Normalized defining polynomial
\( x^{32} - 16 x^{31} + 184 x^{30} - 1520 x^{29} + 10332 x^{28} - 58576 x^{27} + 287336 x^{26} - 1231568 x^{25} + 4678390 x^{24} - 15845360 x^{23} + 48150984 x^{22} - 131699184 x^{21} + 325050908 x^{20} - 724744720 x^{19} + 1460287592 x^{18} - 2657464016 x^{17} + 4362326951 x^{16} - 6446364592 x^{15} + 8552171016 x^{14} - 10150858480 x^{13} + 10733886004 x^{12} - 10060579408 x^{11} + 8306946984 x^{10} - 5997987856 x^{9} + 3753175770 x^{8} - 2012511408 x^{7} + 911516648 x^{6} - 342110896 x^{5} + 103611084 x^{4} - 24348880 x^{3} + 4169416 x^{2} - 463120 x + 25054 \)
Invariants
| Degree: | $32$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 16]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(486797299958529611528622959063550881784243829894751113445376=2^{128}\cdot 3^{16}\cdot 7^{16}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $73.32$ | 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: | $2, 3, 7$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(672=2^{5}\cdot 3\cdot 7\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{672}(1,·)$, $\chi_{672}(643,·)$, $\chi_{672}(265,·)$, $\chi_{672}(139,·)$, $\chi_{672}(533,·)$, $\chi_{672}(407,·)$, $\chi_{672}(29,·)$, $\chi_{672}(671,·)$, $\chi_{672}(547,·)$, $\chi_{672}(293,·)$, $\chi_{672}(167,·)$, $\chi_{672}(169,·)$, $\chi_{672}(43,·)$, $\chi_{672}(433,·)$, $\chi_{672}(307,·)$, $\chi_{672}(575,·)$, $\chi_{672}(197,·)$, $\chi_{672}(71,·)$, $\chi_{672}(461,·)$, $\chi_{672}(335,·)$, $\chi_{672}(337,·)$, $\chi_{672}(211,·)$, $\chi_{672}(601,·)$, $\chi_{672}(475,·)$, $\chi_{672}(97,·)$, $\chi_{672}(365,·)$, $\chi_{672}(239,·)$, $\chi_{672}(629,·)$, $\chi_{672}(503,·)$, $\chi_{672}(505,·)$, $\chi_{672}(379,·)$, $\chi_{672}(125,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $\frac{1}{60511348672248695464963355773231} a^{30} - \frac{15}{60511348672248695464963355773231} a^{29} + \frac{23779599793064579820018313539848}{60511348672248695464963355773231} a^{28} + \frac{30153694930588055309523745082529}{60511348672248695464963355773231} a^{27} + \frac{18925872456754136241532965645777}{60511348672248695464963355773231} a^{26} - \frac{13152989192998056403276900218300}{60511348672248695464963355773231} a^{25} + \frac{12054944133716877641267815830136}{60511348672248695464963355773231} a^{24} - \frac{502937235295994948984637566817}{60511348672248695464963355773231} a^{23} + \frac{22860368549101790269781484104704}{60511348672248695464963355773231} a^{22} - \frac{22289037896768766242774707540028}{60511348672248695464963355773231} a^{21} + \frac{26528937865829498387040239092383}{60511348672248695464963355773231} a^{20} + \frac{8488376099671655550070058591794}{60511348672248695464963355773231} a^{19} + \frac{10251832025664018441661894675355}{60511348672248695464963355773231} a^{18} - \frac{398890246197832650101517711741}{1951978989427377273063334057201} a^{17} - \frac{16020276809829962355440348337201}{60511348672248695464963355773231} a^{16} - \frac{16977199320967618717961798759338}{60511348672248695464963355773231} a^{15} - \frac{378770649379188998359037612855}{1951978989427377273063334057201} a^{14} + \frac{29941203530754308370135288834795}{60511348672248695464963355773231} a^{13} + \frac{18710517012157803328043366588744}{60511348672248695464963355773231} a^{12} - \frac{18601315742558282793717607789694}{60511348672248695464963355773231} a^{11} - \frac{1316672568521107790924268427133}{60511348672248695464963355773231} a^{10} - \frac{4716788377605061330915634114312}{60511348672248695464963355773231} a^{9} - \frac{26634488819958616189528461677122}{60511348672248695464963355773231} a^{8} - \frac{727196147943711237135895093715}{3559491098367570321468432692543} a^{7} + \frac{14945209323592981786737336058929}{60511348672248695464963355773231} a^{6} + \frac{3849395684908105841507223766684}{60511348672248695464963355773231} a^{5} - \frac{49821443160213375622710757937}{60511348672248695464963355773231} a^{4} + \frac{15112691650718200222440539014279}{60511348672248695464963355773231} a^{3} - \frac{9251835327913765311257897242827}{60511348672248695464963355773231} a^{2} - \frac{9108109370765108150804510966372}{60511348672248695464963355773231} a + \frac{4037268821058775356915923742047}{60511348672248695464963355773231}$, $\frac{1}{302133163920537736456562035375742383} a^{31} + \frac{2481}{302133163920537736456562035375742383} a^{30} + \frac{93755858693106293855048256406237227}{302133163920537736456562035375742383} a^{29} + \frac{59505057475764276677005161052089599}{302133163920537736456562035375742383} a^{28} - \frac{100684453519694666223300637896917587}{302133163920537736456562035375742383} a^{27} + \frac{147977220202159403951626992714070707}{302133163920537736456562035375742383} a^{26} + \frac{128929028919331331072909174852799799}{302133163920537736456562035375742383} a^{25} + \frac{1587426574224278011760771099961085}{9746231094210894724405226947604593} a^{24} + \frac{32774906983054065248913532061885294}{302133163920537736456562035375742383} a^{23} + \frac{147562168474345348162716989854554732}{302133163920537736456562035375742383} a^{22} - \frac{30918279394191543293873580625339257}{302133163920537736456562035375742383} a^{21} + \frac{92607665310310530852326529950308478}{302133163920537736456562035375742383} a^{20} - \frac{6131928660045025260438382845771452}{17772539054149278615091884433867199} a^{19} - \frac{84978026885773109396253108937070928}{302133163920537736456562035375742383} a^{18} + \frac{108598106722919913950130060245288638}{302133163920537736456562035375742383} a^{17} - \frac{87626019521716276988823232241945831}{302133163920537736456562035375742383} a^{16} + \frac{55944110197138198951583673262194222}{302133163920537736456562035375742383} a^{15} + \frac{114618170579774024187024100985309633}{302133163920537736456562035375742383} a^{14} + \frac{144037448759499667802568766658586559}{302133163920537736456562035375742383} a^{13} - \frac{51345535702439567829238110954918890}{302133163920537736456562035375742383} a^{12} + \frac{30419212047761848279593388520500013}{302133163920537736456562035375742383} a^{11} - \frac{99806732651642959374094442086541725}{302133163920537736456562035375742383} a^{10} + \frac{93368985713046041019951361122875604}{302133163920537736456562035375742383} a^{9} - \frac{51485231958414072624783438330928379}{302133163920537736456562035375742383} a^{8} - \frac{35379792890642614392235814963448276}{302133163920537736456562035375742383} a^{7} + \frac{57578393672576303626300622210885853}{302133163920537736456562035375742383} a^{6} - \frac{35533424301410054849836255595944999}{302133163920537736456562035375742383} a^{5} + \frac{142394984492668503434656857333162532}{302133163920537736456562035375742383} a^{4} - \frac{63523459803807353522983366726217906}{302133163920537736456562035375742383} a^{3} - \frac{31573277901958340690145385837314904}{302133163920537736456562035375742383} a^{2} + \frac{96417041815049030201461704069169374}{302133163920537736456562035375742383} a - \frac{97088575521189736377587957414242789}{302133163920537736456562035375742383}$
Class group and class number
$C_{4}\times C_{24}\times C_{24}$, which has order $2304$ (assuming GRH)
Unit group
| Rank: | $15$ | 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: | \( 56335773768373.43 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^2\times C_8$ (as 32T37):
| An abelian group of order 32 |
| The 32 conjugacy class representatives for $C_2^2\times C_8$ |
| Character table for $C_2^2\times C_8$ 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 | ${\href{/LocalNumberField/5.8.0.1}{8} }^{4}$ | R | ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }^{4}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ | ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$ |
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 | ||||||
| $7$ | 7.8.4.1 | $x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 7.8.4.1 | $x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 7.8.4.1 | $x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 7.8.4.1 | $x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |