Normalized defining polynomial
\( x^{20} - x^{19} + 8 x^{18} - 15 x^{17} + 71 x^{16} - 176 x^{15} + 673 x^{14} - 1905 x^{13} + 6616 x^{12} - 19951 x^{11} + 66263 x^{10} + 139657 x^{9} + 324184 x^{8} + 653415 x^{7} + 1615873 x^{6} + 2958032 x^{5} + 8353079 x^{4} + 12353145 x^{3} + 46118408 x^{2} + 40353607 x + 282475249 \)
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: | \(2339097430337203010794683455827681=11^{18}\cdot 29^{10}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $46.61$ | 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: | $11, 29$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(319=11\cdot 29\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{319}(1,·)$, $\chi_{319}(260,·)$, $\chi_{319}(262,·)$, $\chi_{319}(202,·)$, $\chi_{319}(204,·)$, $\chi_{319}(144,·)$, $\chi_{319}(146,·)$, $\chi_{319}(86,·)$, $\chi_{319}(28,·)$, $\chi_{319}(30,·)$, $\chi_{319}(289,·)$, $\chi_{319}(291,·)$, $\chi_{319}(233,·)$, $\chi_{319}(173,·)$, $\chi_{319}(175,·)$, $\chi_{319}(115,·)$, $\chi_{319}(117,·)$, $\chi_{319}(57,·)$, $\chi_{319}(59,·)$, $\chi_{319}(318,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{463841} a^{11} - \frac{1}{7} a^{10} + \frac{1}{7} a^{9} - \frac{1}{7} a^{8} + \frac{1}{7} a^{7} - \frac{1}{7} a^{6} + \frac{1}{7} a^{5} - \frac{1}{7} a^{4} + \frac{1}{7} a^{3} - \frac{1}{7} a^{2} + \frac{1}{7} a + \frac{19951}{66263}$, $\frac{1}{3246887} a^{12} - \frac{1}{3246887} a^{11} - \frac{6}{49} a^{10} - \frac{1}{49} a^{9} + \frac{8}{49} a^{8} - \frac{15}{49} a^{7} + \frac{22}{49} a^{6} + \frac{20}{49} a^{5} - \frac{13}{49} a^{4} + \frac{6}{49} a^{3} + \frac{1}{49} a^{2} + \frac{19951}{463841} a + \frac{6616}{66263}$, $\frac{1}{22728209} a^{13} - \frac{1}{22728209} a^{12} + \frac{8}{22728209} a^{11} + \frac{48}{343} a^{10} - \frac{90}{343} a^{9} + \frac{83}{343} a^{8} - \frac{27}{343} a^{7} - \frac{78}{343} a^{6} - \frac{111}{343} a^{5} - \frac{92}{343} a^{4} + \frac{1}{343} a^{3} + \frac{19951}{3246887} a^{2} + \frac{6616}{463841} a + \frac{1905}{66263}$, $\frac{1}{159097463} a^{14} - \frac{1}{159097463} a^{13} + \frac{8}{159097463} a^{12} - \frac{15}{159097463} a^{11} - \frac{433}{2401} a^{10} + \frac{769}{2401} a^{9} + \frac{1002}{2401} a^{8} - \frac{421}{2401} a^{7} + \frac{232}{2401} a^{6} - \frac{778}{2401} a^{5} + \frac{1}{2401} a^{4} + \frac{19951}{22728209} a^{3} + \frac{6616}{3246887} a^{2} + \frac{1905}{463841} a + \frac{673}{66263}$, $\frac{1}{1113682241} a^{15} - \frac{1}{1113682241} a^{14} + \frac{8}{1113682241} a^{13} - \frac{15}{1113682241} a^{12} + \frac{71}{1113682241} a^{11} + \frac{3170}{16807} a^{10} - \frac{6201}{16807} a^{9} - \frac{5223}{16807} a^{8} - \frac{4570}{16807} a^{7} + \frac{1623}{16807} a^{6} + \frac{1}{16807} a^{5} + \frac{19951}{159097463} a^{4} + \frac{6616}{22728209} a^{3} + \frac{1905}{3246887} a^{2} + \frac{673}{463841} a + \frac{176}{66263}$, $\frac{1}{7795775687} a^{16} - \frac{1}{7795775687} a^{15} + \frac{8}{7795775687} a^{14} - \frac{15}{7795775687} a^{13} + \frac{71}{7795775687} a^{12} - \frac{176}{7795775687} a^{11} - \frac{56622}{117649} a^{10} - \frac{38837}{117649} a^{9} - \frac{4570}{117649} a^{8} - \frac{31991}{117649} a^{7} + \frac{1}{117649} a^{6} + \frac{19951}{1113682241} a^{5} + \frac{6616}{159097463} a^{4} + \frac{1905}{22728209} a^{3} + \frac{673}{3246887} a^{2} + \frac{176}{463841} a + \frac{71}{66263}$, $\frac{1}{54570429809} a^{17} - \frac{1}{54570429809} a^{16} + \frac{8}{54570429809} a^{15} - \frac{15}{54570429809} a^{14} + \frac{71}{54570429809} a^{13} - \frac{176}{54570429809} a^{12} + \frac{673}{54570429809} a^{11} - \frac{156486}{823543} a^{10} - \frac{239868}{823543} a^{9} - \frac{31991}{823543} a^{8} + \frac{1}{823543} a^{7} + \frac{19951}{7795775687} a^{6} + \frac{6616}{1113682241} a^{5} + \frac{1905}{159097463} a^{4} + \frac{673}{22728209} a^{3} + \frac{176}{3246887} a^{2} + \frac{71}{463841} a + \frac{15}{66263}$, $\frac{1}{381993008663} a^{18} - \frac{1}{381993008663} a^{17} + \frac{8}{381993008663} a^{16} - \frac{15}{381993008663} a^{15} + \frac{71}{381993008663} a^{14} - \frac{176}{381993008663} a^{13} + \frac{673}{381993008663} a^{12} - \frac{1905}{381993008663} a^{11} + \frac{583675}{5764801} a^{10} - \frac{1679077}{5764801} a^{9} + \frac{1}{5764801} a^{8} + \frac{19951}{54570429809} a^{7} + \frac{6616}{7795775687} a^{6} + \frac{1905}{1113682241} a^{5} + \frac{673}{159097463} a^{4} + \frac{176}{22728209} a^{3} + \frac{71}{3246887} a^{2} + \frac{15}{463841} a + \frac{8}{66263}$, $\frac{1}{2673951060641} a^{19} - \frac{1}{2673951060641} a^{18} + \frac{8}{2673951060641} a^{17} - \frac{15}{2673951060641} a^{16} + \frac{71}{2673951060641} a^{15} - \frac{176}{2673951060641} a^{14} + \frac{673}{2673951060641} a^{13} - \frac{1905}{2673951060641} a^{12} + \frac{6616}{2673951060641} a^{11} + \frac{4085724}{40353607} a^{10} + \frac{1}{40353607} a^{9} + \frac{19951}{381993008663} a^{8} + \frac{6616}{54570429809} a^{7} + \frac{1905}{7795775687} a^{6} + \frac{673}{1113682241} a^{5} + \frac{176}{159097463} a^{4} + \frac{71}{22728209} a^{3} + \frac{15}{3246887} a^{2} + \frac{8}{463841} a + \frac{1}{66263}$
Class group and class number
$C_{275}$, which has order $275$ (assuming GRH)
Unit group
| Rank: | $9$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{71}{1113682241} a^{16} - \frac{21577935}{1113682241} a^{5} \) (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: | \( 15197121.6751 \) (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
| \(\Q(\sqrt{-319}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{29}) \), \(\Q(\sqrt{-11}, \sqrt{29})\), \(\Q(\zeta_{11})^+\), 10.0.48364216424306959.3, \(\Q(\zeta_{11})\), 10.10.4396746947664269.1 |
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 | ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/23.1.0.1}{1} }^{20}$ | R | ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$ |
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 | ||||||
| 29 | Data not computed | ||||||