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Properties

Label 24.0.185...736.1
Degree $24$
Signature $[0, 12]$
Discriminant $1.856\times 10^{46}$
Root discriminant \(84.69\)
Ramified primes $2,3,7,31$
Class number $3528$ (GRH)
Class group [2, 42, 42] (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^22 - 12*x^21 + 165*x^20 + 180*x^19 - 226*x^18 + 648*x^17 - 23244*x^16 - 117288*x^15 - 222138*x^14 + 219384*x^13 + 547477*x^12 + 5606064*x^11 + 28237566*x^10 + 14850228*x^9 + 29776521*x^8 + 106963380*x^7 - 310672020*x^6 - 76948200*x^5 + 606651408*x^4 - 581982480*x^3 + 263674800*x^2 - 63828000*x + 7110000)
 
gp: K = bnfinit(y^24 - 6*y^22 - 12*y^21 + 165*y^20 + 180*y^19 - 226*y^18 + 648*y^17 - 23244*y^16 - 117288*y^15 - 222138*y^14 + 219384*y^13 + 547477*y^12 + 5606064*y^11 + 28237566*y^10 + 14850228*y^9 + 29776521*y^8 + 106963380*y^7 - 310672020*y^6 - 76948200*y^5 + 606651408*y^4 - 581982480*y^3 + 263674800*y^2 - 63828000*y + 7110000, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 6*x^22 - 12*x^21 + 165*x^20 + 180*x^19 - 226*x^18 + 648*x^17 - 23244*x^16 - 117288*x^15 - 222138*x^14 + 219384*x^13 + 547477*x^12 + 5606064*x^11 + 28237566*x^10 + 14850228*x^9 + 29776521*x^8 + 106963380*x^7 - 310672020*x^6 - 76948200*x^5 + 606651408*x^4 - 581982480*x^3 + 263674800*x^2 - 63828000*x + 7110000);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - 6*x^22 - 12*x^21 + 165*x^20 + 180*x^19 - 226*x^18 + 648*x^17 - 23244*x^16 - 117288*x^15 - 222138*x^14 + 219384*x^13 + 547477*x^12 + 5606064*x^11 + 28237566*x^10 + 14850228*x^9 + 29776521*x^8 + 106963380*x^7 - 310672020*x^6 - 76948200*x^5 + 606651408*x^4 - 581982480*x^3 + 263674800*x^2 - 63828000*x + 7110000)
 

\( x^{24} - 6 x^{22} - 12 x^{21} + 165 x^{20} + 180 x^{19} - 226 x^{18} + 648 x^{17} - 23244 x^{16} + \cdots + 7110000 \) 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:   \(18558591262603306248362791442359121427862388736\) \(\medspace = 2^{36}\cdot 3^{28}\cdot 7^{12}\cdot 31^{8}\) 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:  \(84.69\)
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:  $2^{3/2}3^{7/6}7^{1/2}31^{1/2}\approx 150.11231557231721$
Ramified primes:   \(2\), \(3\), \(7\), \(31\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4}a^{8}-\frac{1}{2}a^{5}+\frac{1}{4}a^{2}-\frac{1}{2}$, $\frac{1}{4}a^{9}-\frac{1}{2}a^{6}+\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{10}-\frac{1}{2}a^{7}+\frac{1}{4}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{11}+\frac{1}{4}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{20}a^{12}+\frac{1}{20}a^{11}-\frac{1}{10}a^{10}-\frac{1}{10}a^{9}+\frac{9}{20}a^{6}+\frac{9}{20}a^{5}-\frac{1}{2}a^{3}-\frac{1}{10}a^{2}+\frac{2}{5}a$, $\frac{1}{20}a^{13}+\frac{1}{10}a^{11}+\frac{1}{10}a^{9}+\frac{9}{20}a^{7}-\frac{1}{5}a^{5}-\frac{1}{2}a^{4}-\frac{1}{10}a^{3}-\frac{2}{5}a$, $\frac{1}{20}a^{14}-\frac{1}{10}a^{11}+\frac{1}{20}a^{10}-\frac{1}{20}a^{9}-\frac{1}{20}a^{8}-\frac{1}{2}a^{7}+\frac{2}{5}a^{6}-\frac{2}{5}a^{5}-\frac{7}{20}a^{4}-\frac{1}{4}a^{3}-\frac{1}{5}a^{2}-\frac{3}{10}a$, $\frac{1}{20}a^{15}-\frac{1}{10}a^{11}-\frac{1}{10}a^{7}+\frac{3}{10}a^{5}-\frac{9}{20}a^{3}+\frac{3}{10}a$, $\frac{1}{80}a^{16}-\frac{1}{10}a^{11}+\frac{3}{40}a^{10}-\frac{1}{20}a^{9}+\frac{1}{10}a^{8}+\frac{3}{10}a^{6}-\frac{2}{5}a^{5}-\frac{39}{80}a^{4}-\frac{1}{4}a^{3}-\frac{1}{10}a^{2}-\frac{3}{10}a-\frac{1}{4}$, $\frac{1}{80}a^{17}-\frac{3}{40}a^{11}-\frac{1}{10}a^{9}-\frac{1}{5}a^{7}-\frac{1}{2}a^{6}+\frac{13}{80}a^{5}+\frac{2}{5}a^{3}-\frac{1}{2}a^{2}-\frac{9}{20}a$, $\frac{1}{720}a^{18}-\frac{1}{240}a^{16}-\frac{1}{60}a^{14}+\frac{7}{360}a^{12}-\frac{1}{10}a^{11}-\frac{1}{40}a^{10}+\frac{1}{30}a^{9}-\frac{1}{15}a^{8}-\frac{1}{2}a^{7}-\frac{43}{144}a^{6}-\frac{2}{5}a^{5}+\frac{71}{240}a^{4}-\frac{1}{6}a^{3}-\frac{11}{30}a^{2}+\frac{1}{5}a-\frac{1}{12}$, $\frac{1}{3600}a^{19}+\frac{1}{3600}a^{18}-\frac{1}{300}a^{17}+\frac{1}{240}a^{16}+\frac{1}{60}a^{15}-\frac{1}{300}a^{14}-\frac{29}{1800}a^{13}+\frac{7}{1800}a^{12}+\frac{1}{25}a^{11}+\frac{1}{24}a^{10}-\frac{11}{300}a^{9}-\frac{37}{300}a^{8}+\frac{1297}{3600}a^{7}+\frac{577}{3600}a^{6}+\frac{5}{12}a^{5}-\frac{383}{1200}a^{4}+\frac{37}{300}a^{3}-\frac{19}{300}a^{2}-\frac{1}{15}a-\frac{5}{12}$, $\frac{1}{28800}a^{20}+\frac{1}{14400}a^{19}-\frac{11}{28800}a^{18}-\frac{11}{2400}a^{17}-\frac{1}{192}a^{16}+\frac{1}{600}a^{15}+\frac{47}{2880}a^{14}+\frac{79}{7200}a^{13}-\frac{11}{14400}a^{12}-\frac{7}{600}a^{11}-\frac{31}{400}a^{10}+\frac{1}{200}a^{9}-\frac{1487}{28800}a^{8}-\frac{863}{14400}a^{7}+\frac{8377}{28800}a^{6}-\frac{119}{800}a^{5}-\frac{157}{960}a^{4}+\frac{33}{400}a^{3}+\frac{387}{800}a^{2}+\frac{19}{120}a-\frac{1}{48}$, $\frac{1}{28800}a^{21}+\frac{1}{28800}a^{19}-\frac{7}{14400}a^{18}-\frac{13}{4800}a^{17}-\frac{1}{2400}a^{16}-\frac{53}{14400}a^{15}-\frac{7}{600}a^{14}-\frac{71}{14400}a^{13}-\frac{97}{7200}a^{12}-\frac{29}{1200}a^{11}+\frac{11}{600}a^{10}-\frac{3407}{28800}a^{9}+\frac{1}{75}a^{8}-\frac{9179}{28800}a^{7}+\frac{2057}{14400}a^{6}+\frac{1043}{4800}a^{5}-\frac{599}{2400}a^{4}-\frac{403}{2400}a^{3}+\frac{397}{1200}a^{2}+\frac{79}{240}a+\frac{7}{24}$, $\frac{1}{37\!\cdots\!00}a^{22}+\frac{76\!\cdots\!39}{57\!\cdots\!40}a^{21}-\frac{14\!\cdots\!79}{12\!\cdots\!00}a^{20}+\frac{57\!\cdots\!17}{74\!\cdots\!20}a^{19}-\frac{20\!\cdots\!71}{41\!\cdots\!40}a^{18}+\frac{92\!\cdots\!21}{15\!\cdots\!00}a^{17}-\frac{51\!\cdots\!93}{18\!\cdots\!00}a^{16}-\frac{69\!\cdots\!99}{37\!\cdots\!60}a^{15}+\frac{30\!\cdots\!31}{20\!\cdots\!00}a^{14}-\frac{23\!\cdots\!83}{18\!\cdots\!00}a^{13}-\frac{57\!\cdots\!07}{77\!\cdots\!20}a^{12}-\frac{12\!\cdots\!01}{15\!\cdots\!00}a^{11}+\frac{30\!\cdots\!89}{37\!\cdots\!00}a^{10}-\frac{24\!\cdots\!73}{37\!\cdots\!00}a^{9}+\frac{30\!\cdots\!37}{33\!\cdots\!00}a^{8}+\frac{16\!\cdots\!17}{37\!\cdots\!00}a^{7}-\frac{27\!\cdots\!53}{61\!\cdots\!00}a^{6}+\frac{83\!\cdots\!93}{20\!\cdots\!00}a^{5}-\frac{43\!\cdots\!81}{10\!\cdots\!00}a^{4}+\frac{19\!\cdots\!97}{61\!\cdots\!60}a^{3}+\frac{10\!\cdots\!57}{51\!\cdots\!00}a^{2}+\frac{83\!\cdots\!79}{30\!\cdots\!80}a-\frac{20\!\cdots\!83}{51\!\cdots\!48}$, $\frac{1}{53\!\cdots\!00}a^{23}-\frac{54\!\cdots\!87}{44\!\cdots\!00}a^{22}+\frac{21\!\cdots\!03}{29\!\cdots\!00}a^{21}-\frac{64\!\cdots\!87}{56\!\cdots\!00}a^{20}+\frac{12\!\cdots\!99}{11\!\cdots\!00}a^{19}+\frac{18\!