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Properties

Label 24.0.605...625.1
Degree $24$
Signature $[0, 12]$
Discriminant $6.054\times 10^{42}$
Root discriminant \(60.62\)
Ramified primes $3,5,7,23$
Class number $960$ (GRH)
Class group [8, 120] (GRH)
Galois group $C_2^2\times D_6$ (as 24T30)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 6*x^23 + 78*x^22 - 392*x^21 + 2556*x^20 - 10386*x^19 + 44933*x^18 - 145902*x^17 + 460860*x^16 - 1187632*x^15 + 2848656*x^14 - 5775900*x^13 + 10594843*x^12 - 16507914*x^11 + 22656126*x^10 - 25834416*x^9 + 25004868*x^8 - 19088658*x^7 + 12519519*x^6 - 6912450*x^5 + 4180536*x^4 - 1559448*x^3 + 473328*x^2 + 44928*x + 251136)
 
gp: K = bnfinit(y^24 - 6*y^23 + 78*y^22 - 392*y^21 + 2556*y^20 - 10386*y^19 + 44933*y^18 - 145902*y^17 + 460860*y^16 - 1187632*y^15 + 2848656*y^14 - 5775900*y^13 + 10594843*y^12 - 16507914*y^11 + 22656126*y^10 - 25834416*y^9 + 25004868*y^8 - 19088658*y^7 + 12519519*y^6 - 6912450*y^5 + 4180536*y^4 - 1559448*y^3 + 473328*y^2 + 44928*y + 251136, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 6*x^23 + 78*x^22 - 392*x^21 + 2556*x^20 - 10386*x^19 + 44933*x^18 - 145902*x^17 + 460860*x^16 - 1187632*x^15 + 2848656*x^14 - 5775900*x^13 + 10594843*x^12 - 16507914*x^11 + 22656126*x^10 - 25834416*x^9 + 25004868*x^8 - 19088658*x^7 + 12519519*x^6 - 6912450*x^5 + 4180536*x^4 - 1559448*x^3 + 473328*x^2 + 44928*x + 251136);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - 6*x^23 + 78*x^22 - 392*x^21 + 2556*x^20 - 10386*x^19 + 44933*x^18 - 145902*x^17 + 460860*x^16 - 1187632*x^15 + 2848656*x^14 - 5775900*x^13 + 10594843*x^12 - 16507914*x^11 + 22656126*x^10 - 25834416*x^9 + 25004868*x^8 - 19088658*x^7 + 12519519*x^6 - 6912450*x^5 + 4180536*x^4 - 1559448*x^3 + 473328*x^2 + 44928*x + 251136)
 

\( x^{24} - 6 x^{23} + 78 x^{22} - 392 x^{21} + 2556 x^{20} - 10386 x^{19} + 44933 x^{18} - 145902 x^{17} + \cdots + 251136 \) 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:   \(6053887588336220981365314850586972900390625\) \(\medspace = 3^{28}\cdot 5^{12}\cdot 7^{12}\cdot 23^{8}\) 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:  \(60.62\)
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:  $3^{7/6}5^{1/2}7^{1/2}23^{1/2}\approx 102.22083024940257$
Ramified primes:   \(3\), \(5\), \(7\), \(23\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{2}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{3}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{4}$, $\frac{1}{8}a^{11}-\frac{1}{8}a^{10}-\frac{1}{8}a^{9}-\frac{1}{8}a^{5}+\frac{1}{8}a^{4}+\frac{1}{8}a^{3}$, $\frac{1}{8}a^{12}-\frac{1}{8}a^{9}-\frac{1}{8}a^{6}+\frac{1}{8}a^{3}$, $\frac{1}{8}a^{13}-\frac{1}{8}a^{10}-\frac{1}{8}a^{7}+\frac{1}{8}a^{4}$, $\frac{1}{8}a^{14}-\frac{1}{8}a^{10}-\frac{1}{8}a^{9}-\frac{1}{8}a^{8}+\frac{1}{8}a^{4}+\frac{1}{8}a^{3}$, $\frac{1}{32}a^{15}-\frac{1}{32}a^{14}+\frac{1}{32}a^{13}-\frac{1}{32}a^{12}+\frac{1}{32}a^{11}+\frac{3}{32}a^{10}+\frac{3}{32}a^{9}+\frac{1}{32}a^{8}+\frac{7}{32}a^{7}+\frac{1}{32}a^{6}-\frac{1}{32}a^{5}-\frac{3}{32}a^{4}-\frac{1}{8}a^{