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Properties

Label 32.0.301...000.1
Degree $32$
Signature $[0, 16]$
Discriminant $3.018\times 10^{50}$
Root discriminant \(37.80\)
Ramified primes $2,3,5,41,167$
Class number $18$ (GRH)
Class group [18] (GRH)
Galois group $C_2^6:S_4$ (as 32T96908)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 46*x^30 + 899*x^28 + 9779*x^26 + 65207*x^24 + 277346*x^22 + 763587*x^20 + 1370784*x^18 + 1627006*x^16 + 1296861*x^14 + 701262*x^12 + 257219*x^10 + 63092*x^8 + 10001*x^6 + 959*x^4 + 49*x^2 + 1)
 
gp: K = bnfinit(y^32 + 46*y^30 + 899*y^28 + 9779*y^26 + 65207*y^24 + 277346*y^22 + 763587*y^20 + 1370784*y^18 + 1627006*y^16 + 1296861*y^14 + 701262*y^12 + 257219*y^10 + 63092*y^8 + 10001*y^6 + 959*y^4 + 49*y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 + 46*x^30 + 899*x^28 + 9779*x^26 + 65207*x^24 + 277346*x^22 + 763587*x^20 + 1370784*x^18 + 1627006*x^16 + 1296861*x^14 + 701262*x^12 + 257219*x^10 + 63092*x^8 + 10001*x^6 + 959*x^4 + 49*x^2 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 + 46*x^30 + 899*x^28 + 9779*x^26 + 65207*x^24 + 277346*x^22 + 763587*x^20 + 1370784*x^18 + 1627006*x^16 + 1296861*x^14 + 701262*x^12 + 257219*x^10 + 63092*x^8 + 10001*x^6 + 959*x^4 + 49*x^2 + 1)
 

\( x^{32} + 46 x^{30} + 899 x^{28} + 9779 x^{26} + 65207 x^{24} + 277346 x^{22} + 763587 x^{20} + 1370784 x^{18} + \cdots + 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $32$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 16]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(301756981416745756154944013144439800625000000000000\) \(\medspace = 2^{12}\cdot 3^{24}\cdot 5^{16}\cdot 41^{4}\cdot 167^{8}\) 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:  \(37.80\)
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:  $2^{3/2}3^{3/4}5^{1/2}41^{1/2}167^{1/2}\approx 1192.946866180926$
Ramified primes:   \(2\), \(3\), \(5\), \(41\), \(167\) 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$^{32768}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{15}-\frac{1}{2}a^{13}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{4}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{20}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{21}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{22}-\frac{1}{2}a^{15}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{23}-\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}$, $\frac{1}{30}a^{24}+\frac{1}{6}a^{22}+\frac{7}{30}a^{20}-\frac{1}{15}a^{18}+\frac{2}{15}a^{16}-\frac{1}{2}a^{15}-\frac{7}{30}a^{14}-\frac{1}{2}a^{13}+\frac{1}{15}a^{12}-\frac{1}{2}a^{11}+\frac{1}{15}a^{10}-\frac{1}{2}a^{9}-\frac{11}{30}a^{8}-\frac{4}{15}a^{6}-\frac{1}{2}a^{5}+\frac{7}{30}a^{4}+\frac{1}{6}a^{2}-\frac{1}{2}a+\frac{1}{30}$, $\frac{1}{30}a^{25}+\frac{1}{6}a^{23}+\frac{7}{30}a^{21}-\frac{1}{15}a^{19}+\frac{2}{15}a^{17}-\frac{7}{30}a^{15}-\frac{13}{30}a^{13}+\frac{1}{15}a^{11}-\frac{11}{30}a^{9}+\frac{7}{30}a^{7}-\frac{1}{2}a^{6}-\frac{4}{15}a^{5}-\frac{1}{3}a^{3}-\frac{7}{15}a-\frac{1}{2}$, $\frac{1}{30}a^{26}-\frac{1}{10}a^{22}-\frac{7}{30}a^{20}-\frac{1}{30}a^{18}+\frac{1}{10}a^{16}-\frac{1}{2}a^{15}-\frac{4}{15}a^{14}-\frac{1}{2}a^{13}+\frac{7}{30}a^{12}-\frac{1}{2}a^{11}-\frac{1}{5}a^{10}+\frac{1}{15}a^{8}-\frac{13}{30}a^{6}+\frac{1}{5}a^{2}-\frac{1}{6}$, $\frac{1}{30}a^{27}-\frac{1}{10}a^{23}-\frac{7}{30}a^{21}-\frac{1}{30}a^{19}+\frac{1}{10}a^{17}-\frac{4}{15}a^{15}-\frac{4}{15}a^{13}-\frac{1}{5}a^{11}-\frac{1}{2}a^{10}+\frac{1}{15}a^{9}+\frac{1}{15}a^{7}-\frac{1}{2}a^{5}-\frac{3}{10}a^{3}-\frac{1}{2}a^{2}+\frac{1}{3}a-\frac{1}{2}$, $\frac{1}{60}a^{28}-\frac{1}{60}a^{26}-\frac{1}{60}a^{24}-\frac{3}{20}a^{22}+\frac{1}{12}a^{20}+\frac{1}{5}a^{16}-\frac{7}{30}a^{14}-\frac{2}{5}a^{12}-\frac{1}{2}a^{11}+\frac{9}{20}a^{10}+\frac{23}{60}a^{8}+\frac{9}{20}a^{6}+\frac{1}{3}a^{4}-\frac{1}{2}a^{3}+\frac{7}{30}a^{2}+\frac{7}{60}$, $\frac{1}{60}a^{29}-\frac{1}{60}a^{27}-\frac{1}{60}a^{25}-\frac{3}{20}a^{23}+\frac{1}{12}a^{21}+\frac{1}{5}a^{17}-\frac{7}{30}a^{15}-\frac{2}{5}a^{13}-\frac{1}{2}a^{12}+\frac{9}{20}a^{11}+\frac{23}{60}a^{9}+\frac{9}{20}a^{7}+\frac{1}{3}a^{5}-\frac{1}{2}a^{4}+\frac{7}{30}a^{3}+\frac{7}{60}a$, $\frac{1}{1234630260}a^{30}-\frac{814864}{308657565}a^{28}-\frac{1638437}{123463026}a^{26}-\frac{2038303}{205771710}a^{24}+\frac{44242658}{308657565}a^{22}+\frac{53917505}{246926052}a^{20}-\frac{3209743}{617315130}a^{18}-\frac{71832691}{617315130}a^{16}-\frac{1}{2}a^{15}-\frac{20986427}{617315130}a^{14}+\frac{30269051}{1234630260}a^{12}-\frac{1}{2}a^{11}-\frac{43981384}{308657565}a^{10}-\frac{1}{2}a^{9}-\frac{105448051}{617315130}a^{8}+\frac{110764993}{411543420}a^{6}-\frac{1}{2}a^{5}-\frac{118873246}{308657565}a^{4}-\frac{1}{2}a^{3}-\frac{417748367}{1234630260}a^{2}-\frac{1}{2}a+\frac{106169621}{1234630260}$, $\frac{1}{1234630260}a^{31}-\frac{814864}{308657565}a^{29}-\frac{1638437}{123463026}a^{27}-\frac{2038303}{205771710}a^{25}+\frac{44242658}{308657565}a^{23}+\frac{53917505}{246926052}a^{21}-\frac{3209743}{617315130}a^{19}-\frac{71832691}{617315130}a^{17}-\frac{20986427}{617315130}a^{15}-\frac{1}{2}a^{14}-\frac{587046079}{1234630260}a^{13}-\frac{43981384}{308657565}a^{11}-\frac{105448051}{617315130}a^{9}-\frac{95006717}{411543420}a^{7}-\frac{1}{2}a^{6}+\frac{70911073}{617315130}a^{5}-\frac{1}{2}a^{4}+\frac{199566763}{1234630260}a^{3}-\frac{511145509}{1234630260}a-\frac{1}{2}$ 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:  $2$, $3$

Class group and class number

$C_{18}$, which has order $18$ (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:  $15$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( \frac{9063389651}{137181140} a^{31} + \frac{618142211371}{205771710} a^{29} + \frac{2378588856203}{41154342} a^{27} + \frac{126609793292381}{205771710} a^{25} + \frac{136491494365708}{34295285} a^{23} + \frac{2221875466558009}{137181140} a^{21} + \frac{1718923032115505}{41154342} a^{19} + \frac{4669676456062973}{68590570} a^{17} + \frac{4841924958186281}{68590570} a^{15} + \frac{3869878528831055}{82308684} a^{13} + \frac{1383880386661479}{68590570} a^{11} + \frac{377883484606723}{68590570} a^{9} + \frac{379318992170459}{411543420} a^{7} + \frac{3582071748535}{41154342} a^{5} + \frac{1609329774529}{411543420} a^{3} + \frac{8127275673}{137181140} a + \frac{1}{2} \)  (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:   $\frac{17150544479}{16245135}a^{30}+\frac{156616231412}{3249027}a^{28}+\frac{15151613125358}{16245135}a^{26}+\frac{54185662214648}{5415045}a^{24}+\frac{10\!\cdots\!66}{16245135}a^{22}+\frac{43\!\cdots\!71}{16245135}a^{20}+\frac{11\!