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Properties

Label 24.0.187...816.1
Degree $24$
Signature $[0, 12]$
Discriminant $1.872\times 10^{38}$
Root discriminant \(39.33\)
Ramified primes $2,3,7,79$
Class number $55$ (GRH)
Class group [55] (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 - 2*x^23 - 11*x^22 + 2*x^21 + 122*x^20 - 22*x^19 - 603*x^18 + 126*x^17 + 2838*x^16 - 4554*x^15 - 11143*x^14 + 38030*x^13 - 5743*x^12 - 74196*x^11 + 192820*x^10 + 2152*x^9 - 196612*x^8 + 432432*x^7 - 65552*x^6 - 178400*x^5 + 381072*x^4 - 166208*x^3 + 51840*x^2 - 8704*x + 1024)
 
gp: K = bnfinit(y^24 - 2*y^23 - 11*y^22 + 2*y^21 + 122*y^20 - 22*y^19 - 603*y^18 + 126*y^17 + 2838*y^16 - 4554*y^15 - 11143*y^14 + 38030*y^13 - 5743*y^12 - 74196*y^11 + 192820*y^10 + 2152*y^9 - 196612*y^8 + 432432*y^7 - 65552*y^6 - 178400*y^5 + 381072*y^4 - 166208*y^3 + 51840*y^2 - 8704*y + 1024, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 2*x^23 - 11*x^22 + 2*x^21 + 122*x^20 - 22*x^19 - 603*x^18 + 126*x^17 + 2838*x^16 - 4554*x^15 - 11143*x^14 + 38030*x^13 - 5743*x^12 - 74196*x^11 + 192820*x^10 + 2152*x^9 - 196612*x^8 + 432432*x^7 - 65552*x^6 - 178400*x^5 + 381072*x^4 - 166208*x^3 + 51840*x^2 - 8704*x + 1024);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - 2*x^23 - 11*x^22 + 2*x^21 + 122*x^20 - 22*x^19 - 603*x^18 + 126*x^17 + 2838*x^16 - 4554*x^15 - 11143*x^14 + 38030*x^13 - 5743*x^12 - 74196*x^11 + 192820*x^10 + 2152*x^9 - 196612*x^8 + 432432*x^7 - 65552*x^6 - 178400*x^5 + 381072*x^4 - 166208*x^3 + 51840*x^2 - 8704*x + 1024)
 

\( x^{24} - 2 x^{23} - 11 x^{22} + 2 x^{21} + 122 x^{20} - 22 x^{19} - 603 x^{18} + 126 x^{17} + \cdots + 1024 \) 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:   \(187226864009183111800823017295698722816\) \(\medspace = 2^{24}\cdot 3^{12}\cdot 7^{12}\cdot 79^{8}\) 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:  \(39.33\)
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:  $2\cdot 3^{1/2}7^{1/2}79^{1/2}\approx 81.4616474176652$
Ramified primes:   \(2\), \(3\), \(7\), \(79\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{7}+\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{8}+\frac{1}{4}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{13}-\frac{1}{8}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{6}-\frac{3}{8}a^{5}+\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{16}a^{14}-\frac{1}{16}a^{10}+\frac{1}{8}a^{9}-\frac{1}{4}a^{8}+\frac{1}{8}a^{7}+\frac{1}{16}a^{6}-\frac{1}{8}a^{5}-\frac{1}{8}a^{4}+\frac{1}{4}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{16}a^{15}-\frac{1}{16}a^{11}-\frac{1}{8}a^{10}+\frac{1}{8}a^{8}-\frac{3}{16}a^{7}-\frac{1}{8}a^{6}+\frac{1}{8}a^{5}-\frac{1}{4}a^{3}$, $\frac{1}{16}a^{16}-\frac{1}{16}a^{12}-\frac{1}{8}a^{11}+\frac{1}{8}a^{9}-\frac{3}{16}a^{8}-\frac{1}{8}a^{7}+\frac{1}{8}a^{6}-\frac{1}{4}a^{4}$, $\frac{1}{32}a^{17}-\frac{1}{32}a^{16}-\frac{1}{32}a^{15}-\frac{1}{32}a^{13}+\frac{3}{32}a^{12}