3}-\frac{1}{4}a$, $\frac{1}{224}a^{16}+\frac{3}{224}a^{15}+\frac{5}{224}a^{14}+\frac{1}{32}a^{13}-\frac{3}{224}a^{12}-\frac{5}{224}a^{11}-\frac{9}{224}a^{10}+\frac{25}{224}a^{9}+\frac{3}{224}a^{8}-\frac{23}{224}a^{7}+\frac{51}{224}a^{6}-\frac{11}{224}a^{5}-\frac{1}{28}a^{4}-\frac{27}{56}a^{3}-\frac{3}{28}a^{2}-\frac{3}{7}$, $\frac{1}{224}a^{17}+\frac{3}{224}a^{15}+\frac{13}{224}a^{14}+\frac{11}{224}a^{13}-\frac{3}{224}a^{12}+\frac{13}{224}a^{11}+\frac{17}{224}a^{10}-\frac{23}{224}a^{9}+\frac{3}{224}a^{8}+\frac{29}{224}a^{7}-\frac{45}{224}a^{6}+\frac{9}{112}a^{5}-\frac{7}{32}a^{4}-\frac{23}{56}a^{3}+\frac{1}{14}a^{2}-\frac{5}{28}a+\frac{2}{7}$, $\frac{1}{2016}a^{18}+\frac{1}{672}a^{17}+\frac{13}{2016}a^{15}-\frac{1}{96}a^{14}+\frac{31}{672}a^{13}-\frac{43}{2016}a^{12}-\frac{23}{672}a^{11}+\frac{37}{672}a^{10}+\frac{167}{2016}a^{9}+\frac{47}{672}a^{8}+\frac{3}{224}a^{7}+\frac{5}{1008}a^{6}+\frac{67}{336}a^{5}+\frac{59}{672}a^{4}+\frac{17}{168}a^{3}+\frac{29}{84}a^{2}-\frac{1}{4}a+\frac{5}{21}$, $\frac{1}{8064}a^{19}-\frac{1}{8064}a^{18}-\frac{1}{2688}a^{17}-\frac{1}{576}a^{16}-\frac{1}{126}a^{15}-\frac{13}{672}a^{14}-\frac{95}{4032}a^{13}+\frac{173}{4032}a^{12}+\frac{25}{448}a^{11}+\frac{1}{144}a^{10}-\frac{121}{1008}a^{9}+\frac{19}{672}a^{8}-\frac{149}{1152}a^{7}-\frac{601}{8064}a^{6}-\frac{199}{896}a^{5}-\frac{93}{448}a^{4}+\frac{19}{336}a^{3}-\frac{83}{336}a^{2}-\frac{71}{168}a+\frac{17}{42}$, $\frac{1}{8064}a^{20}-\frac{5}{8064}a^{17}-\frac{1}{1344}a^{16}+\frac{1}{168}a^{15}-\frac{5}{576}a^{14}+\frac{1}{336}a^{13}+\frac{17}{336}a^{12}-\frac{65}{4032}a^{11}-\frac{3}{224}a^{10}-\frac{1}{28}a^{9}-\frac{5}{1152}a^{8}+\frac{11}{48}a^{7}+\frac{55}{336}a^{6}+\frac{461}{2688}a^{5}+\frac{323}{1344}a^{4}+\frac{11}{56}a^{3}-\frac{13}{336}a^{2}-\frac{15}{56}a-\frac{3}{14}$, $\frac{1}{8064}a^{21}-\frac{1}{8064}a^{18}+\frac{1}{1344}a^{17}+\frac{1}{672}a^{16}-\frac{1}{64}a^{15}-\frac{5}{168}a^{14}-\frac{5}{84}a^{13}-\frac{97}{4032}a^{12}-\frac{17}{672}a^{11}-\frac{11}{168}a^{10}-\frac{89}{2688}a^{9}+\frac{1}{28}a^{8}-\frac{2}{21}a^{7}-\frac{59}{1152}a^{6}-\frac{5}{448}a^{5}-\frac{37}{672}a^{4}-\frac{51}{112}a^{3}-\frac{11}{168}a^{2}-\frac{13}{28}a-\frac{1}{3}$, $\frac{1}{54\!\cdots\!44}a^{22}+\frac{367926486773}{45\!\cdots\!12}a^{21}-\frac{93376220483}{13\!\cdots\!32}a^{20}-\frac{456491087369}{68\!\cdots\!68}a^{19}-\frac{723506801557}{78\!\cdots\!92}a^{18}-\frac{89405702356147}{18\!\cdots\!48}a^{17}+\frac{5615698282919}{76\!\cdots\!52}a^{16}+\frac{112247222358617}{19\!\cdots\!48}a^{15}+\frac{19\!\cdots\!15}{45\!\cdots\!12}a^{14}-\frac{52\!\cdots\!27}{13\!\cdots\!36}a^{13}+\frac{46\!\cdots\!29}{27\!\cdots\!72}a^{12}+\frac{108461764888951}{31\!\cdots\!56}a^{11}-\frac{53\!\cdots\!55}{60\!\cdots\!16}a^{10}-\frac{16\!\cdots\!57}{17\!\cdots\!92}a^{9}+\frac{46\!\cdots\!59}{91\!\cdots\!24}a^{8}-\frac{25\!\cdots\!67}{13\!\cdots\!36}a^{7}-\frac{13\!\cdots\!73}{78\!\cdots\!92}a^{6}+\frac{33\!\cdots\!87}{60\!\cdots\!16}a^{5}+\frac{83\!\cdots\!65}{91\!\cdots\!24}a^{4}+\frac{409855589828893}{32\!\cdots\!08}a^{3}-\frac{35\!\cdots\!85}{22\!