Normalized defining polynomial
\( x^{32} - 61 x^{30} + 1681 x^{28} - 27673 x^{26} + 303233 x^{24} - 2332729 x^{22} + 12948977 x^{20} - 52494441 x^{18} + 155540513 x^{16} - 333780761 x^{14} + 508663377 x^{12} - 532743113 x^{10} + 364184961 x^{8} - 149641593 x^{6} + 32083377 x^{4} - 2695209 x^{2} + 73441 \)
Invariants
| Degree: | $32$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[32, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(1169489004184531786140825032159476461065507700131479058120704=2^{32}\cdot 7^{16}\cdot 17^{30}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $75.36$ | 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: | $2, 7, 17$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(476=2^{2}\cdot 7\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{476}(253,·)$, $\chi_{476}(1,·)$, $\chi_{476}(393,·)$, $\chi_{476}(267,·)$, $\chi_{476}(435,·)$, $\chi_{476}(405,·)$, $\chi_{476}(281,·)$, $\chi_{476}(421,·)$, $\chi_{476}(295,·)$, $\chi_{476}(41,·)$, $\chi_{476}(307,·)$, $\chi_{476}(181,·)$, $\chi_{476}(265,·)$, $\chi_{476}(351,·)$, $\chi_{476}(195,·)$, $\chi_{476}(71,·)$, $\chi_{476}(55,·)$, $\chi_{476}(209,·)$, $\chi_{476}(83,·)$, $\chi_{476}(251,·)$, $\chi_{476}(475,·)$, $\chi_{476}(223,·)$, $\chi_{476}(97,·)$, $\chi_{476}(99,·)$, $\chi_{476}(225,·)$, $\chi_{476}(365,·)$, $\chi_{476}(111,·)$, $\chi_{476}(211,·)$, $\chi_{476}(169,·)$, $\chi_{476}(377,·)$, $\chi_{476}(379,·)$, $\chi_{476}(125,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{271} a^{17} - \frac{34}{271} a^{15} - \frac{66}{271} a^{13} - \frac{13}{271} a^{11} + \frac{55}{271} a^{9} - \frac{132}{271} a^{7} - \frac{103}{271} a^{5} - \frac{96}{271} a^{3} + \frac{16}{271} a$, $\frac{1}{27371} a^{18} + \frac{10264}{27371} a^{16} - \frac{608}{27371} a^{14} + \frac{10556}{27371} a^{12} + \frac{1681}{27371} a^{10} - \frac{6636}{27371} a^{8} + \frac{1523}{27371} a^{6} - \frac{9310}{27371} a^{4} + \frac{558}{27371} a^{2} - \frac{37}{101}$, $\frac{1}{27371} a^{19} - \frac{38}{27371} a^{17} - \frac{6163}{27371} a^{15} + \frac{6213}{27371} a^{13} - \frac{1248}{27371} a^{11} + \frac{1545}{27371} a^{9} - \frac{7163}{27371} a^{7} + \frac{11698}{27371} a^{5} + \frac{4194}{27371} a^{3} - \frac{10633}{27371} a$, $\frac{1}{27371} a^{20} + \frac{675}{27371} a^{16} + \frac{10480}{27371} a^{14} - \frac{10685}{27371} a^{12} + \frac{10681}{27371} a^{10} - \frac{12992}{27371} a^{8} - \frac{12541}{27371} a^{6} + \frac{6237}{27371} a^{4} + \frac{10571}{27371} a^{2} + \frac{8}{101}$, $\frac{1}{27371} a^{21} - \frac{32}{27371} a^{17} + \frac{7147}{27371} a^{15} + \frac{8606}{27371} a^{13} - \frac{7499}{27371} a^{11} + \frac{2865}{27371} a^{9} - \frac{1330}{27371} a^{7} - \frac{3055}{27371} a^{5} - \frac{3670}{27371} a^{3} - \frac{9144}{27371} a$, $\frac{1}{27371} a^{22} + \frac{7143}{27371} a^{16} - \frac{10850}{27371} a^{14} + \frac{1841}{27371} a^{12} + \frac{1915}{27371} a^{10} + \frac{5286}{27371} a^{8} - \frac{9061}{27371} a^{6} - \frac{509}{27371} a^{4} + \frac{8712}{27371} a^{2} + \frac{28}{101}$, $\frac{1}{27371} a^{23} - \frac{28}{27371} a^{17} - \frac{13375}{27371} a^{15} + \frac{9820}{27371} a^{13} + \frac{13025}{27371} a^{11} - \frac{5925}{27371} a^{9} + \frac{6897}{27371} a^{7} - \frac{913}{27371} a^{5} + \frac{12853}{27371} a^{3} + \frac{2336}{27371} a$, $\frac{1}{27371} a^{24} + \frac{307}{27371} a^{16} - \frac{7204}{27371} a^{14} + \frac{7512}{27371} a^{12} - \frac{13599}{27371} a^{10} + \frac{12686}{27371} a^{8} - \frac{13011}{27371} a^{6} - \frac{1488}{27371} a^{4} - \frac{9411}{27371} a^{2} - \frac{26}{101}$, $\frac{1}{27371} a^{25} + \frac{4}{27371} a^{17} + \frac{3098}{27371} a^{15} + \frac{139}{27371} a^{13} - \frac{9660}{27371} a^{11} - \frac{3979}{27371} a^{9} - \frac{386}{27371} a^{7} + \frac{2350}{27371} a^{5} - \frac{7694}{27371} a^{3} - \frac{11894}{27371} a$, $\frac{1}{27371} a^{26} - \frac{10587}{27371} a^{16} + \frac{2571}{27371} a^{14} + \frac{2858}{27371} a^{12} - \frac{10703}{27371} a^{10} - \frac{1213}{27371} a^{8} - \frac{3742}{27371} a^{6} + \frac{2175}{27371} a^{4} + \frac{13245}{27371} a^{2} + \frac{47}{101}$, $\frac{1}{27371} a^{27} + \frac{18}{27371} a^{17} - \frac{2176}{27371} a^{15} - \frac{12797}{27371} a^{13} - \frac{11713}{27371} a^{11} + \frac{7271}{27371} a^{9} - \frac{7681}{27371} a^{7} + \frac{4700}{27371} a^{5} + \frac{7892}{27371} a^{3} - \frac{9180}{27371} a$, $\frac{1}{27371} a^{28} + \frac{4669}{27371} a^{16} - \frac{1853}{27371} a^{14} - \frac{10124}{27371} a^{12} + \frac{4384}{27371} a^{10} + \frac{2283}{27371} a^{8} + \frac{4657}{27371} a^{6} + \frac{11246}{27371} a^{4} + \frac{8147}{27371} a^{2} - \frac{41}{101}$, $\frac{1}{27371} a^{29} + \frac{23}{27371} a^{17} - \frac{8115}{27371} a^{15} - \frac{4569}{27371} a^{13} + \frac{10040}{27371} a^{11} - \frac{6908}{27371} a^{9} - \frac{11604}{27371} a^{7} - \frac{2894}{27371} a^{5} - \frac{11144}{27371} a^{3} - \frac{3334}{27371} a$, $\frac{1}{27371} a^{30} + \frac{2152}{27371} a^{16} + \frac{9415}{27371} a^{14} + \frac{13591}{27371} a^{12} + \frac{9171}{27371} a^{10} + \frac{4169}{27371} a^{8} - \frac{10552}{27371} a^{6} + \frac{11389}{27371} a^{4} + \frac{11203}{27371} a^{2} + \frac{43}{101}$, $\frac{1}{27371} a^{31} + \frac{31}{27371} a^{17} - \frac{584}{27371} a^{15} - \frac{10649}{27371} a^{13} + \frac{9373}{27371} a^{11} - \frac{3002}{27371} a^{9} - \frac{4290}{27371} a^{7} + \frac{10884}{27371} a^{5} - \frac{4149}{27371} a^{3} + \frac{5088}{27371} a$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $31$ | 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: | \( 66503911699898060000 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times C_{16}$ (as 32T32):
| An abelian group of order 32 |
| The 32 conjugacy class representatives for $C_2\times C_{16}$ |
| Character table for $C_2\times C_{16}$ 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 | $16^{2}$ | $16^{2}$ | R | $16^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ | R | ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ | $16^{2}$ | $16^{2}$ | $16^{2}$ | $16^{2}$ | $16^{2}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ | ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| 7 | Data not computed | ||||||
| 17 | Data not computed | ||||||