+\frac{3}{32}a^{11}-\frac{1}{8}a^{10}+\frac{3}{32}a^{9}+\frac{3}{32}a^{8}-\frac{1}{32}a^{7}-\frac{7}{16}a^{5}-\frac{1}{2}a^{4}+\frac{1}{8}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{32}a^{18}-\frac{1}{32}a^{15}-\frac{1}{32}a^{14}-\frac{1}{16}a^{13}-\frac{1}{8}a^{12}+\frac{3}{32}a^{11}-\frac{1}{32}a^{10}+\frac{3}{16}a^{9}+\frac{1}{8}a^{8}+\frac{3}{32}a^{7}+\frac{3}{16}a^{6}+\frac{7}{16}a^{5}-\frac{3}{8}a^{4}+\frac{3}{8}a^{3}$, $\frac{1}{32}a^{19}-\frac{1}{32}a^{16}-\frac{1}{32}a^{15}+\frac{3}{32}a^{12}-\frac{1}{32}a^{11}-\frac{1}{8}a^{10}-\frac{1}{8}a^{9}+\frac{3}{32}a^{8}+\frac{1}{16}a^{7}-\frac{1}{4}a^{6}+\frac{3}{8}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{32}a^{20}-\frac{1}{32}a^{15}-\frac{1}{16}a^{13}+\frac{3}{32}a^{11}-\frac{1}{16}a^{9}-\frac{1}{32}a^{8}+\frac{3}{32}a^{7}+\frac{7}{16}a^{5}+\frac{1}{8}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{1088}a^{21}-\frac{3}{544}a^{20}+\frac{11}{1088}a^{19}-\frac{1}{68}a^{18}+\frac{3}{272}a^{17}+\frac{7}{544}a^{16}+\frac{11}{1088}a^{15}-\frac{1}{68}a^{14}-\frac{13}{272}a^{13}+\frac{45}{544}a^{12}+\frac{39}{1088}a^{11}-\frac{2}{17}a^{10}+\frac{167}{1088}a^{9}+\frac{57}{272}a^{8}-\frac{61}{272}a^{7}-\frac{65}{272}a^{6}-\frac{21}{68}a^{5}+\frac{1}{34}a^{4}+\frac{21}{136}a^{3}+\frac{1}{34}a^{2}+\frac{4}{17}a-\frac{6}{17}$, $\frac{1}{5127498112}a^{22}-\frac{421631}{2563749056}a^{21}+\frac{11968925}{5127498112}a^{20}-\frac{303505}{233068096}a^{19}+\frac{38834489}{2563749056}a^{18}+\frac{1323553}{233068096}a^{17}-\frac{158517443}{5127498112}a^{16}-\frac{3660671}{2563749056}a^{15}+\frac{67827311}{2563749056}a^{14}-\frac{73211515}{2563749056}a^{13}+\frac{77729777}{5127498112}a^{12}-\frac{14908215}{150808768}a^{11}+\frac{352370857}{5127498112}a^{10}-\frac{140548761}{1281874528}a^{9}+\frac{154898877}{1281874528}a^{8}+\frac{95165101}{1281874528}a^{7}-\frac{288668769}{1281874528}a^{6}-\frac{10172603}{640937264}a^{5}-\frac{159250449}{320468632}a^{4}+\frac{76259357}{320468632}a^{3}+\frac{79108707}{320468632}a^{2}+\frac{16276042}{40058579}a+\frac{932550}{40058579}$, $\frac{1}{14\!\cdots\!52}a^{23}-\frac{56\!\cdots\!25}{74\!\cdots\!76}a^{22}+\frac{29\!\cdots\!11}{13\!\cdots\!32}a^{21}+\frac{12\!\cdots\!25}{74\!\cdots\!76}a^{20}-\frac{22\!\cdots\!87}{74\!\cdots\!76}a^{19}-\frac{12\!\cdots\!23}{74\!\cdots\!76}a^{18}-\frac{31\!\cdots\!63}{14\!\cdots\!52}a^{17}+\frac{11\!\cdots\!35}{74\!\cdots\!76}a^{16}-\frac{48\!\cdots\!93}{74\!\cdots\!76}a^{15}+\frac{63\!\cdots\!03}{74\!\cdots\!76}a^{14}+\frac{50\!\cdots\!25}{14\!\cdots\!52}a^{13}+\frac{25\!\cdots\!11}{74\!\cdots\!76}a^{12}+\frac{11\!\cdots\!49}{14\!\cdots\!52}a^{11}-\frac{13\!\cdots\!87}{37\!\cdots\!88}a^{10}+\frac{31\!\cdots\!85}{37\!\cdots\!88}a^{9}+\frac{11\!\cdots\!25}{93\!\cdots\!72}a^{8}-\frac{73\!\cdots\!27}{21\!\cdots\!64}a^{7}-\frac{19\!\cdots\!59}{93\!\cdots\!72}a^{6}-\frac{45\!\cdots\!01}{23\!\cdots\!68}a^{5}-\frac{97\!\cdots\!47}{46\!\cdots\!36}a^{4}-\frac{31\!\cdots\!05}{93\!\cdots\!72}a^{3}-\frac{44\!\cdots\!03}{23\!\cdots\!68}a^{2}+\frac{36\!\cdots\!11}{11\!\cdots\!34}a+\frac{17\!\cdots\!29}{58\!\cdots\!17}$ 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

