Normalized defining polynomial
\( x^{24} - x^{23} + 5 x^{22} - 5 x^{21} + 20 x^{20} - 19 x^{19} + 74 x^{18} - 99 x^{17} + 299 x^{16} + \cdots + 1 \)
Invariants
Degree: | $24$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 12]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(161761786626698377317203521728515625\) \(\medspace = 3^{12}\cdot 5^{18}\cdot 7^{20}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(29.31\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
Galois root discriminant: | $3^{1/2}5^{3/4}7^{5/6}\approx 29.3113956234904$ | ||
Ramified primes: | \(3\), \(5\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $24$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(105=3\cdot 5\cdot 7\) | ||
Dirichlet character group: | $\lbrace$$\chi_{105}(64,·)$, $\chi_{105}(1,·)$, $\chi_{105}(2,·)$, $\chi_{105}(4,·)$, $\chi_{105}(8,·)$, $\chi_{105}(76,·)$, $\chi_{105}(79,·)$, $\chi_{105}(16,·)$, $\chi_{105}(17,·)$, $\chi_{105}(19,·)$, $\chi_{105}(23,·)$, $\chi_{105}(68,·)$, $\chi_{105}(92,·)$, $\chi_{105}(94,·)$, $\chi_{105}(31,·)$, $\chi_{105}(32,·)$, $\chi_{105}(34,·)$, $\chi_{105}(38,·)$, $\chi_{105}(46,·)$, $\chi_{105}(47,·)$, $\chi_{105}(83,·)$, $\chi_{105}(53,·)$, $\chi_{105}(61,·)$, $\chi_{105}(62,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{2048}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{2146586119}a^{19}-\frac{80671238}{2146586119}a^{18}-\frac{363514156}{2146586119}a^{17}+\frac{40829208}{2146586119}a^{16}+\frac{329015334}{2146586119}a^{15}+\frac{607502231}{2146586119}a^{14}-\frac{911196026}{2146586119}a^{13}-\frac{744303332}{2146586119}a^{12}+\frac{227825652}{2146586119}a^{11}+\frac{10477406}{2146586119}a^{10}+\frac{981394019}{2146586119}a^{9}+\frac{716121545}{2146586119}a^{8}+\frac{876952384}{2146586119}a^{7}-\frac{537125891}{2146586119}a^{6}+\frac{479576926}{2146586119}a^{5}+\frac{489657714}{2146586119}a^{4}-\frac{256694009}{2146586119}a^{3}+\frac{377586222}{2146586119}a^{2}+\frac{74691672}{2146586119}a-\frac{80780534}{2146586119}$, $\frac{1}{2146586119}a^{20}-\frac{935272120}{2146586119}a^{18}+\frac{674003709}{2146586119}a^{17}-\frac{121919952}{2146586119}a^{16}-\frac{923153692}{2146586119}a^{15}-\frac{226411398}{2146586119}a^{14}+\frac{744303298}{2146586119}a^{13}+\frac{151546479}{2146586119}a^{12}-\frac{642200301}{2146586119}a^{11}-\frac{576258079}{2146586119}a^{10}+\frac{15925612}{2146586119}a^{9}-\frac{748133373}{2146586119}a^{8}-\frac{850996194}{2146586119}a^{7}-\frac{510269076}{2146586119}a^{6}-\frac{617694871}{2146586119}a^{5}+\frac{572504157}{2146586119}a^{4}-\frac{907932223}{2146586119}a^{3}-\frac{909147676}{2146586119}a^{2}+\frac{185130283}{2146586119}a+\frac{560390440}{2146586119}$, $\frac{1}{2146586119}a^{21}-\frac{1067568172}{2146586119}a^{14}+\frac{991307695}{2146586119}a^{7}+\frac{401941670}{2146586119}$, $\frac{1}{2146586119}a^{22}-\frac{1067568172}{2146586119}a^{15}+\frac{991307695}{2146586119}a^{8}+\frac{401941670}{2146586119}a$, $\frac{1}{2146586119}a^{23}-\frac{1067568172}{2146586119}a^{16}+\frac{991307695}{2146586119}a^{9}+\frac{401941670}{2146586119}a^{2}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{26}$, which has order $26$ (assuming GRH)
