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Properties

Label 24.0.378...056.1
Degree $24$
Signature $[0, 12]$
Discriminant $3.785\times 10^{33}$
Root discriminant \(25.07\)
Ramified primes $2,3,23,79$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_2^3\times S_4$ (as 24T400)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 6*x^23 + 18*x^22 - 32*x^21 + 30*x^20 + 4*x^19 - 62*x^18 + 110*x^17 - 118*x^16 + 44*x^15 + 206*x^14 - 696*x^13 + 1249*x^12 - 1392*x^11 + 824*x^10 + 352*x^9 - 1888*x^8 + 3520*x^7 - 3968*x^6 + 512*x^5 + 7680*x^4 - 16384*x^3 + 18432*x^2 - 12288*x + 4096)
 
gp: K = bnfinit(y^24 - 6*y^23 + 18*y^22 - 32*y^21 + 30*y^20 + 4*y^19 - 62*y^18 + 110*y^17 - 118*y^16 + 44*y^15 + 206*y^14 - 696*y^13 + 1249*y^12 - 1392*y^11 + 824*y^10 + 352*y^9 - 1888*y^8 + 3520*y^7 - 3968*y^6 + 512*y^5 + 7680*y^4 - 16384*y^3 + 18432*y^2 - 12288*y + 4096, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 6*x^23 + 18*x^22 - 32*x^21 + 30*x^20 + 4*x^19 - 62*x^18 + 110*x^17 - 118*x^16 + 44*x^15 + 206*x^14 - 696*x^13 + 1249*x^12 - 1392*x^11 + 824*x^10 + 352*x^9 - 1888*x^8 + 3520*x^7 - 3968*x^6 + 512*x^5 + 7680*x^4 - 16384*x^3 + 18432*x^2 - 12288*x + 4096);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - 6*x^23 + 18*x^22 - 32*x^21 + 30*x^20 + 4*x^19 - 62*x^18 + 110*x^17 - 118*x^16 + 44*x^15 + 206*x^14 - 696*x^13 + 1249*x^12 - 1392*x^11 + 824*x^10 + 352*x^9 - 1888*x^8 + 3520*x^7 - 3968*x^6 + 512*x^5 + 7680*x^4 - 16384*x^3 + 18432*x^2 - 12288*x + 4096)
 

\( x^{24} - 6 x^{23} + 18 x^{22} - 32 x^{21} + 30 x^{20} + 4 x^{19} - 62 x^{18} + 110 x^{17} - 118 x^{16} + 44 x^{15} + 206 x^{14} - 696 x^{13} + 1249 x^{12} - 1392 x^{11} + 824 x^{10} + \cdots + 4096 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

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:   \(3785323726785214561740247642669056\) \(\medspace = 2^{24}\cdot 3^{12}\cdot 23^{4}\cdot 79^{8}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(25.07\)
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))
 
Ramified primes:   \(2\), \(3\), \(23\), \(79\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $8$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is a CM field.
Reflex fields:  unavailable$^{2048}$

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{13}-\frac{1}{2}a$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{15}-\frac{1}{4}a^{13}-\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{2}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{3}{8}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{16}a^{16}-\frac{1}{8}a^{14}-\frac{1}{4}a^{13}-\frac{1}{8}a^{12}+\frac{1}{8}a^{10}+\frac{1}{8}a^{9}-\frac{1}{8}a^{8}+\frac{3}{8}a^{6}+\frac{1}{4}a^{5}-\frac{7}{16}a^{4}-\frac{1}{8}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{32}a^{17}-\frac{1}{16}a^{15}-\frac{1}{8}a^{14}-\frac{1}{16}a^{13}-\frac{1}{4}a^{12}-\frac{3}{16}a^{11}-\frac{3}{16}a^{10}-\frac{1}{16}a^{9}-\frac{1}{4}a^{8}+\frac{3}{16}a^{7}-\frac{1}{8}a^{6}-\frac{15}{32}a^{5}-\frac{5}{16}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{128}a^{18}-\frac{1}{64}a^{17}+\frac{1}{64}a^{16}-\frac{1}{16}a^{15}+\frac{7}{64}a^{14}-\frac{1}{32}