Normalized defining polynomial
\( x^{24} - 11 x^{22} + 76 x^{20} - 327 x^{18} + 1031 x^{16} - 2261 x^{14} + 3677 x^{12} - 4001 x^{10} + 3091 x^{8} - 1302 x^{6} + 371 x^{4} - 21 x^{2} + 1 \)
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: | \(169450166032303737749261229339181056\) \(\medspace = 2^{24}\cdot 3^{12}\cdot 13^{20}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(29.37\) | 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: | $2\cdot 3^{1/2}13^{5/6}\approx 29.368160836761707$ | ||
Ramified primes: | \(2\), \(3\), \(13\) | 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: | \(156=2^{2}\cdot 3\cdot 13\) | ||
Dirichlet character group: | $\lbrace$$\chi_{156}(1,·)$, $\chi_{156}(131,·)$, $\chi_{156}(133,·)$, $\chi_{156}(113,·)$, $\chi_{156}(139,·)$, $\chi_{156}(77,·)$, $\chi_{156}(79,·)$, $\chi_{156}(17,·)$, $\chi_{156}(107,·)$, $\chi_{156}(23,·)$, $\chi_{156}(25,·)$, $\chi_{156}(155,·)$, $\chi_{156}(29,·)$, $\chi_{156}(95,·)$, $\chi_{156}(35,·)$, $\chi_{156}(101,·)$, $\chi_{156}(103,·)$, $\chi_{156}(43,·)$, $\chi_{156}(49,·)$, $\chi_{156}(53,·)$, $\chi_{156}(55,·)$, $\chi_{156}(121,·)$, $\chi_{156}(61,·)$, $\chi_{156}(127,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{2048}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{5}a^{18}-\frac{1}{5}a^{12}+\frac{1}{5}a^{6}-\frac{1}{5}$, $\frac{1}{5}a^{19}-\frac{1}{5}a^{13}+\frac{1}{5}a^{7}-\frac{1}{5}a$, $\frac{1}{5}a^{20}-\frac{1}{5}a^{14}+\frac{1}{5}a^{8}-\frac{1}{5}a^{2}$, $\frac{1}{5}a^{21}-\frac{1}{5}a^{15}+\frac{1}{5}a^{9}-\frac{1}{5}a^{3}$, $\frac{1}{3140120548525}a^{22}+\frac{51460653714}{628024109705}a^{20}+\frac{174816507931}{3140120548525}a^{18}+\frac{1453912814519}{3140120548525}a^{16}+\frac{206892109419}{628024109705}a^{14}+\frac{1123625754299}{3140120548525}a^{12}+\frac{803708014511}{3140120548525}a^{10}-\frac{73095569372}{628024109705}a^{8}-\frac{411144212009}{3140120548525}a^{6}-\frac{1000770905871}{3140120548525}a^{4}-\frac{252578346066}{628024109705}a^{2}-\frac{1543772195661}{3140120548525}$, $\frac{1}{3140120548525}a^{23}+\frac{51460653714}{628024109705}a^{21}+\frac{174816507931}{3140120548525}a^{19}+\frac{1453912814519}{3140120548525}a^{17}+\frac{206892109419}{628024109705}a^{15}+\frac{1123625754299}{3140120548525}a^{13}+\frac{803708014511}{3140120548525}a^{11}-\frac{73095569372}{628024109705}a^{9}-\frac{411144212009}{3140120548525}a^{7}-\frac{1000770905871}{3140120548525}a^{5}-\frac{252578346066}{628024109705}a^{3}-\frac{1543772195661}{3140120548525}a$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $5$ |
Class group and class number
$C_{6}$, which has order $6$ (assuming GRH)
Unit group
Rank: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -\frac{245903431384}{3140120548525} a^{23} + \frac{105667011281}{125604821941} a^{21} - \frac{17975844073174}{3140120548525} a^{19} + \frac{75426341030079}{3140120548525} a^{17} - \frac{46337912069137}{628024109705} a^{15} + \frac{486228806334304}{3140120548525} a^{13} - \frac{748357514123674}{3140120548525} a^{11} + \frac{145577986639644}{628024109705} a^{9} - \frac{477964007210989}{3140120548525} a^{7} + \frac{107298786752664}{3140120548525} a^{5} - \frac{1220293417317}{628024109705} a^{3} - \frac{12756519451156}{3140120548525} a \) (order $12$) | 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: | $\frac{704397084041}{3140120548525}a^{23}-\frac{1558739512853}{628024109705}a^{21}+\frac{54030205157291}{3140120548525}a^{19}-\frac{233734316792496}{3140120548525}a^{17}+\frac{148128190524686}{628024109705}a^{15}-\frac{16\!