Unit group
Rank: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -\frac{375299640}{2146586119} a^{23} + \frac{469124550}{2146586119} a^{22} - \frac{1970323110}{2146586119} a^{21} + \frac{2345622750}{2146586119} a^{20} - \frac{7975180155}{2146586119} a^{19} + \frac{9007191360}{2146586119} a^{18} - \frac{29554846650}{2146586119} a^{17} + \frac{44097707700}{2146586119} a^{16} - \frac{121503258450}{2146586119} a^{15} + \frac{170667511290}{2146586119} a^{14} - \frac{450734867640}{2146586119} a^{13} + \frac{603295786795}{2146586119} a^{12} - \frac{1602060338250}{2146586119} a^{11} + \frac{2041630041600}{2146586119} a^{10} - \frac{2668380440400}{2146586119} a^{9} + \frac{3093970232160}{2146586119} a^{8} - \frac{3998911489110}{2146586119} a^{7} + \frac{126194503950}{74020211} a^{6} - \frac{4610521190949}{2146586119} a^{5} + \frac{159971471550}{2146586119} a^{4} + \frac{33401667960}{2146586119} a^{3} + \frac{7036868250}{2146586119} a^{2} + \frac{1407373650}{2146586119} a + \frac{375299640}{2146586119} \) (order $14$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
Fundamental units: | $\frac{231865976}{2146586119}a^{23}-\frac{289832470}{2146586119}a^{22}+\frac{1217296374}{2146586119}a^{21}-\frac{1449162350}{2146586119}a^{20}+\frac{4927024693}{2146586119}a^{19}-\frac{5564783424}{2146586119}a^{18}+\frac{18259445610}{2146586119}a^{17}-\frac{27244252180}{2146586119}a^{16}+\frac{75066609730}{2146586119}a^{15}-\frac{105441052586}{2146586119}a^{14}+\frac{278471037176}{2146586119}a^{13}-\frac{372719573107}{2146586119}a^{12}+\frac{989777885050}{2146586119}a^{11}-\frac{1261350909440}{2146586119}a^{10}+\frac{1648567089360}{2146586119}a^{9}-\frac{1911503106144}{2146586119}a^{8}+\frac{2470589940774}{2146586119}a^{7}-\frac{77964934430}{74020211}a^{6}+\frac{2849625874868}{2146586119}a^{5}-\frac{98832872270}{2146586119}a^{4}-\frac{20636071864}{2146586119}a^{3}-\frac{4347487050}{2146586119}a^{2}-\frac{869497410}{2146586119}a-\frac{231865976}{2146586119}$, $\frac{73451185}{2146586119}a^{23}+\frac{20373725}{2146586119}a^{22}+\frac{273431015}{2146586119}a^{21}+\frac{101922056}{2146586119}a^{20}+\frac{999836345}{2146586119}a^{19}+\frac{480925685}{2146586119}a^{18}+\frac{3652714400}{2146586119}a^{17}-\frac{328623975}{2146586119}a^{16}+\frac{12673238225}{2146586119}a^{15}+\frac{142197790}{2146586119}a^{14}+\frac{45581486865}{2146586119}a^{13}+\frac{6008438720}{2146586119}a^{12}+\frac{164222908950}{2146586119}a^{11}+\frac{41996365400}{2146586119}a^{10}+\frac{22329506300}{2146586119}a^{9}+\frac{67145917535}{2146586119}a^{8}+\frac{25934660315}{2146586119}a^{7}+\frac{289519634845}{2146586119}a^{6}+\frac{60526146476}{2146586119}a^{5}+\frac{1136453266750}{2146586119}a^{4}+\frac{237583293335}{2146586119}a^{3}+\frac{49668196625}{2146586119}a^{2}+\frac{10382390150}{2146586119}a+\frac{23716821}{2146586119}$, $\frac{453702664}{2146586119}a^{23}-\frac{469124550}{2146586119}a^{22}+\frac{2264334450}{2146586119}a^{21}-\frac