5}{31084544} a^{14} - \frac{9}{31084544} a^{13} + \frac{29}{31084544} a^{12} - \frac{65}{31084544} a^{11} - \frac{1583}{4096} a^{10} + \frac{123}{4096} a^{9} + \frac{1737}{4096} a^{8} - \frac{1245}{4096} a^{7} + \frac{1}{4096} a^{6} + \frac{2929}{7771136} a^{5} + \frac{1165}{1942784} a^{4} + \frac{441}{485696} a^{3} + \frac{181}{121424} a^{2} + \frac{65}{30356} a + \frac{29}{7589}$, $\frac{1}{124338176} a^{17} - \frac{1}{124338176} a^{16} + \frac{5}{124338176} a^{15} - \frac{9}{124338176} a^{14} + \frac{29}{124338176} a^{13} - \frac{65}{124338176} a^{12} + \frac{181}{124338176} a^{11} + \frac{123}{16384} a^{10} - \frac{6455}{16384} a^{9} + \frac{6947}{16384} a^{8} + \frac{1}{16384} a^{7} + \frac{2929}{31084544} a^{6} + \frac{1165}{7771136} a^{5} + \frac{441}{1942784} a^{4} + \frac{181}{485696} a^{3} + \frac{65}{121424} a^{2} + \frac{29}{30356} a + \frac{9}{7589}$, $\frac{1}{497352704} a^{18} - \frac{1}{497352704} a^{17} + \frac{5}{497352704} a^{16} - \frac{9}{497352704} a^{15} + \frac{29}{497352704} a^{14} - \frac{65}{497352704} a^{13} + \frac{181}{497352704} a^{12} - \frac{441}{497352704} a^{11} + \frac{26313}{65536} a^{10} - \frac{25821}{65536} a^{9} + \frac{1}{65536} a^{8} + \frac{2929}{124338176} a^{7} + \frac{1165}{31084544} a^{6} + \frac{441}{7771136} a^{5} + \frac{181}{1942784} a^{4} + \frac{65}{485696} a^{3} + \frac{29}{121424} a^{2} + \frac{9}{30356} a + \frac{5}{7589}$, $\frac{1}{1989410816} a^{19} - \frac{1}{1989410816} a^{18} + \frac{5}{1989410816} a^{17} - \frac{9}{1989410816} a^{16} + \frac{29}{1989410816} a^{15} - \frac{65}{1989410816} a^{14} + \frac{181}{1989410816} a^{13} - \frac{441}{1989410816} a^{12} + \frac{1165}{1989410816} a^{11} + \frac{105251}{262144} a^{10} + \frac{1}{262144} a^{9} + \frac{2929}{497352704} a^{8} + \frac{1165}{124338176} a^{7} + \frac{441}{31084544} a^{6} + \frac{181}{7771136} a^{5} + \frac{65}{1942784} a^{4} + \frac{29}{485696} a^{3} + \frac{9}{121424} a^{2} + \frac{5}{30356} a + \frac{1}{7589}$
Class group and class number
$C_{41}$, which has order $41$ (assuming GRH)
Unit group
| Rank: | $9$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{1}{121424} a^{13} - \frac{49661}{121424} a^{2} \) (order $22$) | 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: | \( 3338983.62101 \) (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
| \(\Q(\sqrt{-187}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-11}, \sqrt{17})\), \(\Q(\zeta_{11})^+\), 10.0.3347948534700187.1, \(\Q(\zeta_{11})\), 10.10.304358957700017.1 |
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 | ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| 11 | Data not computed | ||||||
| 17 | Data not computed | ||||||