\cdots\!86}{3140120548525}a^{13}+\frac{26\!\cdots\!26}{3140120548525}a^{11}-\frac{592515736801589}{628024109705}a^{9}+\frac{23\!\cdots\!01}{3140120548525}a^{7}-\frac{10\!\cdots\!11}{3140120548525}a^{5}+\frac{56412490478381}{628024109705}a^{3}-\frac{15971220329046}{3140120548525}a+1$, $\frac{1356732882273}{3140120548525}a^{23}-\frac{2965595826141}{628024109705}a^{21}+\frac{102080071115143}{3140120548525}a^{19}-\frac{436626603605213}{3140120548525}a^{17}+\frac{54770736718139}{125604821941}a^{15}-\frac{29\!\cdots\!78}{3140120548525}a^{13}+\frac{47\!\cdots\!28}{3140120548525}a^{11}-\frac{10\!\cdots\!24}{628024109705}a^{9}+\frac{38\!\cdots\!48}{3140120548525}a^{7}-\frac{15\!\cdots\!33}{3140120548525}a^{5}+\frac{16982131723433}{125604821941}a^{3}-\frac{4468719827608}{3140120548525}a$, $\frac{600549560964}{3140120548525}a^{23}-\frac{1310947964301}{628024109705}a^{21}+\frac{45098195010069}{3140120548525}a^{19}-\frac{192697297051209}{3140120548525}a^{17}+\frac{120770247601668}{628024109705}a^{15}-\frac{13\!\cdots\!24}{3140120548525}a^{13}+\frac{21\!\cdots\!04}{3140120548525}a^{11}-\frac{90078604652013}{125604821941}a^{9}+\frac{17\!\cdots\!09}{3140120548525}a^{7}-\frac{680309729960169}{3140120548525}a^{5}+\frac{39614768904028}{628024109705}a^{3}-\frac{1963501387089}{3140120548525}a$, $\frac{218116010373}{3140120548525}a^{22}-\frac{480340643134}{628024109705}a^{20}+\frac{16620342391423}{3140120548525}a^{18}-\frac{71661750784463}{3140120548525}a^{16}+\frac{45318346934283}{628024109705}a^{14}-\frac{498800568405133}{3140120548525}a^{12}+\frac{815201488225328}{3140120548525}a^{10}-\frac{178453352585392}{628024109705}a^{8}+\frac{693576867788178}{3140120548525}a^{6}-\frac{291167481730908}{3140120548525}a^{4}+\frac{16762586327118}{628024109705}a^{2}-\frac{4744921526313}{3140120548525}$, $\frac{52795263591}{3140120548525}a^{22}-\frac{121942742607}{628024109705}a^{20}+\frac{4325273387946}{3140120548525}a^{18}-\frac{19393889334871}{3140120548525}a^{16}+\frac{2536358044785}{125604821941}a^{14}-\frac{146980831797291}{3140120548525}a^{12}+\frac{252377873956276}{3140120548525}a^{10}-\frac{60109331096313}{628024109705}a^{8}+\frac{250091798957931}{3140120548525}a^{6}-\frac{123245261151761}{3140120548525}a^{4}+\frac{1296024463851}{125604821941}a^{2}-\frac{1837338567651}{3140120548525}$, $a$, $\frac{1909393882417}{3140120548525}a^{23}+\frac{13185531480}{125604821941}a^{22}-\frac{4174182863438}{628024109705}a^{21}-\frac{141732915712}{125604821941}a^{20}+\frac{143702804456922}{3140120548525}a^{19}+\frac{966103169228}{125604821941}a^{18}-\frac{614794010267927}{3140120548525}a^{17}-\frac{4065666277275}{125604821941}a^{16}+\frac{385718008805639}{628024109705}a^{15}+\frac{12554022088692}{125604821941}a^{14}-\frac{41\!\cdots\!12}{3140120548525}a^{13}-\frac{26607179407868}{125604821941}a^{12}+\frac{67\!\cdots\!87}{3140120548525}a^{11}+\frac{41718954698664}{125604821941}a^{10}-\frac{289731340384415}{125604821941}a^{9}-\frac{42366067915672}{125604821941}a^{8}+\frac{55\!\cdots\!92}{3140120548525}a^{7}+\frac{30642628259568}{125604821941}a^{6}-\frac{22\!\cdots\!57}{3140120548525}a^{5}-\frac{10635879483100}{125604821941}a^{4}+\frac{127421385514859}{628024109705}a^{3}+\frac{3400016873472}{125604821941}a^{2}-\frac{21227589648282}{3140120548525}a-\frac{192064117548}{125604821941}$, $\frac{236799467779}{628024109705}a^{23}+\frac{149961967854}{3140120548525}a^{22}-\frac{2585657820389}{628024109705}a^{21}-\frac{318253728976}{628024109705}a^{20}+\frac{17793784045093}{628024109705}a^{19}+\frac{10786135837474}{3140120548525}a^{18}-\frac{76057894864114}{628024109705}a^{17}-\frac{44913500596199}{3140120548525}a^{16}+\frac{238415050407304}{628024109705}a^{15}+\frac{27516749439158}{628024109705}a^{14}-\frac{517873187091888}{628024109705}a^{13}-\frac{287093779889754}{3140120548525}a^{12}+\frac{834165713384839}{628024109705}a^{11}+\frac{443770824968969}{3140120548525}a^{10}-\frac{891015091873169}{628024109705}a^{9}-\frac{17333803098644}{125604821941}a^{8}+\frac{677120029399793}{628024109705}a^{7}+\frac{301731278702264}{3140120548525}a^{6}-\frac{270684563847469}{628024109705}a^{5}-\frac{81198757872234}{3140120548525}a^{4}+\frac{79147692624884}{628024109705}a^{3}+\frac{6363141888848}{628024109705}a^{2}-\frac{2629436246553}{628024109705}a-\frac{1794719610594}{3140120548525}$, $\frac{456763158121}{628024109705}a^{23}+\frac{793803675654}{3140120548525}a^{22}-\frac{4989050554889}{628024109705}a^{21}-\frac{1733477475901}{628024109705}a^{20}+\frac{6863329410548}{125604821941}a^{19}+\frac{59643763410149}{3140120548525}a^{18}-\frac{146585912263386}{628024109705}a^{17}-\frac{254925015074749}{3140120548525}a^{16}+\frac{458766086249679}{628024109705}a^{15}+\frac{159804737582463}{628024109705}a^{14}-\frac{198770214661521}{125604821941}a^{13}-\frac{17\!\cdots\!04}{3140120548525}a^{12}+\frac{15\!\cdots\!76}{628024109705}a^{11}+\frac{27\!\cdots\!44}{3140120548525}a^{10}-\frac{16\!\cdots\!09}{628024109705}a^{9}-\frac{119345085727350}{125604821941}a^{8}+\frac{249815712027337}{125604821941}a^{7}+\frac{22\!\cdots\!39}{3140120548525}a^{6}-\frac{467197891083261}{628024109705}a^{5}-\frac{901144104357509}{3140120548525}a^{4}+\frac{113886280011149}{628024109705}a^{3}+\frac{51113170061483}{628024109705}a^{2}+\frac{745005739755}{125604821941}a-\frac{2600900066419}{3140120548525}$, $\frac{45329639814}{3140120548525}a^{23}+\frac{74247}{2146975}a^{22}-\frac{19841070807}{125604821941}a^{21}-\frac{165303}{429395}a^{20}+\frac{3396470311089}{3140120548525}a^{19}+\frac{5729982}{2146975}a^{18}-\frac{14407558977159}{3140120548525}a^{17}-\frac{24781182}{2146975}a^{16}+\frac{8890172671257}{628024109705}a^{15}+\frac{15617579}{429395}a^{14}-\frac{94015796469869}{3140120548525}a^{13}-\frac{170825097}{2146975}a^{12}+\frac{144285950759904}{3140120548525}a^{11}+\frac{273664292}{2146975}a^{10}-\frac{28102355007354}{628024109705}a^{9}-\frac{11558631}{85879}a^{8}+\frac{89202070