\cdots\!21}{29\!\cdots\!00}a^{18}-\frac{51\!\cdots\!51}{33\!\cdots\!00}a^{17}-\frac{71\!\cdots\!43}{22\!\cdots\!00}a^{16}+\frac{24\!\cdots\!91}{56\!\cdots\!00}a^{15}+\frac{76\!\cdots\!77}{56\!\cdots\!00}a^{14}-\frac{18\!\cdots\!33}{89\!\cdots\!00}a^{13}+\frac{17\!\cdots\!39}{24\!\cdots\!00}a^{12}-\frac{46\!\cdots\!23}{53\!\cdots\!00}a^{11}-\frac{19\!\cdots\!73}{17\!\cdots\!00}a^{10}-\frac{52\!\cdots\!03}{29\!\cdots\!00}a^{9}-\frac{17\!\cdots\!29}{18\!\cdots\!00}a^{8}+\frac{24\!\cdots\!79}{59\!\cdots\!00}a^{7}-\frac{78\!\cdots\!67}{29\!\cdots\!00}a^{6}-\frac{11\!\cdots\!99}{17\!\cdots\!00}a^{5}+\frac{11\!\cdots\!11}{49\!\cdots\!00}a^{4}-\frac{13\!\cdots\!49}{49\!\cdots\!00}a^{3}+\frac{11\!\cdots\!57}{74\!\cdots\!00}a^{2}-\frac{36\!\cdots\!27}{80\!\cdots\!20}a+\frac{90\!\cdots\!43}{49\!\cdots\!84}$ 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:  $2$, $5$

Class group and class number

$C_{2}\times C_{42}\times C_{42}$, which has order $3528$ (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:   \( \frac{2214075669610849513924577}{272127980961322018154720432000} a^{23} + \frac{514418104388624088117093}{108851192384528807261888172800} a^{22} - \frac{75279801252374705382637297}{1632767885767932108928322592000} a^{21} - \frac{67727422402152322897420713}{544255961922644036309440864000} a^{20} + \frac{138279773359243092350971057}{108851192384528807261888172800} a^{19} + \frac{4795890285958270572492533}{2177023847690576145237763456} a^{18} - \frac{153561966618175049467748457}{272127980961322018154720432000} a^{17} + \frac{1342820094280828930578831951}{272127980961322018154720432000} a^{16} - \frac{152053727356342882831209616159}{816383942883966054464161296000} a^{15} - \frac{289130194460553565006904100861}{272127980961322018154720432000} a^{14} - \frac{659562965887668186787353711531}{272127980961322018154720432000} a^{13} + \frac{6491468415305415577418321923}{17007998810082626134670027000} a^{12} + \frac{1275544228529743862121497481269}{272127980961322018154720432000} a^{11} + \frac{26307374472627671961416773648761}{544255961922644036309440864000} a^{10} + \frac{420936000580615504529015628231767}{1632767885767932108928322592000} a^{9} + \frac{147147134070268436477097353110107}{544255961922644036309440864000} a^{8} + \frac{216722336179873883452868395402689}{544255961922644036309440864000} a^{7} + \frac{59875866291331718425397665766457}{54425596192264403630944086400} a^{6} - \frac{102914547835850844316547758846323}{54425596192264403630944086400} a^{5} - \frac{1518289512433765782560412905511}{877832196649425865015227200} a^{4} + \frac{535670785399029451718475894598563}{136063990480661009077360216000} a^{3} - \frac{33212054891278023055628103354771}{13606399048066100907736021600} a^{2} + \frac{1946724617794691701970073009657}{2721279809613220181547204320} a - \frac{13246808382017196785271917647}{136063990480661009077360216} \)  (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:   $\frac{25\!\cdots\!53}{82\!\cdots\!00}a^{23}+\frac{12\!\cdots\!61}{73\!\cdots\!20}a^{22}-\frac{98\!\cdots\!17}{55\!\cdots\!00}a^{21}-\frac{25\!\cdots\!19}{55\!\cdots\!00}a^{20}+\frac{60\!\cdots\!91}{12\!