a^{13}+\frac{1}{64}a^{12}+\frac{11}{64}a^{11}+\frac{9}{64}a^{10}+\frac{1}{32}a^{9}+\frac{7}{64}a^{8}-\frac{1}{4}a^{7}-\frac{15}{128}a^{6}-\frac{11}{32}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{256}a^{19}-\frac{1}{128}a^{17}-\frac{1}{64}a^{16}-\frac{1}{128}a^{15}+\frac{3}{32}a^{14}+\frac{29}{128}a^{13}+\frac{13}{128}a^{12}+\frac{31}{128}a^{11}-\frac{3}{32}a^{10}+\frac{11}{128}a^{9}+\frac{15}{64}a^{8}-\frac{15}{256}a^{7}+\frac{27}{128}a^{6}-\frac{3}{32}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{512}a^{20}-\frac{1}{256}a^{18}-\frac{1}{128}a^{17}-\frac{1}{256}a^{16}+\frac{3}{64}a^{15}+\frac{29}{256}a^{14}+\frac{13}{256}a^{13}+\frac{31}{256}a^{12}-\frac{3}{64}a^{11}-\frac{53}{256}a^{10}-\frac{17}{128}a^{9}+\frac{113}{512}a^{8}+\frac{27}{256}a^{7}+\frac{29}{64}a^{6}-\frac{1}{8}a^{5}-\frac{1}{8}a^{4}+\frac{1}{4}a^{3}+\frac{3}{8}a^{2}-\frac{1}{4}a$, $\frac{1}{1024}a^{21}-\frac{1}{512}a^{19}-\frac{1}{256}a^{18}-\frac{1}{512}a^{17}+\frac{3}{128}a^{16}+\frac{29}{512}a^{15}+\frac{13}{512}a^{14}-\frac{97}{512}a^{13}-\frac{3}{128}a^{12}-\frac{53}{512}a^{11}+\frac{47}{256}a^{10}-\frac{143}{1024}a^{9}+\frac{27}{512}a^{8}-\frac{3}{128}a^{7}-\frac{1}{16}a^{6}-\frac{1}{16}a^{5}+\frac{3}{8}a^{4}+\frac{7}{16}a^{3}+\frac{3}{8}a^{2}-\frac{1}{2}a$, $\frac{1}{698368}a^{22}-\frac{19}{87296}a^{21}-\frac{149}{349184}a^{20}+\frac{3}{15872}a^{19}-\frac{457}{349184}a^{18}+\frac{1303}{87296}a^{17}+\frac{9909}{349184}a^{16}-\frac{19427}{349184}a^{15}+\frac{41407}{349184}a^{14}-\frac{21185}{87296}a^{13}+\frac{41291}{349184}a^{12}+\frac{31567}{174592}a^{11}-\frac{129119}{698368}a^{10}+\frac{41255}{349184}a^{9}-\frac{8379}{43648}a^{8}+\frac{16749}{87296}a^{7}-\frac{2307}{43648}a^{6}+\frac{5309}{10912}a^{5}-\frac{3533}{10912}a^{4}+\frac{179}{496}a^{3}-\frac{149}{682}a^{2}+\frac{37}{1364}a-\frac{339}{682}$, $\frac{1}{101961728}a^{23}-\frac{19}{50980864}a^{22}-\frac{20407}{50980864}a^{21}-\frac{8663}{12745216}a^{20}+\frac{5703}{50980864}a^{19}+\frac{14749}{25490432}a^{18}-\frac{49279}{50980864}a^{17}-\frac{311089}{50980864}a^{16}-\frac{2436523}{50980864}a^{15}+\frac{2629503}{25490432}a^{14}-\frac{8141857}{50980864}a^{13}+\frac{279379}{12745216}a^{12}-\frac{23858367}{101961728}a^{11}+\frac{168327}{3186304}a^{10}+\frac{29709}{1158656}a^{9}+\frac{1579081}{12745216}a^{8}-\frac{73361}{796576}a^{7}-\frac{813111}{3186304}a^{6}-\frac{451997}{1593152}a^{5}+\frac{41535}{199144}a^{4}+\frac{4910}{24893}a^{3}+\frac{2309}{18104}a^{2}+\frac{8456}{24893}a-\frac{9093}{49786}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

$C_{4}$, which has order $4$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $11$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -\frac{290201}{50980864} a^{23} + \frac{163497}{4634624} a^{22} - \frac{2171221}{25490432} a^{21} + \frac{2715659}{25490432} a^{20} - \frac{571313}{25490432} a^{19} - \frac{3867623}{25490432} a^{18} + \frac{7251851}{25490432} a^{17} - \frac{4062931}{12745216} a^{16} + \frac{1472277}{6372608} a^{15} + \frac{4176357}{25490432} a^{14} - \frac{3346569}{2317312} a^{13} + \frac{7550321}{2317312} a^{12} - \frac{205942813}{50980864} a^{11} + \frac{128302637}{50980864} a^{10} + \frac{14494175}{25490432} a^{9} - \frac{23181691}{6372608} a^{8} + \frac{26950403}{3186304} a^{7} - \frac{3669117}{289664} a^{6} + \frac{5971125}{796576} a^{5} + \frac{12539553}{796576} a^{4} - \frac{17844963}{398288} a^{3} + \frac{5194591}{99572} a^{2} - \frac{789966}{24893} a + \frac{266757}{49786} \)  (order $12$) Copy content Toggle raw display