$C_{55}$, which has order $55$ (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{5942368460445318593434215596273378313}{8779154017051044799770197895614529945856} a^{23} - \frac{712872270722320274150239964221843241}{548697126065690299985637368475908121616} a^{22} - \frac{66846759032861029925235359867694696575}{8779154017051044799770197895614529945856} a^{21} + \frac{1864570321249342340976563266105984707}{2194788504262761199942549473903632486464} a^{20} + \frac{366105859790365437979497434272206199683}{4389577008525522399885098947807264972928} a^{19} - \frac{35370597086185831009060254730910121361}{4389577008525522399885098947807264972928} a^{18} - \frac{3660529549126225573580433172951336991359}{8779154017051044799770197895614529945856} a^{17} + \frac{5003455554277614262461299207056047019}{99763113830125509088297703359256022112} a^{16} + \frac{8629515316737360594579663155677548454645}{4389577008525522399885098947807264972928} a^{15} - \frac{1162407953897283793093622491146008922573}{399052455320502036353190813437024088448} a^{14} - \frac{69983955525190908540626185874802182126547}{8779154017051044799770197895614529945856} a^{13} + \frac{6941360988589829824739221297055100654067}{274348563032845149992818684237954060808} a^{12} - \frac{8868227843386634601136707121605355421571}{8779154017051044799770197895614529945856} a^{11} - \frac{230042578148374156716980664325379041350145}{4389577008525522399885098947807264972928} a^{10} + \frac{275468401415819628053890120311723149877161}{2194788504262761199942549473903632486464} a^{9} + \frac{8951592931738728858395741497566672384889}{548697126065690299985637368475908121616} a^{8} - \frac{18150382950185255685332081287313749620289}{129105206133103599996620557288448969792} a^{7} + \frac{303381738167443554665928686292674931447429}{1097394252131380599971274736951816243232} a^{6} - \frac{5568703163483055691571625595990450918241}{548697126065690299985637368475908121616} a^{5} - \frac{427812144644226536937319847511142560586}{3117597307191422159009303229976750691} a^{4} + \frac{132781111548242962937514185938714448762365}{548697126065690299985637368475908121616} a^{3} - \frac{22378283789218835008826861188668054465713}{274348563032845149992818684237954060808} a^{2} + \frac{572916308596124854978500491777004644602}{34293570379105643749102335529744257601} a - \frac{60035977083660855040798992399247968925}{34293570379105643749102335529744257601} \)  (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:   $\frac{27\!\cdots\!71}{99\!\cdots\!12}a^{23}-\frac{17\!\cdots\!67}{43\!\cdots\!28}a^{22}-\frac{74\!\cdots\!29}{21\!\cdots\!64}a^{21}-\frac{54\!\cdots\!83}{43\!\cdots\!28}a^{20}+\frac{68\!\cdots\!13}{19\!\cdots\!24}a^{19}+\frac{30\!\cdots\!49}{21\!\cdots\!64}a^{18}-\frac{34\!