{2326021994}{2146586119}a^{20}+\frac{9053221735}{2146586119}a^{19}-\frac{8830854485}{2146586119}a^{18}+\frac{33474997850}{2146586119}a^{17}-\frac{45685368936}{2146586119}a^{16}+\frac{135733407306}{2146586119}a^{15}-\frac{175077681390}{2146586119}a^{14}+\frac{501794837020}{2146586119}a^{13}-\frac{613488179915}{2146586119}a^{12}+\frac{1784624432840}{2146586119}a^{11}-\frac{2056311007844}{2146586119}a^{10}+\frac{2711972521744}{2146586119}a^{9}-\frac{3065353128400}{2146586119}a^{8}+\frac{4055753681510}{2146586119}a^{7}-\frac{3412965100290}{2146586119}a^{6}+\frac{4662090779985}{2146586119}a^{5}+\frac{974608985625}{2146586119}a^{4}-\frac{31147581020}{2146586119}a^{3}-\frac{226429314}{74020211}a^{2}-\frac{1309369870}{2146586119}a-\frac{355698884}{2146586119}$, $\frac{277925034}{2146586119}a^{23}-\frac{277502460}{2146586119}a^{22}+\frac{1386883689}{2146586119}a^{21}-\frac{1386874315}{2146586119}a^{20}+\frac{5547173300}{2146586119}a^{19}-\frac{5269814635}{2146586119}a^{18}+\frac{20524541210}{2146586119}a^{17}-\frac{27478032097}{2146586119}a^{16}+\frac{82935819409}{2146586119}a^{15}-\frac{105399813329}{2146586119}a^{14}+\frac{306762646569}{2146586119}a^{13}-\frac{369164383115}{2146586119}a^{12}+\frac{26626431840}{52355759}a^{11}-\frac{1235910211240}{2146586119}a^{10}+\frac{1658658109750}{2146586119}a^{9}-\frac{1869676379636}{2146586119}a^{8}+\frac{2486973085376}{2146586119}a^{7}-\frac{2082082410697}{2146586119}a^{6}+\frac{2887026343985}{2146586119}a^{5}+\frac{603532455040}{2146586119}a^{4}+\frac{126198192575}{2146586119}a^{3}-\frac{4005942330}{2146586119}a^{2}+\frac{9469161963}{2146586119}a+\frac{1929369060}{2146586119}$, $\frac{43883946}{2146586119}a^{23}+\frac{20373725}{2146586119}a^{22}+\frac{161453236}{2146586119}a^{21}+\frac{94462895}{2146586119}a^{20}+\frac{589582490}{2146586119}a^{19}+\frac{414015771}{2146586119}a^{18}+\frac{2160882200}{2146586119}a^{17}+\frac{266758502}{2146586119}a^{16}+\frac{7257887339}{2146586119}a^{15}+\frac{1824029644}{2146586119}a^{14}+\frac{26150372460}{2146586119}a^{13}+\frac{9887202440}{2146586119}a^{12}+\frac{94737261582}{2146586119}a^{11}+\frac{47583276989}{2146586119}a^{10}+\frac{3624494445}{2146586119}a^{9}+\frac{56255542475}{2146586119}a^{8}+\frac{4959874424}{2146586119}a^{7}+\frac{195646093660}{2146586119}a^{6}+\frac{40901093885}{2146586119}a^{5}+\frac{703097305375}{2146586119}a^{4}+\frac{236725489820}{2146586119}a^{3}+\frac{30728735085}{2146586119}a^{2}+\frac{10345094345}{2146586119}a+\frac{1342909379}{2146586119}$, $\frac{1402143}{2146586119}a^{23}-\frac{7010715}{2146586119}a^{22}+\frac{7010715}{2146586119}a^{21}-\frac{28042860}{2146586119}a^{20}+\frac{26640717}{2146586119}a^{19}-\frac{103758582}{2146586119}a^{18}+\frac{90954069}{2146586119}a^{17}-\frac{419240757}{2146586119}a^{16}+\frac{532814340}{2146586119}a^{15}-\frac{1550770158}{2146586119}a^{14}+\frac{1866252333}{2146586119}a^{13}-\frac{134605728}{52355759}a^{12}+\frac{6247949208}{2146586119}a^{11}-\frac{19291978399}{2146586119}a^{10}+\frac{9457454535}{2146586119}a^{9}-\frac{12575820567}{2146586119}a^{8}+\frac{10528691787}{2146586119}a^{7}-\frac{14594906487}{2146586119}a^{6}-\frac{3051063168}{2146586119}a^{5}-\frac{637975065}{2146586119}a^{4}-\frac{79603676271}{2146586119}a^{3}-\frac{28042860}{2146586119}a^{2}-\frac{5608572}{2146586119}a-\frac{1402143}{2146586119}$, $\frac{453702664}{2146586119}a^{23}-\frac{567128330}{2146586119}a^{22}+\frac{2381938986}{2146586119}a^{21}-\frac{2835641650}{2146586119}a^{20}+\frac{9641174486}{2146586119}a^{19}-\frac{10888863936}{2146586119}a^{18}+\frac{35729084790}{2146586119}a^{17}-\frac{53310063020}{2146586119}a^{16}+\frac{146886237470}{2146586119}a^{15}-\frac{206321286454}{2146586119}a^{14}+\frac{544896899464}{2146586119}a^{13}-\frac{729325994669}{2146586119}a^{12}+\frac{1936743246950}{2146586119}a^{11}-\frac{2468142492160}{2146586119}a^{10}+\frac{3225825941040}{2146586119}a^{9}-\frac{3740324762016}{2146586119}a^{8}+\frac{4834315310586}{2146586119}a^{7}-\frac{152557520770}{74020211}a^{6}+\frac{5574496667900}{2146586119}a^{5}-\frac{193390760530}{2146586119}a^{4}-\frac{40379537096}{2146586119}a^{3}-\frac{8506924950}{2146586119}a^{2}-\frac{1701384990}{2146586119}a-\frac{453702664}{2146586119}$, $\frac{622021133}{2146586119}a^{23}-\frac{542728904}{2146586119}a^{22}+\frac{3006880641}{2146586119}a^{21}-\frac{2693287293}{2146586119}a^{20}+\frac{11929556280}{2146586119}a^{19}-\frac{10132602090}{2146586119}a^{18}+\frac{44070755590}{2146586119}a^{17}-\frac{55338186896}{2146586119}a^{16}+\frac{176465429745}{2146586119}a^{15}-\frac{210589391850}{2146586119}a^{14}+\frac{651148289840}{2146586119}a^{13}-\frac{732360696405}{2146586119}a^{12}+\frac{2318142187550}{2146586119}a^{11}-\frac{2432398441404}{2146586119}a^{10}+\frac{3288764078233}{2146586119}a^{9}-\frac{3629895436805}{2146586119}a^{8}+\frac{4923256935836}{2146586119}a^{7}-\frac{3827472703634}{2146586119}a^{6}+\frac{5699240414651}{2146586119}a^{5}+\frac{2315225256535}{2146586119}a^{4}+\frac{249126046365}{2146586119}a^{3}-\frac{7368266322}{2146586119}a^{2}+\frac{18622103646}{2146586119}a+\frac{3793914080}{2146586119}$, $\frac{622318097}{2146586119}a^{23}-\frac{600542229}{2146586119}a^{22}+\frac{3085456706}{2146586119}a^{21}-\frac{2997075014}{2146586119}a^{20}+\frac{12314346020}{2146586119}a^{19}-\frac{11360345828}{2146586119}a^{18}+\frac{45543766546}{2146586119}a^{17}-\frac{59889940821}{2146586119}a^{16}+\frac{4477276719}{52355759}a^{15}-\frac{229445738686}{2146586119}a^{14}+\frac{678569211351}{2146586119}a^{13}-\frac{802191174959}{2146586119}a^{12}+\frac{2415114388539}{2146586119}a^{11}-\frac{2680149310216}{2146586119}a^{10}+\frac{3615392307475}{2146586119}a^{9}-\frac{4042600750925}{2146586119}a^{8}+\frac{5402418825109}{2146586119}a^{7}-\frac{4440834500898}{2146586119}a^{6}+\frac{6265562466909}{2146586119}a^{5}+\frac{1624223557610}{2146586119}a^{4}+\frac{273881308115}{2146586119}a^{3}+\frac{70925108695}{2146586119}a^{2}+\frac{12038435620}{2146586119}a-\frac{465008485}{2146586119}$, $\frac{42122034}{2146586119}a^{23}-\frac{93915274}{2146586119}a^{22}+\frac{274491574}{2146586119}a^{21}-\frac{482164724}{2146586119}a^{20}+\frac{1162742905