\cdots\!00}a^{19}+\frac{15\!\cdots\!57}{18\!\cdots\!00}a^{18}-\frac{22\!\cdots\!13}{82\!\cdots\!00}a^{17}+\frac{17\!\cdots\!51}{91\!\cdots\!00}a^{16}-\frac{65\!\cdots\!53}{91\!\cdots\!00}a^{15}-\frac{37\!\cdots\!61}{91\!\cdots\!00}a^{14}-\frac{24\!\cdots\!33}{27\!\cdots\!00}a^{13}+\frac{11\!\cdots\!03}{57\!\cdots\!00}a^{12}+\frac{14\!\cdots\!41}{82\!\cdots\!00}a^{11}+\frac{11\!\cdots\!67}{61\!\cdots\!00}a^{10}+\frac{53\!\cdots\!67}{55\!\cdots\!00}a^{9}+\frac{54\!\cdots\!61}{55\!\cdots\!00}a^{8}+\frac{26\!\cdots\!09}{18\!\cdots\!00}a^{7}+\frac{25\!\cdots\!03}{61\!\cdots\!00}a^{6}-\frac{82\!\cdots\!69}{11\!\cdots\!80}a^{5}-\frac{12\!\cdots\!13}{19\!\cdots\!60}a^{4}+\frac{71\!\cdots\!43}{45\!\cdots\!00}a^{3}-\frac{45\!\cdots\!31}{45\!\cdots\!00}a^{2}+\frac{27\!\cdots\!73}{91\!\cdots\!40}a-\frac{63\!\cdots\!09}{15\!\cdots\!04}$, $\frac{11\!\cdots\!07}{65\!\cdots\!00}a^{23}+\frac{29\!\cdots\!39}{29\!\cdots\!00}a^{22}-\frac{41\!\cdots\!83}{43\!\cdots\!00}a^{21}-\frac{37\!\cdots\!57}{14\!\cdots\!00}a^{20}+\frac{76\!\cdots\!11}{29\!\cdots\!00}a^{19}+\frac{11\!\cdots\!39}{24\!\cdots\!00}a^{18}-\frac{65\!\cdots\!47}{65\!\cdots\!00}a^{17}+\frac{25\!\cdots\!93}{24\!\cdots\!00}a^{16}-\frac{28\!\cdots\!67}{73\!\cdots\!00}a^{15}-\frac{48\!\cdots\!07}{21\!\cdots\!00}a^{14}-\frac{11\!\cdots\!67}{21\!\cdots\!00}a^{13}+\frac{22\!\cdots\!11}{36\!\cdots\!00}a^{12}+\frac{61\!\cdots\!19}{65\!\cdots\!00}a^{11}+\frac{14\!\cdots\!99}{14\!\cdots\!00}a^{10}+\frac{23\!\cdots\!53}{43\!\cdots\!00}a^{9}+\frac{25\!\cdots\!69}{43\!\cdots\!00}a^{8}+\frac{42\!\cdots\!57}{48\!\cdots\!00}a^{7}+\frac{17\!\cdots\!13}{73\!\cdots\!00}a^{6}-\frac{16\!\cdots\!43}{43\!\cdots\!00}a^{5}-\frac{20\!\cdots\!57}{59\!\cdots\!00}a^{4}+\frac{29\!\cdots\!77}{36\!\cdots\!00}a^{3}-\frac{31\!\cdots\!39}{61\!\cdots\!20}a^{2}+\frac{12\!\cdots\!11}{73\!\cdots\!40}a-\frac{62\!\cdots\!69}{24\!\cdots\!68}$, $\frac{10\!\cdots\!27}{49\!\cdots\!00}a^{23}+\frac{26\!\cdots\!91}{19\!\cdots\!00}a^{22}-\frac{11\!\cdots\!29}{99\!\cdots\!00}a^{21}-\frac{98\!\cdots\!69}{29\!\cdots\!00}a^{20}+\frac{65\!\cdots\!83}{19\!\cdots\!00}a^{19}+\frac{17\!\cdots\!71}{29\!\cdots\!00}a^{18}-\frac{60\!\cdots\!47}{49\!\cdots\!00}a^{17}+\frac{64\!\cdots\!61}{49\!\cdots\!00}a^{16}-\frac{24\!\cdots\!83}{49\!\cdots\!00}a^{15}-\frac{41\!\cdots\!93}{14\!\cdots\!00}a^{14}-\frac{32\!\cdots\!41}{49\!\cdots\!00}a^{13}+\frac{27\!\cdots\!31}{37\!\cdots\!00}a^{12}+\frac{61\!\cdots\!59}{49\!\cdots\!00}a^{11}+\frac{12\!\cdots\!91}{99\!\cdots\!00}a^{10}+\frac{67\!\cdots\!19}{99\!\cdots\!00}a^{9}+\frac{21\!\cdots\!91}{29\!\cdots\!00}a^{8}+\frac{10\!\cdots\!59}{99\!\cdots\!00}a^{7}+\frac{87\!\cdots\!43}{29\!\cdots\!00}a^{6}-\frac{48\!\cdots\!09}{99\!\cdots\!00}a^{5}-\frac{15\!\cdots\!73}{32\!\cdots\!80}a^{4}+\frac{24\!\cdots\!33}{24\!\cdots\!00}a^{3}-\frac{15\!\cdots\!11}{24\!\cdots\!00}a^{2}+\frac{92\!\cdots\!87}{49\!\cdots\!40}a-\frac{67\!\cdots\!77}{24\!\cdots\!92}$, $\frac{31\!\cdots\!86}{14\!\cdots\!25}a^{23}+\frac{50\!\cdots\!91}{35\!\cdots\!00}a^{22}-\frac{15\!\cdots\!47}{11\!\cdots\!00}a^{21}-\frac{12\!\cdots\!23}{35\!\cdots\!00}a^{20}+\frac{12\!\cdots\!57}{35\!\cdots\!00}a^{19}+\frac{11\!\cdots\!27}{17\!