\cdots\!49}{16245135}a^{18}+\frac{19\!\cdots\!11}{16245135}a^{16}+\frac{21\!\cdots\!37}{16245135}a^{14}+\frac{15\!\cdots\!42}{16245135}a^{12}+\frac{69\!\cdots\!57}{16245135}a^{10}+\frac{20\!\cdots\!53}{16245135}a^{8}+\frac{130228725618241}{5415045}a^{6}+\frac{42807165933784}{16245135}a^{4}+\frac{482236139308}{3249027}a^{2}+\frac{52618669987}{16245135}$, $\frac{1633135371271}{1234630260}a^{31}-\frac{2502167873}{2166018}a^{30}+\frac{74697184486403}{1234630260}a^{29}-\frac{95403077267}{1805015}a^{28}+\frac{14\!\cdots\!59}{1234630260}a^{27}-\frac{1851032456487}{1805015}a^{26}+\frac{51\!\cdots\!47}{411543420}a^{25}-\frac{3986789294618}{361003}a^{24}+\frac{10\!\cdots\!97}{1234630260}a^{23}-\frac{78624309298216}{1083009}a^{22}+\frac{42\!\cdots\!09}{123463026}a^{21}-\frac{16\!\cdots\!43}{5415045}a^{20}+\frac{11\!\cdots\!69}{123463026}a^{19}-\frac{43\!\cdots\!46}{5415045}a^{18}+\frac{96\!\cdots\!91}{617315130}a^{17}-\frac{74\!\cdots\!12}{5415045}a^{16}+\frac{10\!\cdots\!67}{61731513}a^{15}-\frac{16\!\cdots\!61}{10830090}a^{14}+\frac{15\!\cdots\!77}{1234630260}a^{13}-\frac{60\!\cdots\!72}{5415045}a^{12}+\frac{14\!\cdots\!09}{246926052}a^{11}-\frac{57\!\cdots\!79}{10830090}a^{10}+\frac{22\!\cdots\!23}{1234630260}a^{9}-\frac{874976561311901}{5415045}a^{8}+\frac{71\!\cdots\!99}{205771710}a^{7}-\frac{167602687502693}{5415045}a^{6}+\frac{23\!\cdots\!81}{617315130}a^{5}-\frac{37452338426729}{10830090}a^{4}+\frac{270757002705847}{1234630260}a^{3}-\frac{1070086444516}{5415045}a^{2}+\frac{1487805463703}{308657565}a-\frac{23592384383}{5415045}$, $\frac{93670781767}{308657565}a^{31}+\frac{44797290597}{137181140}a^{30}+\frac{17149778854373}{1234630260}a^{29}+\frac{6144528902659}{411543420}a^{28}+\frac{66583294318889}{246926052}a^{27}+\frac{119104182840503}{411543420}a^{26}+\frac{11\!\cdots\!67}{411543420}a^{25}+\frac{426999961126883}{137181140}a^{24}+\frac{23\!\cdots\!19}{1234630260}a^{23}+\frac{84\!\cdots\!33}{411543420}a^{22}+\frac{19\!\cdots\!79}{246926052}a^{21}+\frac{29\!\cdots\!54}{34295285}a^{20}+\frac{65\!\cdots\!61}{308657565}a^{19}+\frac{23\!\cdots\!43}{102885855}a^{18}+\frac{22\!\cdots\!89}{617315130}a^{17}+\frac{15\!\cdots\!45}{41154342}a^{16}+\frac{25\!\cdots\!47}{617315130}a^{15}+\frac{14\!\cdots\!89}{34295285}a^{14}+\frac{91\!\cdots\!42}{308657565}a^{13}+\frac{12\!\cdots\!71}{411543420}a^{12}+\frac{35\!\cdots\!15}{246926052}a^{11}+\frac{11\!\cdots\!53}{82308684}a^{10}+\frac{54\!\cdots\!33}{1234630260}a^{9}+\frac{60\!\cdots\!59}{137181140}a^{8}+\frac{35\!\cdots\!51}{411543420}a^{7}+\frac{17\!\cdots\!77}{205771710}a^{6}+\frac{119496228533705}{123463026}a^{5}+\frac{12891961171863}{13718114}a^{4}+\frac{34696352870837}{617315130}a^{3}+\frac{22061985333121}{411543420}a^{2}+\frac{1558588600751}{1234630260}a+\frac{121125133322}{102885855}$, $\frac{17150544479}{16245135}a^{31}+\frac{156616231412}{3249027}a^{29}+\frac{15151613125358}{16245135}a^{27}+\frac{54185662214648}{5415045}a^{25}+\frac{10\!\cdots\!66}{16245135}a^{23}+\frac{43\!\cdots\!71}{16245135}a^{21}+\frac{11\!\cdots\!49}{16245135}a^{19}+\frac{19\!\cdots\!11}{16245135}a^{17}+\frac{21\!\cdots\!37}{16245135}a^{15}+\frac{15\!\cdots\!42}{16245135}a^{13}+\frac{69\!\cdots\!57}{16245135}a^{11}+\frac{20\!\cdots\!53}{16245135}a^{9}+\frac{130228725618241}{5415045}a^{7}+\frac{42807165933784}{16245135}a^{5}+\frac{482236139308}{3249027}a^{3}+\frac{52602424852}{16245135}a$, $\frac{387891216103}{308657565}a^{31}-\frac{2502167873}{2166018}a^{30}+\frac{70988331218177}{1234630260}a^{29}-\frac{95403077267}{1805015}a^{28}+\frac{13\!\cdots\!69}{1234630260}a^{27}-\frac{1851032456487}{1805015}a^{26}+\frac{329589560366279}{27436228}a^{25}-\frac{3986789294618}{361003}a^{24}+\frac{97\!\cdots\!09}{1234630260}a^{23}-\frac{78624309298216}{1083009}a^{22}+\frac{40\!\cdots\!09}{1234630260}a^{21}-\frac{16\!\cdots\!43}{5415045}a^{20}+\frac{54\!\cdots\!27}{61731513}a^{19}-\frac{43\!\cdots\!46}{5415045}a^{18}+\frac{46\!\cdots\!67}{308657565}a^{17}-\frac{74\!\cdots\!12}{5415045}a^{16}+\frac{10\!\cdots\!41}{617315130}a^{15}-\frac{16\!\cdots\!61}{10830090}a^{14}+\frac{37\!\cdots\!13}{308657565}a^{13}-\frac{60\!\cdots\!72}{5415045}a^{12}+\frac{70\!\cdots\!23}{1234630260}a^{11}-\frac{57\!\cdots\!79}{10830090}a^{10}+\frac{21\!\cdots\!09}{1234630260}a^{9}-\frac{874976561311901}{5415045}a^{8}+\frac{13\!\cdots\!39}{411543420}a^{7}-\frac{167602687502693}{5415045}a^{6}+\frac{11\!\cdots\!28}{308657565}a^{5}-\frac{37452338426729}{10830090}a^{4}+\frac{13296450669113}{61731513}a^{3}-\frac{1070086444516}{5415045}a^{2}+\frac{1175615274751}{246926052}a-\frac{47168523631}{10830090}$, $\frac{189224225621}{617315130}a^{30}+\frac{1723878571325}{123463026}a^{28}+\frac{166244473161827}{617315130}a^{26}+\frac{591940918888519}{205771710}a^{24}+\frac{11\!\cdots\!29}{617315130}a^{22}+\frac{23\!\cdots\!98}{308657565}a^{20}+\frac{61\!\cdots\!37}{308657565}a^{18}+\frac{10\!\cdots\!34}{308657565}a^{16}+\frac{10\!\cdots\!38}{308657565}a^{14}+\frac{29\!\cdots\!47}{123463026}a^{12}+\frac{13\!\cdots\!89}{123463026}a^{10}+\frac{18\!\cdots\!61}{617315130}a^{8}+\frac{187303545283258}{34295285}a^{6}+\frac{174953978547869}{308657565}a^{4}+\frac{3713941725445}{123463026}a^{2}+\frac{191006898482}{308657565}$, $\frac{162184273706}{102885855}a^{31}+\frac{4935427958269}{68590570}a^{29}+\frac{19091647772657}{13718114}a^{27}+\frac{30\!\cdots\!99}{205771710}a^{25}+\frac{66\!\cdots\!27}{68590570}a^{23}+\frac{16\!\cdots\!05}{41154342}a^{21}+\frac{36\!\cdots\!01}{34295285}a^{19}+\frac{61\!\cdots\!47}{34295285}a^{17}+\frac{19\!\cdots\!53}{102885855}a^{15}+\frac{46\!\cdots\!72}{34295285}a^{13}+\frac{86\!\cdots\!03}{13718114}a^{11}+\frac{38\!\cdots\!67}{205771710}a^{9}+\frac{71\!\cdots\!37}{205771710}a^{7}+\frac{25882053257991}{6859057}a^{5}+\frac{21639948995378}{102885855}a^{3}+\frac{931205434049}{205771710}a$, $\frac{120736423105}{82308684}a^{31}-\frac{55420164407}{137181140}a^{30}+\frac{27630519639683}{411543420}a^{29}-\frac{7615261102903}{411543420}a^{28}+\frac{178781841806183}{137181140}a^{27}-\frac{147965041855759}{411543420}a^{26}+\frac{57\!\cdots\!37}{411543420}a^{25}-\frac{15\!\cdots\!61}{411543420}a^{24}+\frac{38\!\cdots\!57}{411543420}a^{23}-\frac{10\!\cdots\!47}{411543420}a^{22}+\frac{39\!\cdots\!19}{102885855}a^{21}-\frac{22\!\cdots\!35}{20577171}a^{20}+\frac{70\!\cdots\!33}{68590570}a^{19}-\frac{39\!\cdots\!07}{13718114}a^{18}+\frac{18\!\cdots\!18}{102885855}a^{17}-\frac{10\!\cdots\!61}{205771710}a^{16}+\frac{40\!\cdots\!21}{205771710}a^{15}-\frac{22\!\cdots\!07}{41154342}a^{14}+\frac{19\!\cdots\!79}{137181140}a^{13}-\frac{55\!\cdots\!59}{137181140}a^{12}+\frac{28\!