\cdots\!71}{19\!\cdots\!24}a^{17}-\frac{29\!\cdots\!95}{43\!\cdots\!28}a^{16}+\frac{17\!\cdots\!73}{21\!\cdots\!64}a^{15}-\frac{17\!\cdots\!01}{21\!\cdots\!64}a^{14}-\frac{83\!\cdots\!11}{21\!\cdots\!64}a^{13}+\frac{37\!\cdots\!85}{43\!\cdots\!28}a^{12}+\frac{10\!\cdots\!63}{21\!\cdots\!64}a^{11}-\frac{91\!\cdots\!55}{43\!\cdots\!28}a^{10}+\frac{21\!\cdots\!65}{54\!\cdots\!16}a^{9}+\frac{33\!\cdots\!99}{10\!\cdots\!32}a^{8}-\frac{16\!\cdots\!83}{32\!\cdots\!48}a^{7}+\frac{89\!\cdots\!47}{10\!\cdots\!32}a^{6}+\frac{12\!\cdots\!79}{27\!\cdots\!08}a^{5}-\frac{14\!\cdots\!69}{27\!\cdots\!08}a^{4}+\frac{23\!\cdots\!91}{34\!\cdots\!01}a^{3}+\frac{28\!\cdots\!89}{27\!\cdots\!08}a^{2}-\frac{47\!\cdots\!49}{68\!\cdots\!02}a+\frac{75\!\cdots\!33}{34\!\cdots\!01}$, $\frac{86\!\cdots\!89}{74\!\cdots\!76}a^{23}-\frac{26\!\cdots\!47}{74\!\cdots\!76}a^{22}-\frac{78\!\cdots\!11}{74\!\cdots\!76}a^{21}+\frac{12\!\cdots\!57}{74\!\cdots\!76}a^{20}+\frac{52\!\cdots\!29}{37\!\cdots\!88}a^{19}-\frac{33\!\cdots\!11}{18\!\cdots\!44}a^{18}-\frac{51\!\cdots\!85}{74\!\cdots\!76}a^{17}+\frac{62\!\cdots\!55}{67\!\cdots\!16}a^{16}+\frac{12\!\cdots\!87}{37\!\cdots\!88}a^{15}-\frac{15\!\cdots\!65}{16\!\cdots\!04}a^{14}-\frac{60\!\cdots\!65}{74\!\cdots\!76}a^{13}+\frac{44\!\cdots\!69}{74\!\cdots\!76}a^{12}-\frac{38\!\cdots\!91}{74\!\cdots\!76}a^{11}-\frac{66\!\cdots\!85}{74\!\cdots\!76}a^{10}+\frac{11\!\cdots\!19}{37\!\cdots\!88}a^{9}-\frac{38\!\cdots\!89}{18\!\cdots\!44}a^{8}-\frac{25\!\cdots\!09}{93\!\cdots\!72}a^{7}+\frac{84\!\cdots\!59}{10\!\cdots\!32}a^{6}-\frac{48\!\cdots\!85}{93\!\cdots\!72}a^{5}-\frac{52\!\cdots\!89}{24\!\cdots\!28}a^{4}+\frac{33\!\cdots\!59}{46\!\cdots\!36}a^{3}-\frac{26\!\cdots\!23}{46\!\cdots\!36}a^{2}+\frac{10\!\cdots\!70}{58\!\cdots\!17}a-\frac{12\!\cdots\!94}{58\!\cdots\!17}$, $\frac{68\!\cdots\!81}{87\!\cdots\!56}a^{23}-\frac{65\!\cdots\!77}{37\!\cdots\!88}a^{22}-\frac{12\!\cdots\!75}{14\!\cdots\!52}a^{21}+\frac{45\!\cdots\!37}{11\!\cdots\!34}a^{20}+\frac{70\!\cdots\!57}{74\!\cdots\!76}a^{19}-\frac{31\!\cdots\!57}{74\!\cdots\!76}a^{18}-\frac{70\!\cdots\!03}{14\!\cdots\!52}a^{17}+\frac{81\!\cdots\!09}{37\!\cdots\!88}a^{16}+\frac{16\!\cdots\!91}{74\!\cdots\!76}a^{15}-\frac{30\!\cdots\!15}{74\!\cdots\!76}a^{14}-\frac{11\!\cdots\!91}{14\!\cdots\!52}a^{13}+\frac{11\!\cdots\!03}{37\!\cdots\!88}a^{12}-\frac{98\!\cdots\!99}{87\!\cdots\!56}a^{11}-\frac{43\!\cdots\!63}{74\!\cdots\!76}a^{10}+\frac{30\!\cdots\!25}{18\!\cdots\!44}a^{9}-\frac{59\!\cdots\!57}{18\!\cdots\!44}a^{8}-\frac{61\!\cdots\!55}{37\!\cdots\!88}a^{7}+\frac{69\!\cdots\!89}{18\!\cdots\!44}a^{6}-\frac{72\!\cdots\!19}{58\!\cdots\!17}a^{5}-\frac{35\!\cdots\!15}{23\!\cdots\!68}a^{4}+\frac{31\!\cdots\!01}{93\!\cdots\!72}a^{3}-\frac{81\!\cdots\!41}{42\!\cdots\!76}a^{2}+\frac{52\!\cdots\!03}{10\!\cdots\!94}a-\frac{36\!\cdots\!83}{58\!\cdots\!17}$, $\frac{16\!\cdots\!55}{46\!\cdots\!36}a^{23}-\frac{28\!\cdots\!99}{74\!