\cdots\!56}a^{2}+\frac{273125154773995}{16\!\cdots\!04}a-\frac{8156885821801}{28\!\cdots\!82}$, $\frac{1}{34\!\cdots\!12}a^{23}+\frac{828924423976577}{11\!\cdots\!04}a^{22}-\frac{80\!\cdots\!61}{39\!\cdots\!76}a^{21}+\frac{38\!\cdots\!93}{34\!\cdots\!12}a^{20}+\frac{86\!\cdots\!85}{16\!\cdots\!72}a^{19}-\frac{68\!\cdots\!71}{11\!\cdots\!04}a^{18}-\frac{14\!\cdots\!81}{43\!\cdots\!64}a^{17}+\frac{12\!\cdots\!35}{57\!\cdots\!52}a^{16}+\frac{29\!\cdots\!33}{19\!\cdots\!52}a^{15}-\frac{10\!\cdots\!49}{17\!\cdots\!56}a^{14}+\frac{88\!\cdots\!13}{57\!\cdots\!52}a^{13}-\frac{23\!\cdots\!45}{57\!\cdots\!52}a^{12}-\frac{16\!\cdots\!27}{34\!\cdots\!12}a^{11}+\frac{10\!\cdots\!93}{11\!\cdots\!04}a^{10}-\frac{70\!\cdots\!33}{11\!\cdots\!04}a^{9}+\frac{25\!\cdots\!17}{38\!\cdots\!68}a^{8}-\frac{28\!\cdots\!09}{12\!\cdots\!56}a^{7}-\frac{28\!\cdots\!23}{11\!\cdots\!04}a^{6}+\frac{33\!\cdots\!71}{19\!\cdots\!88}a^{5}-\frac{61\!\cdots\!95}{59\!\cdots\!12}a^{4}-\frac{12\!\cdots\!99}{47\!\cdots\!96}a^{3}-\frac{38\!\cdots\!17}{79\!\cdots\!16}a^{2}+\frac{35\!\cdots\!15}{25\!\cdots\!42}a+\frac{24\!\cdots\!53}{21\!\cdots\!54}$ 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

$C_{8}\times C_{120}$, which has order $960$ (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:   \( -\frac{109546684378771357351}{2936090530114931686156928} a^{23} + \frac{146147822918134952513}{734022632528732921539232} a^{22} - \frac{3035912921656331671159}{1101033948793099382308848} a^{21} + \frac{9327306017823453150477}{734022632528732921539232} a^{20} - \frac{125511038746991869123613}{1468045265057465843078464} a^{19} + \frac{716175613585761816747259}{2202067897586198764617696} a^{18} - \frac{4166619843725754747508171}{2936090530114931686156928} a^{17} + \frac{6408676376614059192020757}{1468045265057465843078464} a^{16} - \frac{266145490185445163665697}{19487326527311493492192} a^{15} + \frac{24597326772212216113787163}{734022632528732921539232} a^{14} - \frac{7189382680292209475260845}{91752829066091615192404} a^{13} + \frac{332637112899573551252074163}{2202067897586198764617696} a^{12} - \frac{778757618842943051889998529}{2936090530114931686156928} a^{11} + \frac{285302360389072683370762793}{734022632528732921539232} a^{10} - \frac{1094748747896333307778373717}{2202067897586198764617696} a^{9} + \frac{190290121766949242775097811}{367011316264366460769616} a^{8} - \frac{651874954861666892661523755}{1468045265057465843078464} a^{7} + \frac{108117822456329168703082595}{367011316264366460769616} a^{6} - \frac{501140467856126957818617549}{2936090530114931686156928} a^{5} + \frac{149749266134609129514822423}{1468045265057465843078464} a^{4} - \frac{9440920371722628410591637}{183505658132183230384808} a^{3} + \frac{4975317794414202022876713}{367011316264366460769616} a^{2} + \frac{396569876248865367122355}{183505658132183230384808} a + \frac{360893802753954249260599}{45876414533045807596202} \)  (order $6$) 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:   $\frac{93\!\cdots\!51}{41\!\cdots\!32}a^{23}-\frac{16\!\cdots\!05}{13\!