\cdots\!00}a^{18}-\frac{22\!\cdots\!27}{17\!\cdots\!00}a^{17}+\frac{24\!\cdots\!41}{17\!\cdots\!00}a^{16}-\frac{30\!\cdots\!81}{59\!\cdots\!00}a^{15}-\frac{68\!\cdots\!13}{23\!\cdots\!16}a^{14}-\frac{12\!\cdots\!59}{17\!\cdots\!00}a^{13}+\frac{98\!\cdots\!32}{14\!\cdots\!25}a^{12}+\frac{57\!\cdots\!89}{44\!\cdots\!00}a^{11}+\frac{48\!\cdots\!71}{35\!\cdots\!00}a^{10}+\frac{28\!\cdots\!63}{39\!\cdots\!00}a^{9}+\frac{28\!\cdots\!53}{35\!\cdots\!00}a^{8}+\frac{41\!\cdots\!37}{35\!\cdots\!00}a^{7}+\frac{56\!\cdots\!87}{17\!\cdots\!00}a^{6}-\frac{60\!\cdots\!09}{11\!\cdots\!60}a^{5}-\frac{49\!\cdots\!91}{99\!\cdots\!00}a^{4}+\frac{31\!\cdots\!77}{29\!\cdots\!00}a^{3}-\frac{97\!\cdots\!71}{14\!\cdots\!00}a^{2}+\frac{56\!\cdots\!07}{29\!\cdots\!40}a-\frac{38\!\cdots\!69}{14\!\cdots\!52}$, $\frac{11\!\cdots\!07}{21\!\cdots\!00}a^{23}+\frac{65\!\cdots\!47}{11\!\cdots\!00}a^{22}-\frac{49\!\cdots\!67}{17\!\cdots\!00}a^{21}-\frac{16\!\cdots\!59}{17\!\cdots\!00}a^{20}+\frac{29\!\cdots\!95}{38\!\cdots\!68}a^{19}+\frac{53\!\cdots\!87}{29\!\cdots\!00}a^{18}+\frac{92\!\cdots\!17}{26\!\cdots\!00}a^{17}+\frac{10\!\cdots\!11}{36\!\cdots\!00}a^{16}-\frac{34\!\cdots\!83}{29\!\cdots\!00}a^{15}-\frac{21\!\cdots\!41}{29\!\cdots\!00}a^{14}-\frac{16\!\cdots\!13}{89\!\cdots\!00}a^{13}-\frac{75\!\cdots\!81}{14\!\cdots\!00}a^{12}+\frac{10\!\cdots\!57}{33\!\cdots\!00}a^{11}+\frac{63\!\cdots\!57}{19\!\cdots\!00}a^{10}+\frac{31\!\cdots\!97}{17\!\cdots\!00}a^{9}+\frac{44\!\cdots\!01}{17\!\cdots\!00}a^{8}+\frac{20\!\cdots\!49}{59\!\cdots\!00}a^{7}+\frac{80\!\cdots\!81}{99\!\cdots\!00}a^{6}-\frac{38\!\cdots\!81}{43\!\cdots\!00}a^{5}-\frac{52\!\cdots\!87}{31\!\cdots\!50}a^{4}+\frac{27\!\cdots\!03}{14\!\cdots\!00}a^{3}-\frac{56\!\cdots\!61}{18\!\cdots\!00}a^{2}-\frac{42\!\cdots\!79}{29\!\cdots\!40}a+\frac{22\!\cdots\!51}{99\!\cdots\!68}$, $\frac{13\!\cdots\!89}{72\!\cdots\!00}a^{23}+\frac{10\!\cdots\!87}{97\!\cdots\!00}a^{22}-\frac{57\!\cdots\!09}{53\!\cdots\!00}a^{21}-\frac{46\!\cdots\!29}{16\!\cdots\!00}a^{20}+\frac{31\!\cdots\!03}{10\!\cdots\!00}a^{19}+\frac{81\!\cdots\!03}{16\!\cdots\!00}a^{18}-\frac{10\!\cdots\!29}{72\!\cdots\!00}a^{17}+\frac{27\!\cdots\!69}{24\!\cdots\!00}a^{16}-\frac{10\!\cdots\!67}{24\!\cdots\!00}a^{15}-\frac{59\!\cdots\!59}{24\!\cdots\!00}a^{14}-\frac{13\!\cdots\!09}{24\!\cdots\!00}a^{13}+\frac{20\!\cdots\!61}{20\!\cdots\!00}a^{12}+\frac{79\!\cdots\!93}{72\!\cdots\!00}a^{11}+\frac{54\!\cdots\!39}{48\!\cdots\!00}a^{10}+\frac{95\!\cdots\!77}{16\!\cdots\!00}a^{9}+\frac{22\!\cdots\!01}{37\!\cdots\!00}a^{8}+\frac{48\!\cdots\!59}{53\!\cdots\!00}a^{7}+\frac{27\!\cdots\!21}{10\!\cdots\!60}a^{6}-\frac{21\!\cdots\!21}{48\!\cdots\!00}a^{5}-\frac{34\!\cdots\!51}{87\!\cdots\!00}a^{4}+\frac{12\!\cdots\!93}{13\!\cdots\!00}a^{3}-\frac{76\!\cdots\!19}{13\!\cdots\!00}a^{2}+\frac{13\!\cdots\!33}{80\!\cdots\!20}a-\frac{14\!\cdots\!93}{67\!\cdots\!16}$, $\frac{14\!\cdots\!09}{29\!\cdots\!00}a^{23}+\frac{10\!\cdots\!23}{35\!\cdots\!00}a^{22}-\frac{16\!\cdots\!43}{59\!\cdots\!00}a^{21}-\frac{13\!\cdots\!43}{17\!\cdots\!00}a^{20}+\frac{20\!\cdots\!51}{27\!\cdots\!00}a^{19}+\frac{79\!\cdots\!49}{59\!\cdots\!00}a^{18}-\frac{27\!\cdots\!03}{99\!