\cdots\!76}a^{22}-\frac{14\!\cdots\!63}{33\!\cdots\!08}a^{21}-\frac{21\!\cdots\!63}{74\!\cdots\!76}a^{20}+\frac{15\!\cdots\!85}{37\!\cdots\!88}a^{19}+\frac{11\!\cdots\!07}{37\!\cdots\!88}a^{18}-\frac{78\!\cdots\!87}{37\!\cdots\!88}a^{17}-\frac{10\!\cdots\!15}{74\!\cdots\!76}a^{16}+\frac{37\!\cdots\!17}{37\!\cdots\!88}a^{15}-\frac{26\!\cdots\!27}{37\!\cdots\!88}a^{14}-\frac{19\!\cdots\!17}{37\!\cdots\!88}a^{13}+\frac{71\!\cdots\!17}{74\!\cdots\!76}a^{12}+\frac{35\!\cdots\!61}{37\!\cdots\!88}a^{11}-\frac{19\!\cdots\!83}{74\!\cdots\!76}a^{10}+\frac{83\!\cdots\!49}{18\!\cdots\!44}a^{9}+\frac{11\!\cdots\!85}{18\!\cdots\!44}a^{8}-\frac{10\!\cdots\!85}{18\!\cdots\!44}a^{7}+\frac{17\!\cdots\!41}{18\!\cdots\!44}a^{6}+\frac{10\!\cdots\!55}{93\!\cdots\!72}a^{5}-\frac{13\!\cdots\!93}{23\!\cdots\!68}a^{4}+\frac{35\!\cdots\!87}{46\!\cdots\!36}a^{3}+\frac{31\!\cdots\!21}{46\!\cdots\!36}a^{2}-\frac{13\!\cdots\!87}{11\!\cdots\!34}a+\frac{24\!\cdots\!32}{58\!\cdots\!17}$, $\frac{28\!\cdots\!61}{14\!\cdots\!52}a^{23}-\frac{13\!\cdots\!13}{37\!\cdots\!88}a^{22}-\frac{31\!\cdots\!55}{14\!\cdots\!52}a^{21}+\frac{24\!\cdots\!23}{18\!\cdots\!44}a^{20}+\frac{17\!\cdots\!67}{74\!\cdots\!76}a^{19}-\frac{10\!\cdots\!33}{74\!\cdots\!76}a^{18}-\frac{17\!\cdots\!47}{14\!\cdots\!52}a^{17}+\frac{37\!\cdots\!21}{37\!\cdots\!88}a^{16}+\frac{40\!\cdots\!77}{74\!\cdots\!76}a^{15}-\frac{59\!\cdots\!67}{74\!\cdots\!76}a^{14}-\frac{33\!\cdots\!31}{14\!\cdots\!52}a^{13}+\frac{26\!\cdots\!87}{37\!\cdots\!88}a^{12}-\frac{26\!\cdots\!79}{14\!\cdots\!52}a^{11}-\frac{10\!\cdots\!83}{74\!\cdots\!76}a^{10}+\frac{13\!\cdots\!19}{37\!\cdots\!88}a^{9}+\frac{47\!\cdots\!65}{93\!\cdots\!72}a^{8}-\frac{14\!\cdots\!91}{37\!\cdots\!88}a^{7}+\frac{14\!\cdots\!85}{18\!\cdots\!44}a^{6}-\frac{38\!\cdots\!11}{23\!\cdots\!68}a^{5}-\frac{16\!\cdots\!93}{46\!\cdots\!36}a^{4}+\frac{63\!\cdots\!27}{93\!\cdots\!72}a^{3}-\frac{10\!\cdots\!79}{46\!\cdots\!36}a^{2}+\frac{63\!\cdots\!93}{11\!\cdots\!34}a-\frac{39\!\cdots\!34}{58\!\cdots\!17}$, $\frac{17\!\cdots\!49}{74\!\cdots\!76}a^{23}-\frac{18\!\cdots\!01}{37\!\cdots\!88}a^{22}-\frac{19\!\cdots\!49}{74\!\cdots\!76}a^{21}+\frac{33\!\cdots\!07}{37\!\cdots\!88}a^{20}+\frac{35\!\cdots\!11}{11\!\cdots\!34}a^{19}-\frac{30\!\cdots\!21}{37\!\cdots\!88}a^{18}-\frac{11\!\cdots\!31}{74\!\cdots\!76}a^{17}+\frac{15\!\cdots\!39}{37\!\cdots\!88}a^{16}+\frac{13\!\cdots\!21}{18\!\cdots\!44}a^{15}-\frac{42\!\cdots\!35}{37\!\cdots\!88}a^{14}-\frac{20\!\cdots\!35}{74\!\cdots\!76}a^{13}+\frac{35\!\cdots\!63}{37\!\cdots\!88}a^{12}-\frac{86\!\cdots\!25}{74\!\cdots\!76}a^{11}-\frac{18\!\cdots\!23}{93\!\cdots\!72}a^{10}+\frac{17\!\cdots\!95}{37\!\cdots\!88}a^{9}+\frac{41\!\cdots\!13}{18\!\cdots\!44}a^{8}-\frac{10\!\cdots\!13}{18\!\cdots\!44}a^{7}+\frac{94\!\cdots\!27}{93\!\cdots\!72}a^{6}-\frac{19\!\cdots\!47}{21\!\cdots\!88}a^{5}-\frac{14\!\cdots\!69}{23\!\cdots\!68}a^{4}+\frac{10\!\cdots\!43}{11\!\cdots\!34}a^{3}-\frac{71\!\cdots\!49}{23\!