\cdots\!44}a^{22}+\frac{23\!\cdots\!61}{13\!\cdots\!44}a^{21}-\frac{31\!\cdots\!77}{41\!\cdots\!32}a^{20}+\frac{10\!\cdots\!63}{19\!\cdots\!92}a^{19}-\frac{29\!\cdots\!81}{15\!\cdots\!16}a^{18}+\frac{88\!\cdots\!23}{10\!\cdots\!08}a^{17}-\frac{17\!\cdots\!43}{68\!\cdots\!72}a^{16}+\frac{49\!\cdots\!15}{60\!\cdots\!44}a^{15}-\frac{58\!\cdots\!73}{29\!\cdots\!88}a^{14}+\frac{31\!\cdots\!35}{68\!\cdots\!72}a^{13}-\frac{61\!\cdots\!03}{68\!\cdots\!72}a^{12}+\frac{64\!\cdots\!83}{41\!\cdots\!32}a^{11}-\frac{30\!\cdots\!41}{13\!\cdots\!44}a^{10}+\frac{39\!\cdots\!45}{13\!\cdots\!44}a^{9}-\frac{57\!\cdots\!93}{19\!\cdots\!92}a^{8}+\frac{37\!\cdots\!57}{15\!\cdots\!16}a^{7}-\frac{23\!\cdots\!49}{15\!\cdots\!16}a^{6}+\frac{58\!\cdots\!39}{68\!\cdots\!72}a^{5}-\frac{28\!\cdots\!57}{57\!\cdots\!56}a^{4}+\frac{13\!\cdots\!99}{57\!\cdots\!56}a^{3}-\frac{29\!\cdots\!49}{95\!\cdots\!76}a^{2}-\frac{44\!\cdots\!83}{10\!\cdots\!76}a-\frac{11\!\cdots\!92}{29\!\cdots\!43}$, $\frac{78\!\cdots\!79}{18\!\cdots\!88}a^{23}-\frac{14\!\cdots\!21}{62\!\cdots\!96}a^{22}+\frac{19\!\cdots\!89}{62\!\cdots\!96}a^{21}-\frac{28\!\cdots\!77}{18\!\cdots\!88}a^{20}+\frac{89\!\cdots\!55}{88\!\cdots\!28}a^{19}-\frac{27\!\cdots\!09}{69\!\cdots\!44}a^{18}+\frac{19\!\cdots\!99}{11\!\cdots\!68}a^{17}-\frac{16\!\cdots\!75}{31\!\cdots\!48}a^{16}+\frac{45\!\cdots\!63}{27\!\cdots\!96}a^{15}-\frac{38\!\cdots\!27}{93\!\cdots\!44}a^{14}+\frac{29\!\cdots\!75}{31\!\cdots\!48}a^{13}-\frac{84\!\cdots\!17}{44\!\cdots\!64}a^{12}+\frac{61\!\cdots\!39}{18\!\cdots\!88}a^{11}-\frac{30\!\cdots\!17}{62\!\cdots\!96}a^{10}+\frac{39\!\cdots\!05}{62\!\cdots\!96}a^{9}-\frac{66\!\cdots\!89}{98\!\cdots\!92}a^{8}+\frac{39\!\cdots\!37}{69\!\cdots\!44}a^{7}-\frac{80\!\cdots\!79}{20\!\cdots\!32}a^{6}+\frac{67\!\cdots\!33}{31\!\cdots\!48}a^{5}-\frac{30\!\cdots\!17}{21\!\cdots\!92}a^{4}+\frac{17\!\cdots\!63}{25\!\cdots\!04}a^{3}-\frac{29\!\cdots\!57}{12\!\cdots\!52}a^{2}-\frac{26\!\cdots\!21}{40\!\cdots\!61}a-\frac{34\!\cdots\!09}{27\!\cdots\!74}$, $\frac{24\!\cdots\!81}{54\!\cdots\!24}a^{23}-\frac{10\!\cdots\!47}{38\!\cdots\!68}a^{22}+\frac{15\!\cdots\!97}{44\!\cdots\!64}a^{21}-\frac{68\!\cdots\!53}{38\!\cdots\!68}a^{20}+\frac{43\!\cdots\!75}{38\!\cdots\!68}a^{19}-\frac{41\!\cdots\!29}{89\!\cdots\!76}a^{18}+\frac{18\!\cdots\!45}{95\!\cdots\!92}a^{17}-\frac{12\!\cdots\!01}{19\!\cdots\!84}a^{16}+\frac{37\!\cdots\!77}{19\!\cdots\!84}a^{15}-\frac{97\!\cdots\!43}{19\!\cdots\!84}a^{14}+\frac{22\!\cdots\!81}{19\!\cdots\!84}a^{13}-\frac{45\!\cdots\!21}{19\!\cdots\!84}a^{12}+\frac{16\!\cdots\!27}{38\!\cdots\!68}a^{11}-\frac{24\!\cdots\!27}{38\!\cdots\!68}a^{10}+\frac{31\!\cdots\!27}{38\!\cdots\!68}a^{9}-\frac{16\!\cdots\!29}{18\!\cdots\!08}a^{8}+\frac{10\!\cdots\!13}{12\!\cdots\!56}a^{7}-\frac{21\!\cdots\!03}{38\!\cdots\!68}a^{6}+\frac{64\!\cdots\!31}{22\!\cdots\!32}a^{5}-\frac{30\!\cdots\!05}{15\!\cdots\!32}a^{4}+\frac{13\!\cdots\!49}{15\!\cdots\!32}a^{3}-\frac{35\!\cdots\!33}{79\!\cdots\!16}a^{2}-\frac{31\!\cdots\!85}{19\!\cdots\!04}a-\frac{40\!\cdots\!28}{24\!\cdots\!63}$, $\frac{77\!\cdots\!19}{53\!