Normalized defining polynomial
\( x^{36} + 36 x^{34} + 594 x^{32} + 5953 x^{30} + 40485 x^{28} + 197721 x^{26} + 715780 x^{24} + 1954770 x^{22} + 4059891 x^{20} + 6417344 x^{18} + 7674462 x^{16} + 6854571 x^{14} + 4475587 x^{12} + 2066547 x^{10} + 640764 x^{8} + 122466 x^{6} + 12276 x^{4} + 432 x^{2} + 1 \)
Invariants
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$
Class group and class number
$C_{2}\times C_{28}\times C_{364}$, which has order $20384$ (assuming GRH)
Unit group
| Rank: | $17$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( a^{21} + 21 a^{19} + 189 a^{17} + 952 a^{15} + 2940 a^{13} + 5733 a^{11} + 7007 a^{9} + 5148 a^{7} + 2079 a^{5} + 385 a^{3} + 21 a \) (order $4$) | 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: | \( 816369751172.7767 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| An abelian group of order 36 |
| The 36 conjugacy class representatives for $C_6^2$ |
| Character table for $C_6^2$ is not computed |
Intermediate fields
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 | ${\href{/LocalNumberField/5.3.0.1}{3} }^{12}$ | R | ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ | ${\href{/LocalNumberField/13.6.0.1}{6} }^{6}$ | ${\href{/LocalNumberField/17.3.0.1}{3} }^{12}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{6}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{6}$ | ${\href{/LocalNumberField/29.6.0.1}{6} }^{6}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ | ${\href{/LocalNumberField/37.3.0.1}{3} }^{12}$ | ${\href{/LocalNumberField/41.3.0.1}{3} }^{12}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }^{6}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{6}$ | ${\href{/LocalNumberField/53.6.0.1}{6} }^{6}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.12.12.26 | $x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$ | $2$ | $6$ | $12$ | $C_6\times C_2$ | $[2]^{6}$ |
| 2.12.12.26 | $x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$ | $2$ | $6$ | $12$ | $C_6\times C_2$ | $[2]^{6}$ | |
| 2.12.12.26 | $x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$ | $2$ | $6$ | $12$ | $C_6\times C_2$ | $[2]^{6}$ | |
| 3 | Data not computed | ||||||
| 7 | Data not computed | ||||||