\cdots\!68}a^{2}+\frac{36\!\cdots\!91}{11\!\cdots\!34}a+\frac{12\!\cdots\!83}{58\!\cdots\!17}$, $\frac{85\!\cdots\!09}{74\!\cdots\!76}a^{23}-\frac{16\!\cdots\!13}{74\!\cdots\!76}a^{22}-\frac{96\!\cdots\!55}{74\!\cdots\!76}a^{21}+\frac{15\!\cdots\!51}{74\!\cdots\!76}a^{20}+\frac{53\!\cdots\!83}{37\!\cdots\!88}a^{19}-\frac{34\!\cdots\!59}{18\!\cdots\!44}a^{18}-\frac{53\!\cdots\!37}{74\!\cdots\!76}a^{17}+\frac{78\!\cdots\!39}{74\!\cdots\!76}a^{16}+\frac{12\!\cdots\!13}{37\!\cdots\!88}a^{15}-\frac{93\!\cdots\!99}{18\!\cdots\!44}a^{14}-\frac{10\!\cdots\!33}{74\!\cdots\!76}a^{13}+\frac{29\!\cdots\!89}{67\!\cdots\!16}a^{12}-\frac{18\!\cdots\!95}{74\!\cdots\!76}a^{11}-\frac{67\!\cdots\!47}{74\!\cdots\!76}a^{10}+\frac{79\!\cdots\!21}{37\!\cdots\!88}a^{9}+\frac{44\!\cdots\!73}{18\!\cdots\!44}a^{8}-\frac{26\!\cdots\!13}{10\!\cdots\!94}a^{7}+\frac{87\!\cdots\!79}{18\!\cdots\!44}a^{6}-\frac{21\!\cdots\!73}{93\!\cdots\!72}a^{5}-\frac{55\!\cdots\!85}{23\!\cdots\!68}a^{4}+\frac{17\!\cdots\!03}{42\!\cdots\!76}a^{3}-\frac{64\!\cdots\!47}{46\!\cdots\!36}a^{2}+\frac{18\!\cdots\!12}{58\!\cdots\!17}a-\frac{26\!\cdots\!94}{58\!\cdots\!17}$, $\frac{40\!\cdots\!57}{74\!\cdots\!76}a^{23}-\frac{68\!\cdots\!83}{67\!\cdots\!16}a^{22}-\frac{45\!\cdots\!37}{74\!\cdots\!76}a^{21}+\frac{18\!\cdots\!27}{74\!\cdots\!76}a^{20}+\frac{62\!\cdots\!45}{93\!\cdots\!72}a^{19}-\frac{20\!\cdots\!69}{11\!\cdots\!34}a^{18}-\frac{24\!\cdots\!57}{74\!\cdots\!76}a^{17}+\frac{98\!\cdots\!95}{74\!\cdots\!76}a^{16}+\frac{29\!\cdots\!19}{18\!\cdots\!44}a^{15}-\frac{10\!\cdots\!33}{46\!\cdots\!36}a^{14}-\frac{48\!\cdots\!21}{74\!\cdots\!76}a^{13}+\frac{14\!\cdots\!19}{74\!\cdots\!76}a^{12}+\frac{42\!\cdots\!15}{74\!\cdots\!76}a^{11}-\frac{30\!\cdots\!99}{74\!\cdots\!76}a^{10}+\frac{89\!\cdots\!71}{93\!\cdots\!72}a^{9}+\frac{15\!\cdots\!27}{84\!\cdots\!52}a^{8}-\frac{12\!\cdots\!95}{11\!\cdots\!34}a^{7}+\frac{39\!\cdots\!57}{18\!\cdots\!44}a^{6}+\frac{91\!\cdots\!69}{93\!\cdots\!72}a^{5}-\frac{23\!\cdots\!67}{23\!\cdots\!68}a^{4}+\frac{43\!\cdots\!85}{23\!\cdots\!68}a^{3}-\frac{27\!\cdots\!53}{46\!\cdots\!36}a^{2}+\frac{17\!\cdots\!27}{11\!\cdots\!34}a-\frac{74\!\cdots\!51}{58\!\cdots\!17}$, $\frac{44\!\cdots\!69}{37\!\cdots\!88}a^{23}-\frac{41\!\cdots\!75}{93\!\cdots\!72}a^{22}-\frac{31\!\cdots\!95}{18\!\cdots\!44}a^{21}-\frac{95\!\cdots\!51}{46\!\cdots\!36}a^{20}+\frac{57\!\cdots\!75}{37\!\cdots\!88}a^{19}+\frac{10\!\cdots\!55}{46\!\cdots\!36}a^{18}-\frac{28\!\cdots\!59}{37\!\cdots\!88}a^{17}-\frac{53\!\cdots\!17}{42\!\cdots\!76}a^{16}+\frac{13\!\cdots\!15}{37\!\cdots\!88}a^{15}+\frac{72\!\cdots\!69}{52\!\cdots\!47}a^{14}-\frac{46\!\cdots\!87}{21\!\cdots\!64}a^{13}+\frac{17\!\cdots\!73}{93\!\cdots\!72}a^{12}+\frac{13\!\cdots\!47}{18\!\cdots\!44}a^{11}-\frac{43\!\cdots\!88}{58\!\cdots\!17}a^{10}+\frac{14\!\cdots\!27}{37\!\cdots\!88}a^{9}+\frac{29\!\cdots\!25}{93\!\cdots\!72}a^{8}-\frac{11\!\cdots\!13}{93\!\cdots\!72}a^{7}-\frac{66\!