\cdots\!08}a^{23}-\frac{30\!\cdots\!21}{41\!\cdots\!68}a^{22}+\frac{10\!\cdots\!09}{95\!\cdots\!92}a^{21}-\frac{20\!\cdots\!07}{43\!\cdots\!64}a^{20}+\frac{46\!\cdots\!81}{14\!\cdots\!88}a^{19}-\frac{43\!\cdots\!13}{35\!\cdots\!72}a^{18}+\frac{45\!\cdots\!29}{86\!\cdots\!28}a^{17}-\frac{57\!\cdots\!07}{35\!\cdots\!72}a^{16}+\frac{72\!\cdots\!95}{14\!\cdots\!88}a^{15}-\frac{65\!\cdots\!81}{53\!\cdots\!08}a^{14}+\frac{20\!\cdots\!87}{71\!\cdots\!44}a^{13}-\frac{16\!\cdots\!87}{30\!\cdots\!12}a^{12}+\frac{41\!\cdots\!89}{43\!\cdots\!64}a^{11}-\frac{39\!\cdots\!19}{28\!\cdots\!76}a^{10}+\frac{51\!\cdots\!71}{28\!\cdots\!76}a^{9}-\frac{37\!\cdots\!41}{20\!\cdots\!84}a^{8}+\frac{75\!\cdots\!15}{47\!\cdots\!96}a^{7}-\frac{28\!\cdots\!11}{27\!\cdots\!04}a^{6}+\frac{18\!\cdots\!83}{28\!\cdots\!76}a^{5}-\frac{18\!\cdots\!85}{47\!\cdots\!96}a^{4}+\frac{12\!\cdots\!51}{59\!\cdots\!12}a^{3}-\frac{19\!\cdots\!31}{39\!\cdots\!08}a^{2}+\frac{21\!\cdots\!65}{85\!\cdots\!16}a-\frac{14\!\cdots\!87}{49\!\cdots\!26}$, $\frac{19\!\cdots\!39}{17\!\cdots\!56}a^{23}-\frac{11\!\cdots\!17}{19\!\cdots\!84}a^{22}+\frac{52\!\cdots\!75}{63\!\cdots\!28}a^{21}-\frac{94\!\cdots\!09}{24\!\cdots\!08}a^{20}+\frac{14\!\cdots\!11}{57\!\cdots\!52}a^{19}-\frac{56\!\cdots\!99}{57\!\cdots\!52}a^{18}+\frac{37\!\cdots\!11}{86\!\cdots\!28}a^{17}-\frac{54\!\cdots\!33}{41\!\cdots\!68}a^{16}+\frac{58\!\cdots\!79}{14\!\cdots\!92}a^{15}-\frac{88\!\cdots\!93}{86\!\cdots\!28}a^{14}+\frac{76\!\cdots\!33}{31\!\cdots\!64}a^{13}-\frac{14\!\cdots\!25}{31\!\cdots\!64}a^{12}+\frac{32\!\cdots\!37}{40\!\cdots\!92}a^{11}-\frac{68\!\cdots\!87}{57\!\cdots\!52}a^{10}+\frac{12\!\cdots\!05}{82\!\cdots\!36}a^{9}-\frac{91\!\cdots\!45}{57\!\cdots\!52}a^{8}+\frac{78\!\cdots\!23}{57\!\cdots\!52}a^{7}-\frac{56\!\cdots\!87}{63\!\cdots\!28}a^{6}+\frac{69\!\cdots\!27}{14\!\cdots\!88}a^{5}-\frac{13\!\cdots\!31}{47\!\cdots\!96}a^{4}+\frac{10\!\cdots\!39}{79\!\cdots\!16}a^{3}-\frac{10\!\cdots\!65}{29\!\cdots\!56}a^{2}-\frac{19\!\cdots\!49}{19\!\cdots\!04}a-\frac{10\!\cdots\!89}{49\!\cdots\!26}$, $\frac{28\!\cdots\!73}{17\!\cdots\!56}a^{23}-\frac{52\!\cdots\!81}{57\!\cdots\!52}a^{22}+\frac{71\!\cdots\!61}{57\!\cdots\!52}a^{21}-\frac{10\!\cdots\!25}{17\!\cdots\!56}a^{20}+\frac{32\!\cdots\!03}{82\!\cdots\!36}a^{19}-\frac{87\!\cdots\!93}{57\!\cdots\!52}a^{18}+\frac{56\!\cdots\!53}{86\!\cdots\!28}a^{17}-\frac{59\!\cdots\!69}{28\!\cdots\!76}a^{16}+\frac{61\!\cdots\!93}{95\!\cdots\!92}a^{15}-\frac{13\!\cdots\!59}{86\!\cdots\!28}a^{14}+\frac{35\!\cdots\!81}{95\!\cdots\!92}a^{13}-\frac{20\!\cdots\!39}{28\!\cdots\!76}a^{12}+\frac{21\!\cdots\!73}{17\!\cdots\!56}a^{11}-\frac{10\!\cdots\!65}{57\!\cdots\!52}a^{10}+\frac{46\!\cdots\!35}{19\!\cdots\!84}a^{9}-\frac{14\!\cdots\!67}{57\!\cdots\!52}a^{8}+\frac{12\!\cdots\!33}{57\!\cdots\!52}a^{7}-\frac{97\!\cdots\!83}{66\!\cdots\!96}a^{6}+\frac{11\!\cdots\!29}{14\!\cdots\!88}a^{5}-\frac{76\!\cdots\!09}{15\!\cdots\!32}a^{4}+\frac{53\!\cdots\!55}{23\!\cdots\!48}a^{3}-\frac{68\!\cdots\!97}{99\!\cdots\!52}a^{2}-\frac{12\!\cdots\!63}{59\!\cdots\!12}a-\frac{42\!