}{2146586119}a^{19}-\frac{1899539727}{2146586119}a^{18}+\frac{4347569540}{2146586119}a^{17}-\frac{8244693270}{2146586119}a^{16}+\frac{18635598825}{2146586119}a^{15}-\frac{32735462909}{2146586119}a^{14}+\frac{69965919710}{2146586119}a^{13}-\frac{118132723330}{2146586119}a^{12}+\frac{248429023197}{2146586119}a^{11}-\frac{408363749579}{2146586119}a^{10}+\frac{532100664914}{2146586119}a^{9}-\frac{651139195681}{2146586119}a^{8}+\frac{19603133602}{52355759}a^{7}-\frac{21154815294}{52355759}a^{6}+\frac{942480030351}{2146586119}a^{5}-\frac{546636624521}{2146586119}a^{4}+\frac{41197936615}{2146586119}a^{3}-\frac{1666012516}{2146586119}a^{2}+\frac{5284849193}{2146586119}a+\frac{1088431761}{2146586119}$, $\frac{1059259537}{2146586119}a^{23}-\frac{1147125480}{2146586119}a^{22}+\frac{5354407974}{2146586119}a^{21}-\frac{5698107934}{2146586119}a^{20}+\frac{21467436752}{2146586119}a^{19}-\frac{21695514938}{2146586119}a^{18}+\frac{79419659210}{2146586119}a^{17}-\frac{110648046919}{2146586119}a^{16}+\frac{323063092005}{2146586119}a^{15}-\frac{425271535226}{2146586119}a^{14}+\frac{1195250147496}{2146586119}a^{13}-\frac{1493411517203}{2146586119}a^{12}+\frac{4250277704866}{2146586119}a^{11}-\frac{122382982064}{52355759}a^{10}+\frac{6615205792705}{2146586119}a^{9}-\frac{7508357838152}{2146586119}a^{8}+\frac{9879484598534}{2146586119}a^{7}-\frac{8494856133910}{2146586119}a^{6}+\frac{11368091910132}{2146586119}a^{5}+\frac{1681950463875}{2146586119}a^{4}-\frac{77527962984}{2146586119}a^{3}+\frac{73397097730}{2146586119}a^{2}+\frac{21845287393}{2146586119}a-\frac{881576200}{2146586119}$ (assuming GRH) | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 16114643.667937363 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 16114643.667937363 \cdot 26}{14\cdot\sqrt{161761786626698377317203521728515625}}\cr\approx \mathstrut & 0.281699756642720 \end{aligned}\] (assuming GRH)
Galois group
$C_2\times C_{12}$ (as 24T2):
An abelian group of order 24 |
The 24 conjugacy class representatives for $C_2\times C_{12}$ |
Character table for $C_2\times C_{12}$ 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 | ${\href{/padicField/2.12.0.1}{12} }^{2}$ | R | R | R | ${\href{/padicField/11.6.0.1}{6} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{6}$ | ${\href{/padicField/17.12.0.1}{12} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{4}$ | ${\href{/padicField/23.12.0.1}{12} }^{2}$ | ${\href{/padicField/29.1.0.1}{1} }^{24}$ | ${\href{/padicField/31.6.0.1}{6} }^{4}$ | ${\href{/padicField/37.12.0.1}{12} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{12}$ | ${\href{/padicField/43.4.0.1}{4} }^{6}$ | ${\href{/padicField/47.12.0.1}{12} }^{2}$ | ${\href{/padicField/53.12.0.1}{12} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.12.6.1 | $x^{12} + 18 x^{8} + 81 x^{4} - 486 x^{2} + 1458$ | $2$ | $6$ | $6$ | $C_{12}$ | $[\ ]_{2}^{6}$ |
3.12.6.1 | $x^{12} + 18 x^{8} + 81 x^{4} - 486 x^{2} + 1458$ | $2$ | $6$ | $6$ | $C_{12}$ | $[\ ]_{2}^{6}$ | |
\(5\) | Deg $24$ | $4$ | $6$ | $18$ | |||
\(7\) | Deg $24$ | $6$ | $4$ | $20$ |