\cdots\!69}{411543420}a^{11}-\frac{15\!\cdots\!11}{82308684}a^{10}+\frac{86\!\cdots\!53}{411543420}a^{9}-\frac{24\!\cdots\!03}{411543420}a^{8}+\frac{16\!\cdots\!21}{41154342}a^{7}-\frac{398577990226387}{34295285}a^{6}+\frac{30772358251644}{6859057}a^{5}-\frac{134385930162298}{102885855}a^{4}+\frac{105318720926689}{411543420}a^{3}-\frac{10266788496609}{137181140}a^{2}+\frac{1157850186353}{205771710}a-\frac{169815854663}{102885855}$, $\frac{340940031841}{82308684}a^{31}-\frac{1449924014423}{1234630260}a^{30}+\frac{77992458563197}{411543420}a^{29}-\frac{66367958970019}{1234630260}a^{28}+\frac{15\!\cdots\!03}{411543420}a^{27}-\frac{12\!\cdots\!23}{1234630260}a^{26}+\frac{54\!\cdots\!97}{137181140}a^{25}-\frac{308575027299439}{27436228}a^{24}+\frac{10\!\cdots\!77}{411543420}a^{23}-\frac{91\!\cdots\!23}{1234630260}a^{22}+\frac{11\!\cdots\!19}{102885855}a^{21}-\frac{95\!\cdots\!53}{308657565}a^{20}+\frac{59\!\cdots\!37}{205771710}a^{19}-\frac{10\!\cdots\!27}{123463026}a^{18}+\frac{10\!\cdots\!83}{205771710}a^{17}-\frac{87\!\cdots\!51}{617315130}a^{16}+\frac{11\!\cdots\!83}{205771710}a^{15}-\frac{48\!\cdots\!92}{308657565}a^{14}+\frac{16\!\cdots\!33}{411543420}a^{13}-\frac{14\!\cdots\!87}{1234630260}a^{12}+\frac{15\!\cdots\!51}{82308684}a^{11}-\frac{67\!\cdots\!53}{1234630260}a^{10}+\frac{23\!\cdots\!83}{411543420}a^{9}-\frac{20\!\cdots\!89}{1234630260}a^{8}+\frac{15\!\cdots\!65}{13718114}a^{7}-\frac{33\!\cdots\!04}{102885855}a^{6}+\frac{12\!\cdots\!64}{102885855}a^{5}-\frac{224704159556939}{61731513}a^{4}+\frac{290105438525003}{411543420}a^{3}-\frac{257444175522557}{1234630260}a^{2}+\frac{3193060033511}{205771710}a-\frac{1421216994022}{308657565}$, $\frac{226033419227}{205771710}a^{31}+\frac{2066184603673}{41154342}a^{29}+\frac{200159136189449}{205771710}a^{27}+\frac{21\!\cdots\!57}{205771710}a^{25}+\frac{14\!\cdots\!83}{205771710}a^{23}+\frac{29\!\cdots\!19}{102885855}a^{21}+\frac{25\!\cdots\!37}{34295285}a^{19}+\frac{13\!\cdots\!54}{102885855}a^{17}+\frac{14\!\cdots\!63}{102885855}a^{15}+\frac{69\!\cdots\!17}{68590570}a^{13}+\frac{97\!\cdots\!61}{205771710}a^{11}+\frac{29\!\cdots\!89}{205771710}a^{9}+\frac{27\!\cdots\!22}{102885855}a^{7}+\frac{306255647308981}{102885855}a^{5}+\frac{6942347411407}{41154342}a^{3}+\frac{380180033653}{102885855}a$, $\frac{3568435184}{61731513}a^{31}+\frac{37132476133}{205771710}a^{30}+\frac{640489478269}{246926052}a^{29}+\frac{3365445405293}{411543420}a^{28}+\frac{60508507510237}{1234630260}a^{27}+\frac{21486138813129}{137181140}a^{26}+\frac{209293605677801}{411543420}a^{25}+\frac{681910194764981}{411543420}a^{24}+\frac{39\!\cdots\!79}{1234630260}a^{23}+\frac{43\!\cdots\!73}{411543420}a^{22}+\frac{15\!\cdots\!17}{1234630260}a^{21}+\frac{17\!\cdots\!99}{411543420}a^{20}+\frac{17\!\cdots\!41}{617315130}a^{19}+\frac{36\!\cdots\!76}{34295285}a^{18}+\frac{24\!\cdots\!69}{617315130}a^{17}+\frac{35\!\cdots\!43}{205771710}a^{16}+\frac{19\!\cdots\!69}{617315130}a^{15}+\frac{34\!\cdots\!83}{205771710}a^{14}+\frac{722922781167832}{61731513}a^{13}+\frac{72\!\cdots\!63}{68590570}a^{12}+\frac{331989928769279}{1234630260}a^{11}+\frac{17\!\cdots\!37}{411543420}a^{10}-\frac{17\!\cdots\!09}{1234630260}a^{9}+\frac{42\!