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:   $\frac{40555}{12745216}a^{23}-\frac{194089}{4634624}a^{22}+\frac{3607173}{25490432}a^{21}-\frac{6490327}{25490432}a^{20}+\frac{1331661}{6372608}a^{19}+\frac{3178139}{25490432}a^{18}-\frac{6992917}{12745216}a^{17}+\frac{20594913}{25490432}a^{16}-\frac{19040545}{25490432}a^{15}+\frac{7047529}{25490432}a^{14}+\frac{1854399}{1158656}a^{13}-\frac{13116367}{2317312}a^{12}+\frac{121459677}{12745216}a^{11}-\frac{475431371}{50980864}a^{10}+\frac{5046705}{1593152}a^{9}+\frac{15123755}{3186304}a^{8}-\frac{45993421}{3186304}a^{7}+\frac{61003}{2263}a^{6}-\frac{23637937}{796576}a^{5}-\frac{1282907}{796576}a^{4}+\frac{1822977}{24893}a^{3}-\frac{12850479}{99572}a^{2}+\frac{2920321}{24893}a-\frac{40477}{803}$, $\frac{1183995}{101961728}a^{23}-\frac{736389}{12745216}a^{22}+\frac{6502739}{50980864}a^{21}-\frac{3432733}{25490432}a^{20}-\frac{710583}{50980864}a^{19}+\frac{3380715}{12745216}a^{18}-\frac{21172733}{50980864}a^{17}+\frac{1827197}{4634624}a^{16}-\frac{12706615}{50980864}a^{15}-\frac{2756029}{6372608}a^{14}+\frac{120877685}{50980864}a^{13}-\frac{3969677}{822272}a^{12}+\frac{537366099}{101961728}a^{11}-\frac{118703131}{50980864}a^{10}-\frac{51054143}{25490432}a^{9}+\frac{38509689}{6372608}a^{8}-\frac{79204525}{6372608}a^{7}+\frac{27888699}{1593152}a^{6}-\frac{881355}{144832}a^{5}-\frac{24212777}{796576}a^{4}+\frac{26848109}{398288}a^{3}-\frac{6629661}{99572}a^{2}+\frac{2896475}{99572}a+\frac{49987}{24893}$, $\frac{1348585}{101961728}a^{23}-\frac{2446053}{50980864}a^{22}+\frac{328157}{4634624}a^{21}-\frac{60867}{6372608}a^{20}-\frac{7506013}{50980864}a^{19}+\frac{6005075}{25490432}a^{18}-\frac{8664083}{50980864}a^{17}+\frac{499859}{50980864}a^{16}+\frac{7208565}{50980864}a^{15}-\frac{1685371}{2317312}a^{14}+\frac{102714715}{50980864}a^{13}-\frac{7805121}{3186304}a^{12}+\frac{71380865}{101961728}a^{11}+\frac{67525839}{25490432}a^{10}-\frac{102588561}{25490432}a^{9}+\frac{29325991}{6372608}a^{8}-\frac{44216279}{6372608}a^{7}+\frac{237007}{51392}a^{6}+\frac{17589027}{1593152}a^{5}-\frac{1798383}{49786}a^{4}+\frac{14538257}{398288}a^{3}-\frac{104279}{24893}a^{2}-\frac{3165559}{99572}a+\frac{739364}{24893}$, $\frac{232585}{101961728}a^{23}-\frac{1292087}{50980864}a^{22}+\frac{4067353}{50980864}a^{21}-\frac{1702265}{12745216}a^{20}+\frac{4257423}{50980864}a^{19}+\frac{2735185}{25490432}a^{18}-\frac{1427637}{4634624}a^{17}+\frac{20088399}{50980864}a^{16}-\frac{17115083}{50980864}a^{15}+\frac{2180811}{25490432}a^{14}+\frac{53476295}{50980864}a^{13}-\frac{40795775}{12745216}a^{12}+\frac{45822587}{9269248}a^{11}-\frac{4787149}{1158656}a^{10}+\frac{8701519}{12745216}a^{9}+\frac{37999765}{12745216}a^{8}-\frac{25547041}{3186304}a^{7}+\frac{45418589}{3186304}a^{6}-\frac{22703305}{1593152}a^{5}-\frac{2618027