\cdots\!97}{23\!\cdots\!68}a^{6}+\frac{22\!\cdots\!29}{46\!\cdots\!36}a^{5}-\frac{44\!\cdots\!85}{42\!\cdots\!76}a^{4}-\frac{33\!\cdots\!47}{46\!\cdots\!36}a^{3}+\frac{48\!\cdots\!51}{23\!\cdots\!68}a^{2}-\frac{80\!\cdots\!23}{11\!\cdots\!34}a+\frac{68\!\cdots\!23}{58\!\cdots\!17}$, $\frac{33\!\cdots\!47}{74\!\cdots\!76}a^{23}-\frac{38\!\cdots\!55}{43\!\cdots\!28}a^{22}-\frac{38\!\cdots\!23}{74\!\cdots\!76}a^{21}+\frac{63\!\cdots\!33}{74\!\cdots\!76}a^{20}+\frac{52\!\cdots\!67}{93\!\cdots\!72}a^{19}-\frac{26\!\cdots\!39}{46\!\cdots\!36}a^{18}-\frac{21\!\cdots\!23}{74\!\cdots\!76}a^{17}+\frac{19\!\cdots\!29}{74\!\cdots\!76}a^{16}+\frac{25\!\cdots\!45}{18\!\cdots\!44}a^{15}-\frac{43\!\cdots\!17}{23\!\cdots\!68}a^{14}-\frac{41\!\cdots\!51}{74\!\cdots\!76}a^{13}+\frac{11\!\cdots\!87}{67\!\cdots\!16}a^{12}+\frac{52\!\cdots\!65}{74\!\cdots\!76}a^{11}-\frac{27\!\cdots\!21}{74\!\cdots\!76}a^{10}+\frac{74\!\cdots\!19}{93\!\cdots\!72}a^{9}+\frac{14\!\cdots\!23}{93\!\cdots\!72}a^{8}-\frac{44\!\cdots\!75}{42\!\cdots\!76}a^{7}+\frac{31\!\cdots\!39}{18\!\cdots\!44}a^{6}+\frac{31\!\cdots\!95}{93\!\cdots\!72}a^{5}-\frac{23\!\cdots\!65}{23\!\cdots\!68}a^{4}+\frac{30\!\cdots\!79}{21\!\cdots\!88}a^{3}-\frac{19\!\cdots\!79}{46\!\cdots\!36}a^{2}+\frac{11\!\cdots\!77}{11\!\cdots\!34}a-\frac{73\!\cdots\!40}{58\!\cdots\!17}$, $\frac{11\!\cdots\!21}{13\!\cdots\!32}a^{23}+\frac{31\!\cdots\!45}{37\!\cdots\!88}a^{22}-\frac{31\!\cdots\!33}{14\!\cdots\!52}a^{21}-\frac{12\!\cdots\!31}{93\!\cdots\!72}a^{20}+\frac{17\!\cdots\!75}{67\!\cdots\!16}a^{19}+\frac{96\!\cdots\!09}{74\!\cdots\!76}a^{18}+\frac{63\!\cdots\!73}{13\!\cdots\!32}a^{17}-\frac{24\!\cdots\!61}{37\!\cdots\!88}a^{16}-\frac{24\!\cdots\!49}{74\!\cdots\!76}a^{15}+\frac{19\!\cdots\!79}{74\!\cdots\!76}a^{14}-\frac{32\!\cdots\!29}{14\!\cdots\!52}a^{13}-\frac{44\!\cdots\!59}{37\!\cdots\!88}a^{12}+\frac{38\!\cdots\!87}{14\!\cdots\!52}a^{11}+\frac{18\!\cdots\!47}{74\!\cdots\!76}a^{10}-\frac{20\!\cdots\!29}{37\!\cdots\!88}a^{9}+\frac{18\!\cdots\!91}{18\!\cdots\!44}a^{8}+\frac{40\!\cdots\!17}{37\!\cdots\!88}a^{7}-\frac{24\!\cdots\!07}{18\!\cdots\!44}a^{6}+\frac{13\!\cdots\!29}{93\!\cdots\!72}a^{5}+\frac{60\!\cdots\!97}{58\!\cdots\!17}a^{4}-\frac{10\!\cdots\!41}{93\!\cdots\!72}a^{3}+\frac{46\!\cdots\!01}{46\!\cdots\!36}a^{2}-\frac{15\!\cdots\!97}{11\!\cdots\!34}a+\frac{12\!\cdots\!10}{58\!\cdots\!17}$ 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:  \( 699275455.9318979 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 699275455.9318979 \cdot 55}{12\cdot\sqrt{187226864009183111800823017295698722816}}\cr\approx \mathstrut & 0.886755860339702 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^24 - 2*x^23 - 11*x^22 + 2*x^21 + 122*x^20 - 22*x^19 - 603*x^18 + 126*x^17 + 2838*x^16 - 4554*x^15 - 11143*x^14 + 38030*x^13 - 5743*x^12 - 74196*x^11 + 192820*x^10 + 2152*x^9 - 196612*x^8 + 432432*x^7 - 65552*x^6 - 178400*x^5 + 381072*x^4 - 166208*x^3 + 51840*x^2 - 8704*x + 1024)
 