866229}{3140120548525}a^{7}+\frac{206532127}{2146975}a^{6}-\frac{20731134212694}{3140120548525}a^{5}-\frac{72378887}{2146975}a^{4}+\frac{235776312837}{628024109705}a^{3}+\frac{2611739}{429395}a^{2}+\frac{704397084041}{3140120548525}a+\frac{313483}{2146975}$, $\frac{6788509273824}{3140120548525}a^{23}+\frac{1654697599466}{3140120548525}a^{22}-\frac{14868328554044}{628024109705}a^{21}-\frac{3645814082111}{628024109705}a^{20}+\frac{512281640427214}{3140120548525}a^{19}+\frac{126036254387756}{3140120548525}a^{18}-\frac{21\!\cdots\!94}{3140120548525}a^{17}-\frac{542894974615071}{3140120548525}a^{16}+\frac{13\!\cdots\!26}{628024109705}a^{15}+\frac{342594293118509}{628024109705}a^{14}-\frac{15\!\cdots\!44}{3140120548525}a^{13}-\frac{37\!\cdots\!76}{3140120548525}a^{12}+\frac{24\!\cdots\!64}{3140120548525}a^{11}+\frac{61\!\cdots\!26}{3140120548525}a^{10}-\frac{51\!\cdots\!08}{628024109705}a^{9}-\frac{13\!\cdots\!22}{628024109705}a^{8}+\frac{19\!\cdots\!54}{3140120548525}a^{7}+\frac{51\!\cdots\!91}{3140120548525}a^{6}-\frac{78\!\cdots\!54}{3140120548525}a^{5}-\frac{21\!\cdots\!86}{3140120548525}a^{4}+\frac{419384760779151}{628024109705}a^{3}+\frac{114045274374079}{628024109705}a^{2}-\frac{22613437228734}{3140120548525}a-\frac{19185921206936}{3140120548525}$ (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: | \( 67326381.43157153 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 67326381.43157153 \cdot 6}{12\cdot\sqrt{169450166032303737749261229339181056}}\cr\approx \mathstrut & 0.309594116286471 \end{aligned}\] (assuming GRH)
Galois group
$C_2^2\times C_6$ (as 24T3):
An abelian group of order 24 |
The 24 conjugacy class representatives for $C_2^2\times C_6$ |
Character table for $C_2^2\times C_6$ 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 | ${\href{/padicField/5.2.0.1}{2} }^{12}$ | ${\href{/padicField/7.6.0.1}{6} }^{4}$ | ${\href{/padicField/11.6.0.1}{6} }^{4}$ | R | ${\href{/padicField/17.6.0.1}{6} }^{4}$ | ${\href{/padicField/19.6.0.1}{6} }^{4}$ | ${\href{/padicField/23.6.0.1}{6} }^{4}$ | ${\href{/padicField/29.6.0.1}{6} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{12}$ | ${\href{/padicField/37.6.0.1}{6} }^{4}$ | ${\href{/padicField/41.6.0.1}{6} }^{4}$ | ${\href{/padicField/43.6.0.1}{6} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{12}$ | ${\href{/padicField/53.2.0.1}{2} }^{12}$ | ${\href{/padicField/59.6.0.1}{6} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.12.26 | $x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$ | $2$ | $6$ | $12$ | $C_6\times C_2$ | $[2]^{6}$ |
2.12.12.26 | $x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$ | $2$ | $6$ | $12$ | $C_6\times C_2$ | $[2]^{6}$ | |
\(3\) | 3.12.6.2 | $x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
3.12.6.2 | $x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
\(13\) | 13.6.5.2 | $x^{6} + 13$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
13.6.5.2 | $x^{6} + 13$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
13.6.5.2 | $x^{6} + 13$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
13.6.5.2 | $x^{6} + 13$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |