Normalized defining polynomial
\( x^{32} - 2 x^{31} - 63 x^{30} + 124 x^{29} + 1712 x^{28} - 3300 x^{27} - 26300 x^{26} + 49300 x^{25} + 251803 x^{24} - 454306 x^{23} - 1565501 x^{22} + 2676696 x^{21} + 6411521 x^{20} - 10146346 x^{19} - 17289475 x^{18} + 24432604 x^{17} + 30419036 x^{16} - 36405466 x^{15} - 34405772 x^{14} + 32406588 x^{13} + 24527010 x^{12} - 16635294 x^{11} - 10672826 x^{10} + 4758978 x^{9} + 2732732 x^{8} - 704616 x^{7} - 385884 x^{6} + 43964 x^{5} + 26192 x^{4} - 328 x^{3} - 624 x^{2} - 32 x + 1 \)
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: | \(99276740263879938750515115508224780490603194567662317338624=2^{48}\cdot 3^{16}\cdot 17^{30}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $69.77$ | 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: | $2, 3, 17$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(408=2^{3}\cdot 3\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{408}(1,·)$, $\chi_{408}(5,·)$, $\chi_{408}(13,·)$, $\chi_{408}(145,·)$, $\chi_{408}(25,·)$, $\chi_{408}(157,·)$, $\chi_{408}(325,·)$, $\chi_{408}(41,·)$, $\chi_{408}(173,·)$, $\chi_{408}(29,·)$, $\chi_{408}(49,·)$, $\chi_{408}(317,·)$, $\chi_{408}(169,·)$, $\chi_{408}(373,·)$, $\chi_{408}(65,·)$, $\chi_{408}(197,·)$, $\chi_{408}(329,·)$, $\chi_{408}(205,·)$, $\chi_{408}(269,·)$, $\chi_{408}(209,·)$, $\chi_{408}(377,·)$, $\chi_{408}(217,·)$, $\chi_{408}(349,·)$, $\chi_{408}(229,·)$, $\chi_{408}(401,·)$, $\chi_{408}(233,·)$, $\chi_{408}(125,·)$, $\chi_{408}(113,·)$, $\chi_{408}(245,·)$, $\chi_{408}(361,·)$, $\chi_{408}(121,·)$, $\chi_{408}(253,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $\frac{1}{103} a^{30} + \frac{6}{103} a^{29} + \frac{10}{103} a^{28} + \frac{45}{103} a^{27} - \frac{47}{103} a^{26} + \frac{24}{103} a^{25} + \frac{12}{103} a^{24} + \frac{41}{103} a^{23} - \frac{22}{103} a^{22} - \frac{51}{103} a^{21} - \frac{35}{103} a^{20} + \frac{25}{103} a^{19} + \frac{23}{103} a^{18} - \frac{37}{103} a^{17} - \frac{28}{103} a^{16} - \frac{42}{103} a^{15} + \frac{40}{103} a^{14} - \frac{22}{103} a^{13} + \frac{39}{103} a^{12} - \frac{25}{103} a^{11} - \frac{17}{103} a^{10} - \frac{10}{103} a^{9} + \frac{44}{103} a^{8} - \frac{35}{103} a^{7} + \frac{35}{103} a^{6} + \frac{30}{103} a^{5} + \frac{39}{103} a^{4} + \frac{15}{103} a^{3} - \frac{8}{103} a^{2} - \frac{17}{103} a - \frac{33}{103}$, $\frac{1}{18595153207371425174771807875695193912595750497013354815149170411332360844177} a^{31} - \frac{22610290070126632467835052994283550696884750235291070402542670524902901987}{18595153207371425174771807875695193912595750497013354815149170411332360844177} a^{30} - \frac{3194005919637459590358100502284474475246755004301098568735381136543139312496}{18595153207371425174771807875695193912595750497013354815149170411332360844177} a^{29} + \frac{5566542139268489509565066345807044172450966804090392715808683104479259989644}{18595153207371425174771807875695193912595750497013354815149170411332360844177} a^{28} + \frac{5535133761778632727746204766636434720489691424443353838764438215484342587952}{18595153207371425174771807875695193912595750497013354815149170411332360844177} a^{27} - \frac{4950899736849586254696790998036863862460357858982255586538625243050864977392}{18595153207371425174771807875695193912595750497013354815149170411332360844177} a^{26} + \frac{12096298359660539115524857934659619638977930836636281420576845053409737225}{180535468032732283250211726948497028277628645602071405972322042828469522759} a^{25} + \frac{5847775748596274016764460131888979465510118664750682110233895451540098194764}{18595153207371425174771807875695193912595750497013354815149170411332360844177} a^{24} - \frac{9165228622571101727935731653136789226096558484900539648666528365982955671230}{18595153207371425174771807875695193912595750497013354815149170411332360844177} a^{23} - \frac{2884911080423419144664978714961991250415271518263418988220056936792707913983}{18595153207371425174771807875695193912595750497013354815149170411332360844177} a^{22} - \frac{8030791075482978361075492535581113142432790117954585067748084811134726578369}{18595153207371425174771807875695