}{398288}a^{4}+\frac{1089147}{24893}a^{3}-\frac{13044697}{199144}a^{2}+\frac{116945}{2263}a-\frac{851467}{49786}$, $\frac{19365}{4634624}a^{23}-\frac{76957}{2317312}a^{22}+\frac{117057}{1158656}a^{21}-\frac{24043}{144832}a^{20}+\frac{263637}{2317312}a^{19}+\frac{124817}{1158656}a^{18}-\frac{879949}{2317312}a^{17}+\frac{1196407}{2317312}a^{16}-\frac{1081097}{2317312}a^{15}+\frac{47953}{579328}a^{14}+\frac{3015721}{2317312}a^{13}-\frac{1139815}{289664}a^{12}+\frac{28950785}{4634624}a^{11}-\frac{6472553}{1158656}a^{10}+\frac{3734453}{2317312}a^{9}+\frac{4267497}{1158656}a^{8}-\frac{5878797}{579328}a^{7}+\frac{40654}{2263}a^{6}-\frac{1286975}{72416}a^{5}-\frac{56615}{9052}a^{4}+\frac{1852855}{36208}a^{3}-\frac{1509913}{18104}a^{2}+\frac{640699}{9052}a-\frac{2094}{73}$, $\frac{280105}{50980864}a^{23}-\frac{26529}{1158656}a^{22}+\frac{39707}{796576}a^{21}-\frac{607095}{12745216}a^{20}-\frac{391463}{25490432}a^{19}+\frac{617075}{6372608}a^{18}-\frac{4016549}{25490432}a^{17}+\frac{3647009}{25490432}a^{16}-\frac{2370099}{25490432}a^{15}-\frac{2648779}{12745216}a^{14}+\frac{2260955}{2317312}a^{13}-\frac{2069073}{1158656}a^{12}+\frac{97468229}{50980864}a^{11}-\frac{17545159}{25490432}a^{10}-\frac{17302447}{25490432}a^{9}+\frac{31680903}{12745216}a^{8}-\frac{31497419}{6372608}a^{7}+\frac{914577}{144832}a^{6}-\frac{1135293}{796576}a^{5}-\frac{5173177}{398288}a^{4}+\frac{9610619}{398288}a^{3}-\frac{4769203}{199144}a^{2}+\frac{1007673}{99572}a+\frac{1596}{803}$, $\frac{27641}{822272}a^{23}-\frac{141289}{822272}a^{22}+\frac{32183}{74752}a^{21}-\frac{117829}{205568}a^{20}+\frac{46019}{205568}a^{19}+\frac{266221}{411136}a^{18}-\frac{310417}{205568}a^{17}+\frac{371527}{205568}a^{16}-\frac{596139}{411136}a^{15}-\frac{13235}{18688}a^{14}+\frac{1416911}{205568}a^{13}-\frac{6795015}{411136}a^{12}+\frac{18355527}{822272}a^{11}-\frac{13583083}{822272}a^{10}+\frac{793049}{822272}a^{9}+\frac{958475}{51392}a^{8}-\frac{8910423}{205568}a^{7}+\frac{6918189}{102784}a^{6}-\frac{1196509}{25696}a^{5}-\frac{202471}{3212}a^{4}+\frac{2828743}{12848}a^{3}-\frac{235871}{803}a^{2}+\frac{683527}{3212}a-\frac{113777}{1606}$, $\frac{2378969}{101961728}a^{23}-\frac{5732669}{50980864}a^{22}+\frac{13710447}{50980864}a^{21}-\frac{2280053}{6372608}a^{20}+\frac{7182291}{50980864}a^{19}+\frac{10003179}{25490432}a^{18}-\frac{47868835}{50980864}a^{17}+\frac{60130451}{50980864}a^{16}-\frac{4310625}{4634624}a^{15}-\frac{12614801}{25490432}a^{14}+\frac{225838571}{50980864}a^{13}-\frac{8253467}{796576}a^{12}+\frac{1433502129}{101961728}a^{11}-\frac{274917653}{25490432}a^{10}+\frac{29386871}{25490432}a^{9}+\frac{74729483}{6372608}a^{8}-\frac{179139903}{6372608}a^{7}+\frac{67002473}{1593152}a^{6}-\frac{45806791}{1593152}a^{5}-\frac{15325435}{398288}a^{4}+\frac{54302393}{398288}a^{3}-\frac{18347093}{99572}a^{2}+\frac{14071291}{99572}a-\frac{1343283}{24893}$, $\frac{378493}{50980864}a^{23}-\frac{708155}{50980864}a^{22}-\frac{390581}{25490432}a^{21}+\frac{2653055}{25490432}a^{20}-\frac{4455371}{25490432}a^{19}+\frac{2123571}{25490432}a^{18}+\frac{3615957}{25490432}a^{17}-\frac{4073257}{12745216}a^{16}+\frac{209265}{579328}a^{15}-\frac{14886389}{25490