\cdots\!63}{11\!\cdots\!82}$, $\frac{25\!\cdots\!55}{43\!\cdots\!64}a^{23}-\frac{60\!\cdots\!35}{20\!\cdots\!84}a^{22}+\frac{21\!\cdots\!61}{49\!\cdots\!72}a^{21}-\frac{79\!\cdots\!03}{43\!\cdots\!64}a^{20}+\frac{13\!\cdots\!75}{10\!\cdots\!92}a^{19}-\frac{23\!\cdots\!39}{49\!\cdots\!26}a^{18}+\frac{88\!\cdots\!79}{43\!\cdots\!64}a^{17}-\frac{43\!\cdots\!65}{71\!\cdots\!44}a^{16}+\frac{13\!\cdots\!65}{71\!\cdots\!44}a^{15}-\frac{97\!\cdots\!93}{21\!\cdots\!32}a^{14}+\frac{37\!\cdots\!65}{35\!\cdots\!72}a^{13}-\frac{69\!\cdots\!59}{35\!\cdots\!72}a^{12}+\frac{20\!\cdots\!67}{61\!\cdots\!52}a^{11}-\frac{66\!\cdots\!13}{14\!\cdots\!88}a^{10}+\frac{82\!\cdots\!73}{14\!\cdots\!88}a^{9}-\frac{26\!\cdots\!79}{47\!\cdots\!96}a^{8}+\frac{33\!\cdots\!89}{79\!\cdots\!16}a^{7}-\frac{38\!\cdots\!09}{17\!\cdots\!32}a^{6}+\frac{59\!\cdots\!63}{49\!\cdots\!72}a^{5}-\frac{24\!\cdots\!77}{39\!\cdots\!08}a^{4}+\frac{18\!\cdots\!23}{59\!\cdots\!12}a^{3}+\frac{73\!\cdots\!85}{59\!\cdots\!12}a^{2}-\frac{72\!\cdots\!63}{14\!\cdots\!78}a-\frac{34\!\cdots\!57}{42\!\cdots\!23}$, $\frac{22\!\cdots\!07}{34\!\cdots\!12}a^{23}-\frac{38\!\cdots\!33}{11\!\cdots\!04}a^{22}+\frac{54\!\cdots\!73}{11\!\cdots\!04}a^{21}-\frac{16\!\cdots\!15}{80\!\cdots\!84}a^{20}+\frac{18\!\cdots\!93}{12\!\cdots\!56}a^{19}-\frac{68\!\cdots\!89}{12\!\cdots\!56}a^{18}+\frac{10\!\cdots\!03}{43\!\cdots\!64}a^{17}-\frac{40\!\cdots\!67}{57\!\cdots\!52}a^{16}+\frac{43\!\cdots\!91}{19\!\cdots\!88}a^{15}-\frac{91\!\cdots\!91}{17\!\cdots\!56}a^{14}+\frac{10\!\cdots\!77}{82\!\cdots\!36}a^{13}-\frac{13\!\cdots\!99}{57\!\cdots\!52}a^{12}+\frac{19\!\cdots\!85}{49\!\cdots\!16}a^{11}-\frac{66\!\cdots\!81}{11\!\cdots\!04}a^{10}+\frac{83\!\cdots\!61}{11\!\cdots\!04}a^{9}-\frac{12\!\cdots\!65}{16\!\cdots\!72}a^{8}+\frac{10\!\cdots\!11}{16\!\cdots\!72}a^{7}-\frac{15\!\cdots\!91}{38\!\cdots\!68}a^{6}+\frac{15\!\cdots\!13}{57\!\cdots\!52}a^{5}-\frac{10\!\cdots\!09}{59\!\cdots\!12}a^{4}+\frac{13\!\cdots\!57}{15\!\cdots\!32}a^{3}-\frac{21\!\cdots\!61}{23\!\cdots\!48}a^{2}-\frac{19\!\cdots\!29}{14\!\cdots\!78}a-\frac{79\!\cdots\!29}{49\!\cdots\!26}$, $\frac{70\!\cdots\!25}{43\!\cdots\!64}a^{23}-\frac{15\!\cdots\!23}{17\!\cdots\!36}a^{22}+\frac{56\!\cdots\!11}{47\!\cdots\!64}a^{21}-\frac{67\!\cdots\!53}{12\!\cdots\!04}a^{20}+\frac{53\!\cdots\!61}{14\!\cdots\!88}a^{19}-\frac{13\!\cdots\!91}{95\!\cdots\!92}a^{18}+\frac{53\!\cdots\!33}{86\!\cdots\!28}a^{17}-\frac{27\!\cdots\!17}{14\!\cdots\!88}a^{16}+\frac{29\!\cdots\!11}{47\!\cdots\!96}a^{15}-\frac{64\!\cdots\!73}{43\!\cdots\!64}a^{14}+\frac{18\!\cdots\!84}{52\!\cdots\!69}a^{13}-\frac{98\!\cdots\!59}{14\!\cdots\!88}a^{12}+\frac{66\!\cdots\!73}{53\!\cdots\!08}a^{11}-\frac{66\!\cdots\!25}{35\!\cdots\!72}a^{10}+\frac{33\!\cdots\!67}{13\!\cdots\!56}a^{9}-\frac{76\!\cdots\!89}{28\!\cdots\!76}a^{8}+\frac{39\!\cdots\!15}{15\!\cdots\!32}a^{7}-\frac{52\!\cdots\!57}{28\!\cdots\!76}a^{6}+\frac{12\!\cdots\!47}{99\!\cdots\!44}a^{5}-\frac{49\!\cdots\!71}{68\!\cdots\!28}a^{4}+\frac{16\!\cdots\!07}{39\!\cdots\!08}a^{3}-\frac{17\!\cdots\!05}{11\!