193912595750497013354815149170411332360844177} a^{21} + \frac{3573179702012726612640218043229215901069527934141921900337160696147180139319}{18595153207371425174771807875695193912595750497013354815149170411332360844177} a^{20} - \frac{8207810049521939628351676082629798894901260961310513920283542345029705525807}{18595153207371425174771807875695193912595750497013354815149170411332360844177} a^{19} - \frac{1948536405078487711486652176206503329919373561387617777740561622433390367689}{18595153207371425174771807875695193912595750497013354815149170411332360844177} a^{18} - \frac{7127350475588935706307162838589555929729128675536012408198904993215718737912}{18595153207371425174771807875695193912595750497013354815149170411332360844177} a^{17} - \frac{1514918549016798495302270582898659571590058536943731626726495526779100121486}{18595153207371425174771807875695193912595750497013354815149170411332360844177} a^{16} + \frac{1474927990796847364573961266953306783589334111179334620852879580047609743075}{18595153207371425174771807875695193912595750497013354815149170411332360844177} a^{15} - \frac{6826043001632407906025402665352763089173697028545301901820429652573028521768}{18595153207371425174771807875695193912595750497013354815149170411332360844177} a^{14} + \frac{1558162036069820703180370835060579542062757341092070779302271567996498825618}{18595153207371425174771807875695193912595750497013354815149170411332360844177} a^{13} + \frac{4533696547770937361118436759967075912762246248432842166312289886882379702379}{18595153207371425174771807875695193912595750497013354815149170411332360844177} a^{12} + \frac{5749624920258384032511186657743903072211494372832268158883619906382570077990}{18595153207371425174771807875695193912595750497013354815149170411332360844177} a^{11} - \frac{4909457713190823664663777311860963051739659455574618792941334536265116463194}{18595153207371425174771807875695193912595750497013354815149170411332360844177} a^{10} + \frac{6256077937570137105575035201360617841365243016878161869130437491644305366542}{18595153207371425174771807875695193912595750497013354815149170411332360844177} a^{9} - \frac{4414716234716840115697995585082761353033544249657023013420552093147012432948}{18595153207371425174771807875695193912595750497013354815149170411332360844177} a^{8} - \frac{7760583626802437655596688782793725220526594999921058632703971753628963584045}{18595153207371425174771807875695193912595750497013354815149170411332360844177} a^{7} + \frac{7219659535652935545147862543304763108672583251098680856647662372090284600275}{18595153207371425174771807875695193912595750497013354815149170411332360844177} a^{6} + \frac{3689881822486332147224815476939072540997259472794151254250299921687839867354}{18595153207371425174771807875695193912595750497013354815149170411332360844177} a^{5} - \frac{7807456765750753331893235891225097710895525976353927101912841204411049254951}{18595153207371425174771807875695193912595750497013354815149170411332360844177} a^{4} + \frac{4011826847619235782626932074627163471876648355637244958103898268269505866624}{18595153207371425174771807875695193912595750497013354815149170411332360844177} a^{3} - \frac{8480284443903739647292164972604128868409155588742223680120385947125958728482}{18595153207371425174771807875695193912595750497013354815149170411332360844177} a^{2} + \frac{3478037610612905792379898034396213304703085213044136627417715562322661437303}{18595153207371425174771807875695193912595750497013354815149170411332360844177} a + \frac{2680365414374408584440818241394849085298697112478690215131842362792471976042}{18595153207371425174771807875695193912595750497013354815149170411332360844177}$
Class group and class number
$C_{2}$, which has order $2$ (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: | \( 10078657704407090000 \) (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 | R | $16^{2}$ | $16^{2}$ | $16^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ | 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 |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| 3 | Data not computed | ||||||
| 17 | Data not computed | ||||||