\cdots\!00}a^{17}+\frac{24\!\cdots\!81}{89\!\cdots\!00}a^{16}-\frac{32\!\cdots\!61}{29\!\cdots\!00}a^{15}-\frac{43\!\cdots\!07}{69\!\cdots\!00}a^{14}-\frac{99\!\cdots\!17}{69\!\cdots\!00}a^{13}+\frac{70\!\cdots\!17}{37\!\cdots\!00}a^{12}+\frac{28\!\cdots\!71}{99\!\cdots\!00}a^{11}+\frac{51\!\cdots\!31}{17\!\cdots\!00}a^{10}+\frac{30\!\cdots\!71}{19\!\cdots\!00}a^{9}+\frac{29\!\cdots\!17}{17\!\cdots\!00}a^{8}+\frac{40\!\cdots\!79}{17\!\cdots\!00}a^{7}+\frac{37\!\cdots\!21}{59\!\cdots\!00}a^{6}-\frac{44\!\cdots\!61}{39\!\cdots\!20}a^{5}-\frac{11\!\cdots\!11}{96\!\cdots\!00}a^{4}+\frac{32\!\cdots\!11}{14\!\cdots\!00}a^{3}-\frac{35\!\cdots\!79}{29\!\cdots\!40}a^{2}+\frac{96\!\cdots\!73}{29\!\cdots\!40}a-\frac{18\!\cdots\!71}{49\!\cdots\!84}$, $\frac{17\!\cdots\!33}{26\!\cdots\!00}a^{23}+\frac{12\!\cdots\!71}{35\!\cdots\!00}a^{22}-\frac{22\!\cdots\!99}{59\!\cdots\!00}a^{21}-\frac{17\!\cdots\!79}{17\!\cdots\!00}a^{20}+\frac{36\!\cdots\!23}{35\!\cdots\!00}a^{19}+\frac{10\!\cdots\!87}{59\!\cdots\!00}a^{18}-\frac{14\!\cdots\!13}{26\!\cdots\!00}a^{17}+\frac{35\!\cdots\!53}{89\!\cdots\!00}a^{16}-\frac{13\!\cdots\!59}{89\!\cdots\!00}a^{15}-\frac{83\!\cdots\!07}{99\!\cdots\!00}a^{14}-\frac{16\!\cdots\!73}{89\!\cdots\!00}a^{13}+\frac{30\!\cdots\!07}{74\!\cdots\!00}a^{12}+\frac{77\!\cdots\!77}{20\!\cdots\!00}a^{11}+\frac{68\!\cdots\!43}{17\!\cdots\!00}a^{10}+\frac{12\!\cdots\!09}{59\!\cdots\!00}a^{9}+\frac{36\!\cdots\!61}{17\!\cdots\!00}a^{8}+\frac{54\!\cdots\!07}{17\!\cdots\!00}a^{7}+\frac{51\!\cdots\!59}{59\!\cdots\!00}a^{6}-\frac{21\!\cdots\!09}{13\!\cdots\!00}a^{5}-\frac{25\!\cdots\!81}{19\!\cdots\!80}a^{4}+\frac{36\!\cdots\!71}{11\!\cdots\!00}a^{3}-\frac{61\!\cdots\!13}{29\!\cdots\!40}a^{2}+\frac{61\!\cdots\!51}{99\!\cdots\!80}a-\frac{10\!\cdots\!69}{12\!\cdots\!46}$, $\frac{40\!\cdots\!29}{13\!\cdots\!00}a^{23}+\frac{96\!\cdots\!53}{56\!\cdots\!00}a^{22}-\frac{76\!\cdots\!33}{44\!\cdots\!00}a^{21}-\frac{35\!\cdots\!51}{78\!\cdots\!25}a^{20}+\frac{70\!\cdots\!89}{14\!\cdots\!00}a^{19}+\frac{36\!\cdots\!03}{44\!\cdots\!00}a^{18}-\frac{14\!\cdots\!77}{67\!\cdots\!00}a^{17}+\frac{20\!\cdots\!81}{11\!\cdots\!00}a^{16}-\frac{17\!\cdots\!49}{24\!\cdots\!00}a^{15}-\frac{11\!\cdots\!99}{28\!\cdots\!00}a^{14}-\frac{50\!\cdots\!23}{56\!\cdots\!00}a^{13}+\frac{17\!\cdots\!53}{11\!\cdots\!00}a^{12}+\frac{23\!\cdots\!13}{13\!\cdots\!00}a^{11}+\frac{12\!\cdots\!48}{70\!\cdots\!25}a^{10}+\frac{42\!\cdots\!03}{44\!\cdots\!00}a^{9}+\frac{69\!\cdots\!86}{70\!\cdots\!25}a^{8}+\frac{36\!\cdots\!31}{24\!\cdots\!00}a^{7}+\frac{18\!\cdots\!63}{44\!\cdots\!00}a^{6}-\frac{42\!\cdots\!37}{60\!\cdots\!00}a^{5}-\frac{15\!\cdots\!61}{24\!\cdots\!00}a^{4}+\frac{27\!\cdots\!21}{18\!\cdots\!00}a^{3}-\frac{34\!\cdots\!21}{37\!\cdots\!00}a^{2}+\frac{39\!\cdots\!81}{14\!\cdots\!30}a-\frac{28\!\cdots\!13}{74\!\cdots\!76}$, $\frac{16\!\cdots\!37}{53\!\cdots\!00}a^{23}+\frac{59\!\cdots\!61}{35\!\cdots\!00}a^{22}-\frac{67\!\cdots\!59}{39\!\cdots\!00}a^{21}-\frac{39\!\cdots\!17}{87\!\cdots\!00}a^{20}+\frac{11\!\cdots\!87}{23\!\cdots\!20}a^{19}+\frac{47\!\cdots\!97}{59\!\cdots\!00}a^{18}-\frac{41\!\cdots\!41}{17\!\cdots\!00}a^{17}+\frac{31\!\cdots\!47}{17\!