\cdots\!49}{411543420}a^{8}-\frac{14237553426263}{27436228}a^{7}+\frac{125454406577591}{82308684}a^{6}-\frac{46710504459917}{617315130}a^{5}+\frac{9017637340957}{68590570}a^{4}-\frac{1401054831137}{308657565}a^{3}+\frac{123963674411}{20577171}a^{2}-\frac{112660907767}{1234630260}a+\frac{45997979437}{411543420}$, $\frac{2793332517}{137181140}a^{31}+\frac{4677804151}{205771710}a^{30}+\frac{97887716909}{102885855}a^{29}+\frac{423009158011}{411543420}a^{28}+\frac{3903672153257}{205771710}a^{27}+\frac{2692323547867}{137181140}a^{26}+\frac{14525718181363}{68590570}a^{25}+\frac{85078575268157}{411543420}a^{24}+\frac{150243120183604}{102885855}a^{23}+\frac{108394400144689}{82308684}a^{22}+\frac{889578957848723}{137181140}a^{21}+\frac{718618315826923}{137181140}a^{20}+\frac{19\!\cdots\!58}{102885855}a^{19}+\frac{894420075865987}{68590570}a^{18}+\frac{73\!\cdots\!71}{205771710}a^{17}+\frac{20\!\cdots\!27}{102885855}a^{16}+\frac{30\!\cdots\!29}{68590570}a^{15}+\frac{13\!\cdots\!01}{68590570}a^{14}+\frac{15\!\cdots\!41}{411543420}a^{13}+\frac{77576392410736}{6859057}a^{12}+\frac{20\!\cdots\!42}{102885855}a^{11}+\frac{16\!\cdots\!47}{411543420}a^{10}+\frac{463685111078053}{68590570}a^{9}+\frac{121921767325219}{137181140}a^{8}+\frac{584830493817581}{411543420}a^{7}+\frac{45702299298829}{411543420}a^{6}+\frac{5912401359371}{34295285}a^{5}+\frac{265987184637}{34295285}a^{4}+\frac{4403556791897}{411543420}a^{3}+\frac{22561297747}{68590570}a^{2}+\frac{107262854263}{411543420}a+\frac{3243561349}{411543420}$, $\frac{173981365193}{137181140}a^{31}-\frac{63179968511}{137181140}a^{30}+\frac{23839558152611}{411543420}a^{29}-\frac{432635991566}{20577171}a^{28}+\frac{92294273474239}{82308684}a^{27}-\frac{27896558499609}{68590570}a^{26}+\frac{49\!\cdots\!47}{411543420}a^{25}-\frac{897590591793421}{205771710}a^{24}+\frac{10\!\cdots\!31}{137181140}a^{23}-\frac{19\!\cdots\!63}{68590570}a^{22}+\frac{11\!\cdots\!34}{34295285}a^{21}-\frac{48\!\cdots\!09}{411543420}a^{20}+\frac{17\!\cdots\!09}{205771710}a^{19}-\frac{31\!\cdots\!98}{102885855}a^{18}+\frac{50\!\cdots\!04}{34295285}a^{17}-\frac{35\!\cdots\!67}{68590570}a^{16}+\frac{10\!\cdots\!01}{68590570}a^{15}-\frac{58\!\cdots\!83}{102885855}a^{14}+\frac{46\!\cdots\!67}{411543420}a^{13}-\frac{16\!\cdots\!63}{411543420}a^{12}+\frac{72\!\cdots\!63}{137181140}a^{11}-\frac{12\!\cdots\!59}{68590570}a^{10}+\frac{21\!\cdots\!91}{137181140}a^{9}-\frac{56\!\cdots\!27}{102885855}a^{8}+\frac{61\!\cdots\!27}{205771710}a^{7}-\frac{851229075053137}{82308684}a^{6}+\frac{337990590633088}{102885855}a^{5}-\frac{116461889483189}{102885855}a^{4}+\frac{76800004399267}{411543420}a^{3}-\frac{26252991456983}{411543420}a^{2}+\frac{281935479473}{68590570}a-\frac{573806117963}{411543420}$, $\frac{3020700666553}{1234630260}a^{31}-\frac{171571747341}{68590570}a^{30}+\frac{137857256714801}{1234630260}a^{29}-\frac{3914420153537}{34295285}a^{28}+\frac{26\!\cdots\!29}{1234630260}a^{27}-\frac{454045166209043}{205771710}a^{26}+\frac{95\!\cdots\!97}{411543420}a^{25}-\frac{973121384031979}{41154342}a^{24}+\frac{18\!\cdots\!91}{1234630260}a^{23}-\frac{15\!\cdots\!91}{102885855}a^{22}+\frac{19\!\cdots\!43}{308657565}a^{21}-\frac{43\!\cdots\!89}{68590570}a^{20}+\frac{50\!\cdots\!44}{308657565}a^{19}-\frac{11\!\cdots\!37}{68590570}a^{18}+\frac{85\!\cdots\!28}{308657565}a^{17}-\frac{57\!\cdots\!19}{205771710}a^{16}+\frac{18\!\cdots\!91}{61731513}a^{15}-\frac{20\!