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 - 2*x^23 - 11*x^22 + 2*x^21 + 122*x^20 - 22*x^19 - 603*x^18 + 126*x^17 + 2838*x^16 - 4554*x^15 - 11143*x^14 + 38030*x^13 - 5743*x^12 - 74196*x^11 + 192820*x^10 + 2152*x^9 - 196612*x^8 + 432432*x^7 - 65552*x^6 - 178400*x^5 + 381072*x^4 - 166208*x^3 + 51840*x^2 - 8704*x + 1024, 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 - 2*x^23 - 11*x^22 + 2*x^21 + 122*x^20 - 22*x^19 - 603*x^18 + 126*x^17 + 2838*x^16 - 4554*x^15 - 11143*x^14 + 38030*x^13 - 5743*x^12 - 74196*x^11 + 192820*x^10 + 2152*x^9 - 196612*x^8 + 432432*x^7 - 65552*x^6 - 178400*x^5 + 381072*x^4 - 166208*x^3 + 51840*x^2 - 8704*x + 1024);
 
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 - 2*x^23 - 11*x^22 + 2*x^21 + 122*x^20 - 22*x^19 - 603*x^18 + 126*x^17 + 2838*x^16 - 4554*x^15 - 11143*x^14 + 38030*x^13 - 5743*x^12 - 74196*x^11 + 192820*x^10 + 2152*x^9 - 196612*x^8 + 432432*x^7 - 65552*x^6 - 178400*x^5 + 381072*x^4 - 166208*x^3 + 51840*x^2 - 8704*x + 1024);
 
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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-21}) \), 3.3.316.1, \(\Q(\sqrt{3}, \sqrt{-7})\), \(\Q(\sqrt{-3}, \sqrt{-7})\), \(\Q(i, \sqrt{7})\), \(\Q(\zeta_{12})\), \(\Q(\sqrt{3}, \sqrt{7})\), \(\Q(i, \sqrt{21})\), \(\Q(\sqrt{-3}, \sqrt{7})\), 6.0.2696112.1, 6.6.924766416.2, 6.0.399424.1, 6.6.137002432.1, 6.0.34250608.1, 6.6.10784448.1, 6.0.3699065664.2, 8.0.49787136.1, 12.0.13683086786583760896.2, 12.0.855192924161485056.1, 12.0.18769666373914624.1, 12.0.116304318664704.2, 12.12.13683086786583760896.1, 12.0.13683086786583760896.3, 12.0.13683086786583760896.1

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 R R ${\href{/padicField/5.6.0.1}{6} }^{4}$ R ${\href{/padicField/11.2.0.1}{2} }^{12}$ ${\href{/padicField/13.6.0.1}{6} }^{4}$ ${\href{/padicField/17.2.0.1}{2} }^{12}$ ${\href{/padicField/19.2.0.1}{2} }^{12}$ ${\href{/padicField/23.2.0.1}{2} }^{12}$ ${\href{/padicField/29.2.0.1}{2} }^{12}$ ${\href{/padicField/31.2.0.1}{2} }^{12}$ ${\href{/padicField/37.2.0.1}{2} }^{8}{,}\,{\href{/padicField/37.1.0.1}{1} }^{8}$ ${\href{/padicField/41.2.0.1}{2} }^{12}$ ${\href{/padicField/43.2.0.1}{2} }^{12}$ ${\href{/padicField/47.6.0.1}{6} }^{4}$ ${\href{/padicField/53.2.0.1}{2} }^{12}$ ${\href{/padicField/59.6.0.1}{6} }^{4}$

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

# 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}$
\(7\) Copy content Toggle raw display 7.12.6.1$x^{12} + 44 x^{10} + 10 x^{9} + 786 x^{8} + 22 x^{7} + 6899 x^{6} - 3434 x^{5} + 31050 x^{4} - 28440 x^{3} + 84557 x^{2} - 48082 x + 107648$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
7.12.6.1$x^{12} + 44 x^{10} + 10 x^{9} + 786 x^{8} + 22 x^{7} + 6899 x^{6} - 3434 x^{5} + 31050 x^{4} - 28440 x^{3} + 84557 x^{2} - 48082 x + 107648$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
\(79\) 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}$