\cdots\!00}a^{16}-\frac{12\!\cdots\!53}{17\!\cdots\!00}a^{15}-\frac{13\!\cdots\!61}{35\!\cdots\!80}a^{14}-\frac{15\!\cdots\!63}{17\!\cdots\!00}a^{13}+\frac{19\!\cdots\!19}{11\!\cdots\!00}a^{12}+\frac{94\!\cdots\!77}{53\!\cdots\!00}a^{11}+\frac{12\!\cdots\!89}{71\!\cdots\!60}a^{10}+\frac{11\!\cdots\!91}{11\!\cdots\!00}a^{9}+\frac{38\!\cdots\!67}{39\!\cdots\!00}a^{8}+\frac{16\!\cdots\!31}{11\!\cdots\!00}a^{7}+\frac{78\!\cdots\!83}{19\!\cdots\!00}a^{6}-\frac{12\!\cdots\!07}{17\!\cdots\!00}a^{5}-\frac{63\!\cdots\!51}{99\!\cdots\!00}a^{4}+\frac{14\!\cdots\!87}{99\!\cdots\!00}a^{3}-\frac{13\!\cdots\!79}{14\!\cdots\!00}a^{2}+\frac{74\!\cdots\!27}{29\!\cdots\!40}a-\frac{75\!\cdots\!77}{24\!\cdots\!92}$, $\frac{22\!\cdots\!87}{16\!\cdots\!00}a^{23}+\frac{13\!\cdots\!99}{24\!\cdots\!00}a^{22}-\frac{37\!\cdots\!57}{44\!\cdots\!00}a^{21}-\frac{11\!\cdots\!21}{69\!\cdots\!00}a^{20}+\frac{11\!\cdots\!57}{49\!\cdots\!00}a^{19}+\frac{37\!\cdots\!07}{14\!\cdots\!00}a^{18}-\frac{11\!\cdots\!39}{33\!\cdots\!00}a^{17}+\frac{34\!\cdots\!51}{44\!\cdots\!00}a^{16}-\frac{54\!\cdots\!23}{17\!\cdots\!00}a^{15}-\frac{72\!\cdots\!51}{44\!\cdots\!00}a^{14}-\frac{16\!\cdots\!77}{56\!\cdots\!00}a^{13}+\frac{16\!\cdots\!77}{49\!\cdots\!00}a^{12}+\frac{72\!\cdots\!27}{84\!\cdots\!00}a^{11}+\frac{34\!\cdots\!83}{44\!\cdots\!00}a^{10}+\frac{17\!\cdots\!87}{44\!\cdots\!00}a^{9}+\frac{59\!\cdots\!89}{29\!\cdots\!00}a^{8}+\frac{24\!\cdots\!49}{80\!\cdots\!00}a^{7}+\frac{80\!\cdots\!19}{59\!\cdots\!00}a^{6}-\frac{48\!\cdots\!39}{11\!\cdots\!00}a^{5}-\frac{49\!\cdots\!89}{29\!\cdots\!00}a^{4}+\frac{82\!\cdots\!37}{93\!\cdots\!00}a^{3}-\frac{10\!\cdots\!99}{14\!\cdots\!00}a^{2}+\frac{35\!\cdots\!49}{18\!\cdots\!90}a-\frac{15\!\cdots\!91}{99\!\cdots\!68}$ 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:  \( 94328517321.12169 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 94328517321.12169 \cdot 3528}{6\cdot\sqrt{18558591262603306248362791442359121427862388736}}\cr\approx \mathstrut & 1.54136768986528 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^24 - 6*x^22 - 12*x^21 + 165*x^20 + 180*x^19 - 226*x^18 + 648*x^17 - 23244*x^16 - 117288*x^15 - 222138*x^14 + 219384*x^13 + 547477*x^12 + 5606064*x^11 + 28237566*x^10 + 14850228*x^9 + 29776521*x^8 + 106963380*x^7 - 310672020*x^6 - 76948200*x^5 + 606651408*x^4 - 581982480*x^3 + 263674800*x^2 - 63828000*x + 7110000)
 
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^22 - 12*x^21 + 165*x^20 + 180*x^19 - 226*x^18 + 648*x^17 - 23244*x^16 - 117288*x^15 - 222138*x^14 + 219384*x^13 + 547477*x^12 + 5606064*x^11 + 28237566*x^10 + 14850228*x^9 + 29776521*x^8 + 106963380*x^7 - 310672020*x^6 - 76948200*x^5 + 606651408*x^4 - 581982480*x^3 + 263674800*x^2 - 63828000*x + 7110000, 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^22 - 12*x^21 + 165*x^20 + 180*x^19 - 226*x^18 + 648*x^17 - 23244*x^16 - 117288*x^15 - 222138*x^14 + 219384*x^13 + 547477*x^12 + 5606064*x^11 + 28237566*x^10 + 14850228*x^9 + 29776521*x^8 + 106963380*x^7 - 310672020*x^6 - 76948200*x^5 + 606651408*x^4 - 581982480*x^3 + 263674800*x^2 - 63828000*x + 7110000);
 