\cdots\!24}a^{2}+\frac{32\!\cdots\!67}{59\!\cdots\!12}a-\frac{18\!\cdots\!41}{14\!\cdots\!78}$, $\frac{90\!\cdots\!77}{24\!\cdots\!08}a^{23}-\frac{15\!\cdots\!93}{57\!\cdots\!52}a^{22}+\frac{17\!\cdots\!57}{57\!\cdots\!52}a^{21}-\frac{29\!\cdots\!19}{17\!\cdots\!56}a^{20}+\frac{58\!\cdots\!97}{57\!\cdots\!52}a^{19}-\frac{26\!\cdots\!61}{57\!\cdots\!52}a^{18}+\frac{15\!\cdots\!27}{86\!\cdots\!28}a^{17}-\frac{60\!\cdots\!55}{95\!\cdots\!92}a^{16}+\frac{14\!\cdots\!37}{74\!\cdots\!48}a^{15}-\frac{44\!\cdots\!93}{86\!\cdots\!28}a^{14}+\frac{11\!\cdots\!53}{95\!\cdots\!92}a^{13}-\frac{70\!\cdots\!39}{28\!\cdots\!76}a^{12}+\frac{10\!\cdots\!41}{24\!\cdots\!08}a^{11}-\frac{61\!\cdots\!95}{91\!\cdots\!04}a^{10}+\frac{16\!\cdots\!95}{19\!\cdots\!84}a^{9}-\frac{18\!\cdots\!71}{19\!\cdots\!84}a^{8}+\frac{16\!\cdots\!03}{19\!\cdots\!84}a^{7}-\frac{11\!\cdots\!15}{19\!\cdots\!84}a^{6}+\frac{45\!\cdots\!91}{14\!\cdots\!88}a^{5}-\frac{10\!\cdots\!91}{47\!\cdots\!96}a^{4}+\frac{10\!\cdots\!03}{11\!\cdots\!88}a^{3}-\frac{78\!\cdots\!25}{14\!\cdots\!78}a^{2}-\frac{19\!\cdots\!29}{85\!\cdots\!16}a-\frac{10\!\cdots\!31}{49\!\cdots\!26}$, $\frac{80\!\cdots\!51}{17\!\cdots\!56}a^{23}-\frac{20\!\cdots\!89}{82\!\cdots\!36}a^{22}+\frac{67\!\cdots\!19}{19\!\cdots\!88}a^{21}-\frac{62\!\cdots\!73}{40\!\cdots\!92}a^{20}+\frac{59\!\cdots\!43}{57\!\cdots\!52}a^{19}-\frac{75\!\cdots\!09}{19\!\cdots\!84}a^{18}+\frac{14\!\cdots\!63}{86\!\cdots\!28}a^{17}-\frac{14\!\cdots\!35}{28\!\cdots\!76}a^{16}+\frac{58\!\cdots\!99}{36\!\cdots\!36}a^{15}-\frac{33\!\cdots\!13}{86\!\cdots\!28}a^{14}+\frac{25\!\cdots\!85}{28\!\cdots\!76}a^{13}-\frac{48\!\cdots\!09}{28\!\cdots\!76}a^{12}+\frac{49\!\cdots\!87}{17\!\cdots\!56}a^{11}-\frac{23\!\cdots\!23}{57\!\cdots\!52}a^{10}+\frac{27\!\cdots\!71}{57\!\cdots\!52}a^{9}-\frac{26\!\cdots\!69}{57\!\cdots\!52}a^{8}+\frac{30\!\cdots\!49}{91\!\cdots\!04}a^{7}-\frac{42\!\cdots\!19}{27\!\cdots\!12}a^{6}+\frac{24\!\cdots\!53}{49\!\cdots\!72}a^{5}-\frac{14\!\cdots\!23}{47\!\cdots\!96}a^{4}+\frac{11\!\cdots\!53}{23\!\cdots\!48}a^{3}+\frac{20\!\cdots\!23}{14\!\cdots\!78}a^{2}-\frac{73\!\cdots\!51}{59\!\cdots\!12}a-\frac{45\!\cdots\!77}{49\!\cdots\!26}$ 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:  \( 24830472736.92889 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 24830472736.92889 \cdot 960}{6\cdot\sqrt{6053887588336220981365314850586972900390625}}\cr\approx \mathstrut & 6.11288484634663 \end{aligned}\] (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 + 78*x^22 - 392*x^21 + 2556*x^20 - 10386*x^19 + 44933*x^18 - 145902*x^17 + 460860*x^16 - 1187632*x^15 + 2848656*x^14 - 5775900*x^13 + 10594843*x^12 - 16507914*x^11 + 22656126*x^10 - 25834416*x^9 + 25004868*x^8 - 19088658*x^7 + 12519519*x^6 - 6912450*x^5 + 4180536*x^4 - 1559448*x^3 + 473328*x^2 + 44928*x + 251136)
 
