Normalized defining polynomial
\( x^{32} - x^{31} - 49 x^{30} + 48 x^{29} + 1025 x^{28} - 977 x^{27} - 12023 x^{26} + 11046 x^{25} + 87635 x^{24} - 76589 x^{23} - 417686 x^{22} + 341097 x^{21} + 1341822 x^{20} - 1000725 x^{19} - 2965533 x^{18} + 1964808 x^{17} + 4562969 x^{16} - 2594590 x^{15} - 4899810 x^{14} + 2282032 x^{13} + 3639474 x^{12} - 1295443 x^{11} - 1825949 x^{10} + 442548 x^{9} + 590580 x^{8} - 76972 x^{7} - 112748 x^{6} + 3238 x^{5} + 10596 x^{4} + 496 x^{3} - 304 x^{2} - 16 x + 1 \)
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: | \(53818028735688890166227692218037660486426411285400390625=3^{16}\cdot 5^{16}\cdot 17^{30}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $55.16$ | 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: | $3, 5, 17$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(255=3\cdot 5\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{255}(1,·)$, $\chi_{255}(131,·)$, $\chi_{255}(4,·)$, $\chi_{255}(11,·)$, $\chi_{255}(14,·)$, $\chi_{255}(16,·)$, $\chi_{255}(146,·)$, $\chi_{255}(19,·)$, $\chi_{255}(151,·)$, $\chi_{255}(154,·)$, $\chi_{255}(29,·)$, $\chi_{255}(164,·)$, $\chi_{255}(166,·)$, $\chi_{255}(41,·)$, $\chi_{255}(44,·)$, $\chi_{255}(176,·)$, $\chi_{255}(49,·)$, $\chi_{255}(56,·)$, $\chi_{255}(64,·)$, $\chi_{255}(194,·)$, $\chi_{255}(196,·)$, $\chi_{255}(71,·)$, $\chi_{255}(74,·)$, $\chi_{255}(76,·)$, $\chi_{255}(209,·)$, $\chi_{255}(94,·)$, $\chi_{255}(224,·)$, $\chi_{255}(229,·)$, $\chi_{255}(106,·)$, $\chi_{255}(116,·)$, $\chi_{255}(169,·)$, $\chi_{255}(121,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $\frac{1}{101} a^{30} + \frac{1}{101} a^{29} - \frac{18}{101} a^{28} + \frac{41}{101} a^{27} - \frac{21}{101} a^{26} - \frac{32}{101} a^{25} + \frac{30}{101} a^{24} - \frac{23}{101} a^{23} - \frac{17}{101} a^{22} - \frac{25}{101} a^{21} + \frac{12}{101} a^{20} + \frac{26}{101} a^{19} + \frac{33}{101} a^{18} - \frac{5}{101} a^{17} - \frac{34}{101} a^{16} + \frac{44}{101} a^{15} + \frac{2}{101} a^{14} - \frac{34}{101} a^{13} - \frac{7}{101} a^{12} + \frac{48}{101} a^{11} + \frac{34}{101} a^{10} + \frac{29}{101} a^{9} - \frac{37}{101} a^{8} + \frac{26}{101} a^{7} + \frac{22}{101} a^{6} - \frac{20}{101} a^{5} - \frac{40}{101} a^{4} - \frac{48}{101} a^{3} + \frac{48}{101} a^{2} + \frac{8}{101} a - \frac{7}{101}$, $\frac{1}{7097793691045944993173260257257102677872866428523} a^{31} + \frac{23534427774858016893658310083855648654158253861}{7097793691045944993173260257257102677872866428523} a^{30} - \frac{2017915299257623649042539877064414337538160700978}{7097793691045944993173260257257102677872866428523} a^{29} - \frac{716073040426939999052188884029360055053250826932}{7097793691045944993173260257257102677872866428523} a^{28} - \frac{829019427359065784324047980475461454786034048741}{7097793691045944993173260257257102677872866428523} a^{27} - \frac{1855668104771687097924354472058069306279299607058}{7097793691045944993173260257257102677872866428523} a^{26} - \frac{2483741920075166854570764834923808511269135467871}{7097793691045944993173260257257102677872866428523} a^{25} + \frac{281021823238663430539741989639677509334709688003}{7097793691045944993173260257257102677872866428523} a^{24} - \frac{1315453739534597420129312245171379465907841894527}{7097793691045944993173260257257102677872866428523} a^{23} + \frac{211821222779971105733660121687247762949773593092}{7097793691045944993173260257257102677872866428523} a^{22} - \frac{1770496799184591195576961692043745804371428583764}{7097793691045944993173260257257102677872866428523} a^{21} + \frac{120272577512544207421880237187269084081475954633}{7097793691045944993173260257257102677872866428523} a^{20} + \frac{1861845387971323579506787628416111636995738320881}{7097793691045944993173260257257102677872866428523} a^{19} + \frac{588751976373293221844784246513290789500688241725}{7097793691045944993173260257257102677872866428523} a^{18} + \frac{1545480270274037612001613698476150694226228240439}{7097793691045944993173260257257102677872866428523} a^{17} + \frac{3469826602588665714950125316840091962743478252077}{7097793691045944993173260257257102677872866428523} a^{16} + \frac{969159906122054888162908754803583618548048467242}{7097793691045944993173260257257102677872866428523} a^{15} - \frac{2486945810405552483659331168811607984604431089077}{7097793691045944993173260257257102677872866428523} a^{14} - \frac{776686674249708999895196437331415114897550515829}{7097793691045944993173260257257102677872866428523} a^{13} - \frac{1991528463746846780628304383640872709090346750363}{7097793691045944993173260257257102677872866428523} a^{12} + \frac{3218959885647392799289452218941735883432495053649}{7097793691045944993173260257257102677872866428523} a^{11} + \frac{2739100256171103778200907065187556344641507571007}{7097793691045944993173260257257102677872866428523} a^{10} + \frac{634723999143184264481846369976076073395592656896}{7097793691045944993173260257257102677872866428523} a^{9} - \frac{2109185041469950454554052279228057744552598740294}{7097793691045944993173260257257102677872866428523} a^{8} + \frac{1117040073061261950098097306000345467788361342103}{7097793691045944993173260257257102677872866428523} a^{7} + \frac{23610476056158162128430223828145958190781175078}{70275185059860841516566933240169333444285806223} a^{6} - \frac{2836246446468769201765148444616996063845089329980}{7097793691045944993173260257257102677872866428523} a^{5} + \frac{2757880483016368328820556626689721589166931479591}{7097793691045944993173260257257102677872866428523} a^{4} + \frac{2315691497274716143875724303757682954091838592248}{7097793691045944993173260257257102677872866428523} a^{3} + \frac{3325655545349794482873486592916205523008104435721}{7097793691045944993173260257257102677872866428523} a^{2} - \frac{1463616906092701957931550734988169059014528156810}{7097793691045944993173260257257102677872866428523} a + \frac{2978340815081082675067683767793436086991003218877}{7097793691045944993173260257257102677872866428523}$
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: | \( 471152520983870500 \) (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 | ${\href{/LocalNumberField/2.8.0.1}{8} }^{4}$ | R | R | $16^{2}$ | $16^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ | 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 |
|---|---|---|---|---|---|---|---|
| 3 | Data not computed | ||||||
| 5 | Data not computed | ||||||
| 17 | Data not computed | ||||||