432}a^{14}+\frac{14822391}{25490432}a^{13}+\frac{22364553}{25490432}a^{12}-\frac{186572791}{50980864}a^{11}+\frac{269448027}{50980864}a^{10}-\frac{83685341}{25490432}a^{9}+\frac{8982729}{12745216}a^{8}+\frac{1207689}{796576}a^{7}-\frac{12793463}{1593152}a^{6}+\frac{15992247}{796576}a^{5}-\frac{16437583}{796576}a^{4}-\frac{4159901}{398288}a^{3}+\frac{10579645}{199144}a^{2}-\frac{1640168}{24893}a+\frac{774657}{24893}$, $\frac{106389}{25490432}a^{23}-\frac{14793}{6372608}a^{22}-\frac{708369}{25490432}a^{21}+\frac{524097}{6372608}a^{20}-\frac{762797}{6372608}a^{19}+\frac{182803}{6372608}a^{18}+\frac{9715}{72416}a^{17}-\frac{3131175}{12745216}a^{16}+\frac{1047901}{3186304}a^{15}-\frac{134061}{411136}a^{14}+\frac{864099}{6372608}a^{13}+\frac{7679099}{6372608}a^{12}-\frac{6985247}{2317312}a^{11}+\frac{4462553}{1158656}a^{10}-\frac{62497527}{25490432}a^{9}-\frac{1997381}{12745216}a^{8}+\frac{1768435}{796576}a^{7}-\frac{6795073}{796576}a^{6}+\frac{667031}{49786}a^{5}-\frac{4257589}{398288}a^{4}-\frac{5283199}{398288}a^{3}+\frac{8720531}{199144}a^{2}-\frac{103384}{2263}a+\frac{741428}{24893}$, $\frac{70913}{6372608}a^{23}-\frac{55139}{579328}a^{22}+\frac{971233}{3186304}a^{21}-\frac{6501467}{12745216}a^{20}+\frac{284147}{796576}a^{19}+\frac{2194959}{6372608}a^{18}-\frac{938445}{796576}a^{17}+\frac{9912043}{6372608}a^{16}-\frac{2186161}{1593152}a^{15}+\frac{1935499}{6372608}a^{14}+\frac{2210717}{579328}a^{13}-\frac{6984273}{579328}a^{12}+\frac{122442217}{6372608}a^{11}-\frac{27049067}{1593152}a^{10}+\frac{828093}{199144}a^{9}+\frac{144850641}{12745216}a^{8}-\frac{192034719}{6372608}a^{7}+\frac{15801233}{289664}a^{6}-\frac{22248047}{398288}a^{5}-\frac{452720}{24893}a^{4}+\frac{31889207}{199144}a^{3}-\frac{51292273}{199144}a^{2}+\frac{21227917}{99572}a-\frac{125519}{1606}$ Copy content Toggle raw display (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:  \( 42401055.974884115 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 42401055.974884115 \cdot 4}{12\cdot\sqrt{3785323726785214561740247642669056}}\cr\approx \mathstrut & 0.869685691062020 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^24 - 6*x^23 + 18*x^22 - 32*x^21 + 30*x^20 + 4*x^19 - 62*x^18 + 110*x^17 - 118*x^16 + 44*x^15 + 206*x^14 - 696*x^13 + 1249*x^12 - 1392*x^11 + 824*x^10 + 352*x^9 - 1888*x^8 + 3520*x^7 - 3968*x^6 + 512*x^5 + 7680*x^4 - 16384*x^3 + 18432*x^2 - 12288*x + 4096)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^24 - 6*x^23 + 18*x^22 - 32*x^21 + 30*x^20 + 4*x^19 - 62*x^18 + 110*x^17 - 118*x^16 + 44*x^15 + 206*x^14 - 696*x^13 + 1249*x^12 - 1392*x^11 + 824*x^10 + 352*x^9 - 1888*x^8 + 3520*x^7 - 3968*x^6 + 512*x^5 + 7680*x^4 - 16384*x^3 + 18432*x^2 - 12288*x + 4096, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^24 - 6*x^23 + 18*x^22 - 32*x^21 + 30*x^20 + 4*x^19 - 62*x^18 + 110*x^17 - 118*x^16 + 44*x^15 + 206*x^14 - 696*x^13 + 1249*x^12 - 1392*x^11 + 824*x^10 + 352*x^9 - 1888*x^8 + 3520*x^7 - 3968*x^6 + 512*x^5 + 7680*x^4 - 16384*x^3 + 18432*x^2 - 12288*x + 4096);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - 6*x^23 + 18*x^22 - 32*x^21 + 30*x^20 + 4*x^19 - 62*x^18 + 110*x^17 - 118*x^16 + 44*x^15 + 206*x^14 - 696*x^13 + 1249*x^12 - 1392*x^11 + 824*x^10 + 352*x^9 - 1888*x^8 + 3520*x^7 - 3968*x^6 + 512*x^5 + 7680*x^4 - 16384*x^3 + 18432*x^2 - 12288*x + 4096);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$C_2^3\times S_4$ (as 24T400):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 192
The 40 conjugacy class representatives for $C_2^3\times S_4$
Character table for $C_2^3\times S_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), 3.3.316.1, \(\Q(\zeta_{12})\), 6.0.2296688.1, 6.0.399424.1, 6.0.2696112.1, 6.0.248042304.1, 6.6.9186752.1, 6.6.62010576.1, 6.6.10784448.1, 12.12.61524984573628416.1, 12.0.84396412309504.1, 12.0.61524984573628416.1, 12.0.3845311535851776.1, 12.0.61524984573628416.3, 12.0.116304318664704.2, 12.0.61524984573628416.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Sibling fields

Degree 24 siblings: data not computed
Degree 32 siblings: data not computed

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 ${\href{/padicField/5.6.0.1}{6} }^{4}$ ${\href{/padicField/7.6.0.1}{6} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ ${\href{/padicField/13.3.0.1}{3} }^{8}$ ${\href{/padicField/17.4.0.1}{4} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$ ${\href{/padicField/19.2.0.1}{2} }^{12}$ R ${\href{/padicField/29.4.0.1}{4} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$ ${\href{/padicField/31.2.0.1}{2} }^{12}$ ${\href{/padicField/37.2.0.1}{2} }^{12}$ ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$ ${\href{/padicField/43.2.0.1}{2} }^{12}$ ${\href{/padicField/47.6.0.1}{6} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ ${\href{/padicField/59.6.0.1}{6} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.4.4.1$x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$$2$$2$$4$$C_2^2$$[2]^{2}$
\(3\) Copy content Toggle raw display 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}$
\(23\) Copy content Toggle raw display 23.4.0.1$x^{4} + 3 x^{2} + 19 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.2.1$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.0.1$x^{4} + 3 x^{2} + 19 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.0.1$x^{4} + 3 x^{2} + 19 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.2.1$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.0.1$x^{4} + 3 x^{2} + 19 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
\(79\) Copy content Toggle raw display 79.2.0.1$x^{2} + 78 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
79.2.0.1$x^{2} + 78 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
79.2.0.1$x^{2} + 78 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
79.2.0.1$x^{2} + 78 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
79.4.2.1$x^{4} + 156 x^{3} + 6248 x^{2} + 12792 x + 486412$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
79.4.2.1$x^{4} + 156 x^{3} + 6248 x^{2} + 12792 x + 486412$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
79.4.2.1$x^{4} + 156 x^{3} + 6248 x^{2} + 12792 x + 486412$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
79.4.2.1$x^{4} + 156 x^{3} + 6248 x^{2} + 12792 x + 486412$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$