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 + 78*x^22 - 392*x^21 + 2556*x^20 - 10386*x^19 + 44933*x^18 - 145902*x^17 + 460860*x^16 - 1187632*x^15 + 2848656*x^14 - 5775900*x^13 + 10594843*x^12 - 16507914*x^11 + 22656126*x^10 - 25834416*x^9 + 25004868*x^8 - 19088658*x^7 + 12519519*x^6 - 6912450*x^5 + 4180536*x^4 - 1559448*x^3 + 473328*x^2 + 44928*x + 251136, 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 + 78*x^22 - 392*x^21 + 2556*x^20 - 10386*x^19 + 44933*x^18 - 145902*x^17 + 460860*x^16 - 1187632*x^15 + 2848656*x^14 - 5775900*x^13 + 10594843*x^12 - 16507914*x^11 + 22656126*x^10 - 25834416*x^9 + 25004868*x^8 - 19088658*x^7 + 12519519*x^6 - 6912450*x^5 + 4180536*x^4 - 1559448*x^3 + 473328*x^2 + 44928*x + 251136);
 
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 + 78*x^22 - 392*x^21 + 2556*x^20 - 10386*x^19 + 44933*x^18 - 145902*x^17 + 460860*x^16 - 1187632*x^15 + 2848656*x^14 - 5775900*x^13 + 10594843*x^12 - 16507914*x^11 + 22656126*x^10 - 25834416*x^9 + 25004868*x^8 - 19088658*x^7 + 12519519*x^6 - 6912450*x^5 + 4180536*x^4 - 1559448*x^3 + 473328*x^2 + 44928*x + 251136);
 
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

$C_2^2\times D_6$ (as 24T30):

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 48
The 24 conjugacy class representatives for $C_2^2\times D_6$
Character table for $C_2^2\times D_6$ is not computed

Intermediate fields

\(\Q(\sqrt{21}) \), \(\Q(\sqrt{105}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-35}) \), 3.3.621.1, \(\Q(\sqrt{5}, \sqrt{-7})\), \(\Q(\sqrt{5}, \sqrt{21})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{-7})\), \(\Q(\sqrt{-7}, \sqrt{-15})\), \(\Q(\sqrt{-3}, \sqrt{-35})\), \(\Q(\sqrt{-15}, \sqrt{21})\), 6.0.1156923.1, 6.0.144615375.1, 6.6.396824589.1, 6.6.49603073625.1, 6.6.48205125.1, 6.0.132274863.5, 6.0.16534357875.4, 8.0.121550625.1, 12.0.273384990338574515625.1, 12.0.20913606686390625.1, 12.12.2460464913047170640625.1, 12.0.157469754435018921.1, 12.0.2460464913047170640625.1, 12.0.2460464913047170640625.2, 12.0.2460464913047170640625.3

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
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.2.0.1}{2} }^{12}$ R R R ${\href{/padicField/11.6.0.1}{6} }^{4}$ ${\href{/padicField/13.6.0.1}{6} }^{4}$ ${\href{/padicField/17.6.0.1}{6} }^{4}$ ${\href{/padicField/19.2.0.1}{2} }^{12}$ R ${\href{/padicField/29.2.0.1}{2} }^{12}$ ${\href{/padicField/31.6.0.1}{6} }^{4}$ ${\href{/padicField/37.2.0.1}{2} }^{12}$ ${\href{/padicField/41.2.0.1}{2} }^{12}$ ${\href{/padicField/43.2.0.1}{2} }^{12}$ ${\href{/padicField/47.2.0.1}{2} }^{12}$ ${\href{/padicField/53.6.0.1}{6} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{12}$

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
\(3\) Copy content Toggle raw display 3.12.14.6$x^{12} + 6 x^{8} + 15 x^{6} + 9 x^{4} + 18 x^{2} + 9$$6$$2$$14$$D_6$$[3/2]_{2}^{2}$
3.12.14.6$x^{12} + 6 x^{8} + 15 x^{6} + 9 x^{4} + 18 x^{2} + 9$$6$$2$$14$$D_6$$[3/2]_{2}^{2}$
\(5\) Copy content Toggle raw display 5.12.6.1$x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
5.12.6.1$x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
\(7\) Copy content Toggle raw display 7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(23\) Copy content Toggle raw display 23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$