\cdots\!97}{68590570}a^{14}+\frac{25\!\cdots\!91}{1234630260}a^{13}-\frac{14\!\cdots\!79}{68590570}a^{12}+\frac{23\!\cdots\!23}{246926052}a^{11}-\frac{20\!\cdots\!67}{205771710}a^{10}+\frac{35\!\cdots\!93}{1234630260}a^{9}-\frac{19\!\cdots\!37}{68590570}a^{8}+\frac{18\!\cdots\!49}{34295285}a^{7}-\frac{18\!\cdots\!86}{34295285}a^{6}+\frac{35\!\cdots\!43}{617315130}a^{5}-\frac{599134744692928}{102885855}a^{4}+\frac{400903366373713}{1234630260}a^{3}-\frac{66927458551237}{205771710}a^{2}+\frac{871658266655}{123463026}a-\frac{1450655962147}{205771710}$, $\frac{443316988432}{308657565}a^{31}+\frac{40627183755001}{617315130}a^{29}+\frac{789815381711321}{617315130}a^{27}+\frac{28\!\cdots\!89}{205771710}a^{25}+\frac{11\!\cdots\!97}{123463026}a^{23}+\frac{23\!\cdots\!01}{617315130}a^{21}+\frac{31\!\cdots\!31}{308657565}a^{19}+\frac{54\!\cdots\!88}{308657565}a^{17}+\frac{61\!\cdots\!89}{308657565}a^{15}+\frac{45\!\cdots\!07}{308657565}a^{13}+\frac{43\!\cdots\!93}{617315130}a^{11}+\frac{27\!\cdots\!53}{123463026}a^{9}+\frac{29\!\cdots\!79}{68590570}a^{7}+\frac{14\!\cdots\!61}{308657565}a^{5}+\frac{85036883181664}{308657565}a^{3}+\frac{3773896704613}{617315130}a$ 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:  \( 617565056600.4795 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 617565056600.4795 \cdot 18}{6\cdot\sqrt{301756981416745756154944013144439800625000000000000}}\cr\approx \mathstrut & 0.629293432948417 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^32 + 46*x^30 + 899*x^28 + 9779*x^26 + 65207*x^24 + 277346*x^22 + 763587*x^20 + 1370784*x^18 + 1627006*x^16 + 1296861*x^14 + 701262*x^12 + 257219*x^10 + 63092*x^8 + 10001*x^6 + 959*x^4 + 49*x^2 + 1)
 
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^32 + 46*x^30 + 899*x^28 + 9779*x^26 + 65207*x^24 + 277346*x^22 + 763587*x^20 + 1370784*x^18 + 1627006*x^16 + 1296861*x^14 + 701262*x^12 + 257219*x^10 + 63092*x^8 + 10001*x^6 + 959*x^4 + 49*x^2 + 1, 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^32 + 46*x^30 + 899*x^28 + 9779*x^26 + 65207*x^24 + 277346*x^22 + 763587*x^20 + 1370784*x^18 + 1627006*x^16 + 1296861*x^14 + 701262*x^12 + 257219*x^10 + 63092*x^8 + 10001*x^6 + 959*x^4 + 49*x^2 + 1);
 
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^32 + 46*x^30 + 899*x^28 + 9779*x^26 + 65207*x^24 + 277346*x^22 + 763587*x^20 + 1370784*x^18 + 1627006*x^16 + 1296861*x^14 + 701262*x^12 + 257219*x^10 + 63092*x^8 + 10001*x^6 + 959*x^4 + 49*x^2 + 1);
 
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

$C_2^6:S_4$ (as 32T96908):

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 1536
The 80 conjugacy class representatives for $C_2^6:S_4$ are not computed
Character table for $C_2^6:S_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), 4.4.22545.1, \(\Q(\sqrt{-3}, \sqrt{5})\), 8.0.12706925625.1, 8.0.166714864200.1, 8.8.4167871605000.1, 8.0.508277025.1, 8.8.12706925625.1, 8.8.166714864200.1, 8.0.4167871605000.1, 16.0.17371153715765276025000000.1, 16.0.17371153715765276025000000.3, 16.0.161465958839281640625.1, 16.0.27793845945224441640000.1, 16.0.17371153715765276025000000.2, 16.16.17371153715765276025000000.1, 16.0.17371153715765276025000000.4

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 32 siblings: data not computed