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^22 - 12*x^21 + 165*x^20 + 180*x^19 - 226*x^18 + 648*x^17 - 23244*x^16 - 117288*x^15 - 222138*x^14 + 219384*x^13 + 547477*x^12 + 5606064*x^11 + 28237566*x^10 + 14850228*x^9 + 29776521*x^8 + 106963380*x^7 - 310672020*x^6 - 76948200*x^5 + 606651408*x^4 - 581982480*x^3 + 263674800*x^2 - 63828000*x + 7110000);
 
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 $C_2^2\times D_6$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-14}) \), \(\Q(\sqrt{42}) \), 3.3.837.1, \(\Q(\sqrt{-3}, \sqrt{-14})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{-3}, \sqrt{-7})\), \(\Q(\sqrt{2}, \sqrt{-7})\), \(\Q(\sqrt{-6}, \sqrt{-14})\), \(\Q(\sqrt{2}, \sqrt{21})\), \(\Q(\sqrt{-6}, \sqrt{-7})\), 6.0.1076073984.4, 6.6.358691328.1, 6.6.720885501.1, 6.0.240295167.4, 6.0.2101707.2, 6.0.123031125504.4, 6.6.369093376512.1, 8.0.796594176.2, 12.0.136229920585028993286144.5, 12.0.1157935219041632256.2, 12.0.519675905552021001.1, 12.0.136229920585028993286144.1, 12.0.15136657842780999254016.2, 12.12.136229920585028993286144.2, 12.0.136229920585028993286144.2

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
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 ${\href{/padicField/5.2.0.1}{2} }^{12}$ R ${\href{/padicField/11.6.0.1}{6} }^{4}$ ${\href{/padicField/13.2.0.1}{2} }^{12}$ ${\href{/padicField/17.6.0.1}{6} }^{4}$ ${\href{/padicField/19.6.0.1}{6} }^{4}$ ${\href{/padicField/23.6.0.1}{6} }^{4}$ ${\href{/padicField/29.6.0.1}{6} }^{4}$ R ${\href{/padicField/37.2.0.1}{2} }^{12}$ ${\href{/padicField/41.2.0.1}{2} }^{12}$ ${\href{/padicField/43.2.0.1}{2} }^{12}$ ${\href{/padicField/47.2.0.1}{2} }^{12}$ ${\href{/padicField/53.2.0.1}{2} }^{12}$ ${\href{/padicField/59.2.0.1}{2} }^{12}$

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
\(2\) Copy content Toggle raw display 2.4.6.1$x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$$2$$2$$6$$C_2^2$$[3]^{2}$
\(3\) Copy content Toggle raw display 3.12.14.11$x^{12} + 12 x^{11} + 72 x^{10} + 280 x^{9} + 792 x^{8} + 1728 x^{7} + 2918 x^{6} + 3684 x^{5} + 3156 x^{4} + 1376 x^{3} - 36 x^{2} - 168 x + 25$$6$$2$$14$$D_6$$[3/2]_{2}^{2}$
3.12.14.11$x^{12} + 12 x^{11} + 72 x^{10} + 280 x^{9} + 792 x^{8} + 1728 x^{7} + 2918 x^{6} + 3684 x^{5} + 3156 x^{4} + 1376 x^{3} - 36 x^{2} - 168 x + 25$$6$$2$$14$$D_6$$[3/2]_{2}^{2}$
\(7\) Copy content Toggle raw display 7.6.3.2$x^{6} + 12 x^{5} + 57 x^{4} + 176 x^{3} + 699 x^{2} + 420 x + 1787$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} + 12 x^{5} + 57 x^{4} + 176 x^{3} + 699 x^{2} + 420 x + 1787$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} + 12 x^{5} + 57 x^{4} + 176 x^{3} + 699 x^{2} + 420 x + 1787$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} + 12 x^{5} + 57 x^{4} + 176 x^{3} + 699 x^{2} + 420 x + 1787$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
\(31\) Copy content Toggle raw display 31.2.0.1$x^{2} + 29 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} + 29 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} + 29 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} + 29 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
31.4.2.1$x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$