Minimal sibling: 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 R ${\href{/padicField/7.8.0.1}{8} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{8}$ ${\href{/padicField/13.6.0.1}{6} }^{4}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ ${\href{/padicField/17.8.0.1}{8} }^{4}$ ${\href{/padicField/19.3.0.1}{3} }^{8}{,}\,{\href{/padicField/19.1.0.1}{1} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{8}$ ${\href{/padicField/29.4.0.1}{4} }^{8}$ ${\href{/padicField/31.6.0.1}{6} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ ${\href{/padicField/37.6.0.1}{6} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ R ${\href{/padicField/43.6.0.1}{6} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ ${\href{/padicField/47.6.0.1}{6} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{8}$ ${\href{/padicField/59.6.0.1}{6} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$

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 2.4.6.1$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}$
2.6.0.1$x^{6} + x^{4} + x^{3} + x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.6.0.1$x^{6} + x^{4} + x^{3} + x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.6.0.1$x^{6} + x^{4} + x^{3} + x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.6.0.1$x^{6} + x^{4} + x^{3} + x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
\(3\) Copy content Toggle raw display 3.8.6.2$x^{8} + 6 x^{5} + 6 x^{4} + 18 x^{2} + 18 x + 9$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 6 x^{5} + 6 x^{4} + 18 x^{2} + 18 x + 9$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 6 x^{5} + 6 x^{4} + 18 x^{2} + 18 x + 9$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 6 x^{5} + 6 x^{4} + 18 x^{2} + 18 x + 9$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
\(5\) Copy content Toggle raw display 5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(41\) Copy content Toggle raw display 41.4.2.1$x^{4} + 1962 x^{3} + 998289 x^{2} + 35245368 x + 7080121$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 1962 x^{3} + 998289 x^{2} + 35245368 x + 7080121$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.6.0.1$x^{6} + 4 x^{4} + 33 x^{3} + 39 x^{2} + 6 x + 6$$1$$6$$0$$C_6$$[\ ]^{6}$
41.6.0.1$x^{6} + 4 x^{4} + 33 x^{3} + 39 x^{2} + 6 x + 6$$1$$6$$0$$C_6$$[\ ]^{6}$
41.6.0.1$x^{6} + 4 x^{4} + 33 x^{3} + 39 x^{2} + 6 x + 6$$1$$6$$0$$C_6$$[\ ]^{6}$
41.6.0.1$x^{6} + 4 x^{4} + 33 x^{3} + 39 x^{2} + 6 x + 6$$1$$6$$0$$C_6$$[\ ]^{6}$
\(167\) Copy content Toggle raw display 167.4.2.1$x^{4} + 332 x^{3} + 27900 x^{2} + 57104 x + 4628096$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
167.4.2.1$x^{4} + 332 x^{3} + 27900 x^{2} + 57104 x + 4628096$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
167.4.0.1$x^{4} + 3 x^{2} + 120 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
167.4.0.1$x^{4} + 3 x^{2} + 120 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
167.4.0.1$x^{4} + 3 x^{2} + 120 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
167.4.0.1$x^{4} + 3 x^{2} + 120 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
167.4.2.1$x^{4} + 332 x^{3} + 27900 x^{2} + 57104 x + 4628096$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
167.4.2.1$x^{4} + 332 x^{3} + 27900 x^{2} + 57104 x + 4628096$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$