Properties

Label 32.0.664...000.1
Degree $32$
Signature $[0, 16]$
Discriminant $6.649\times 10^{48}$
Root discriminant \(33.55\)
Ramified primes $2,3,5,11$
Class number $24$ (GRH)
Class group [2, 12] (GRH)
Galois group $D_4\times C_2^3$ (as 32T273)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 8*x^31 + 22*x^30 - 40*x^29 + 178*x^28 - 712*x^27 + 1264*x^26 + 844*x^25 - 9359*x^24 + 23304*x^23 - 31738*x^22 + 20712*x^21 + 4242*x^20 - 53384*x^19 + 309318*x^18 - 1090772*x^17 + 2225756*x^16 - 2949980*x^15 + 2866688*x^14 - 2095500*x^13 + 1173576*x^12 - 535544*x^11 + 258862*x^10 - 119572*x^9 + 42617*x^8 - 13676*x^7 + 3860*x^6 - 916*x^5 + 264*x^4 - 80*x^3 + 20*x^2 - 4*x + 1)
 
gp: K = bnfinit(y^32 - 8*y^31 + 22*y^30 - 40*y^29 + 178*y^28 - 712*y^27 + 1264*y^26 + 844*y^25 - 9359*y^24 + 23304*y^23 - 31738*y^22 + 20712*y^21 + 4242*y^20 - 53384*y^19 + 309318*y^18 - 1090772*y^17 + 2225756*y^16 - 2949980*y^15 + 2866688*y^14 - 2095500*y^13 + 1173576*y^12 - 535544*y^11 + 258862*y^10 - 119572*y^9 + 42617*y^8 - 13676*y^7 + 3860*y^6 - 916*y^5 + 264*y^4 - 80*y^3 + 20*y^2 - 4*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 8*x^31 + 22*x^30 - 40*x^29 + 178*x^28 - 712*x^27 + 1264*x^26 + 844*x^25 - 9359*x^24 + 23304*x^23 - 31738*x^22 + 20712*x^21 + 4242*x^20 - 53384*x^19 + 309318*x^18 - 1090772*x^17 + 2225756*x^16 - 2949980*x^15 + 2866688*x^14 - 2095500*x^13 + 1173576*x^12 - 535544*x^11 + 258862*x^10 - 119572*x^9 + 42617*x^8 - 13676*x^7 + 3860*x^6 - 916*x^5 + 264*x^4 - 80*x^3 + 20*x^2 - 4*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - 8*x^31 + 22*x^30 - 40*x^29 + 178*x^28 - 712*x^27 + 1264*x^26 + 844*x^25 - 9359*x^24 + 23304*x^23 - 31738*x^22 + 20712*x^21 + 4242*x^20 - 53384*x^19 + 309318*x^18 - 1090772*x^17 + 2225756*x^16 - 2949980*x^15 + 2866688*x^14 - 2095500*x^13 + 1173576*x^12 - 535544*x^11 + 258862*x^10 - 119572*x^9 + 42617*x^8 - 13676*x^7 + 3860*x^6 - 916*x^5 + 264*x^4 - 80*x^3 + 20*x^2 - 4*x + 1)
 

\( x^{32} - 8 x^{31} + 22 x^{30} - 40 x^{29} + 178 x^{28} - 712 x^{27} + 1264 x^{26} + 844 x^{25} + \cdots + 1 \) 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:   \(6649076259173893054484111016591360000000000000000\) \(\medspace = 2^{72}\cdot 3^{16}\cdot 5^{16}\cdot 11^{8}\) 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:  \(33.55\)
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:  $2^{9/4}3^{1/2}5^{1/2}11^{1/2}\approx 61.10256790565329$
Ramified primes:   \(2\), \(3\), \(5\), \(11\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $16$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{5}a^{22}-\frac{1}{5}a^{21}+\frac{2}{5}a^{20}+\frac{2}{5}a^{19}-\frac{1}{5}a^{18}+\frac{1}{5}a^{17}-\frac{1}{5}a^{16}+\frac{2}{5}a^{14}-\frac{2}{5}a^{13}+\frac{1}{5}a^{12}-\frac{2}{5}a^{11}-\frac{2}{5}a^{10}-\frac{2}{5}a^{9}+\frac{2}{5}a^{8}+\frac{2}{5}a^{7}-\frac{2}{5}a^{6}+\frac{2}{5}a^{4}+\frac{2}{5}a^{3}-\frac{2}{5}a^{2}-\frac{2}{5}a+\frac{1}{5}$, $\frac{1}{5}a^{23}+\frac{1}{5}a^{21}-\frac{1}{5}a^{20}+\frac{1}{5}a^{19}-\frac{1}{5}a^{16}+\frac{2}{5}a^{15}-\frac{1}{5}a^{13}-\frac{1}{5}a^{12}+\frac{1}{5}a^{11}+\frac{1}{5}a^{10}-\frac{1}{5}a^{8}-\frac{2}{5}a^{6}+\frac{2}{5}a^{5}-\frac{1}{5}a^{4}+\frac{1}{5}a^{2}-\frac{1}{5}a+\frac{1}{5}$, $\frac{1}{10}a^{24}+\frac{2}{5}a^{20}-\frac{1}{5}a^{19}-\frac{2}{5}a^{18}-\frac{1}{5}a^{17}-\frac{1}{5}a^{16}+\frac{1}{5}a^{14}-\frac{2}{5}a^{13}-\frac{1}{5}a^{11}+\frac{1}{5}a^{10}-\frac{2}{5}a^{9}-\frac{1}{5}a^{8}-\frac{2}{5}a^{7}+\frac{2}{5}a^{6}+\frac{2}{5}a^{5}-\frac{1}{5}a^{4}+\frac{2}{5}a^{3}-\frac{2}{5}a^{2}-\frac{1}{5}a-\frac{1}{10}$, $\frac{1}{10}a^{25}+\frac{2}{5}a^{21}-\frac{1}{5}a^{20}-\frac{2}{5}a^{19}-\frac{1}{5}a^{18}-\frac{1}{5}a^{17}+\frac{1}{5}a^{15}-\frac{2}{5}a^{14}-\frac{1}{5}a^{12}+\frac{1}{5}a^{11}-\frac{2}{5}a^{10}-\frac{1}{5}a^{9}-\frac{2}{5}a^{8}+\frac{2}{5}a^{7}+\frac{2}{5}a^{6}-\frac{1}{5}a^{5}+\frac{2}{5}a^{4}-\frac{2}{5}a^{3}-\frac{1}{5}a^{2}-\frac{1}{10}a$, $\frac{1}{4130}a^{26}+\frac{76}{2065}a^{25}+\frac{171}{4130}a^{24}-\frac{157}{2065}a^{23}-\frac{21}{295}a^{22}-\frac{185}{413}a^{21}+\frac{802}{2065}a^{20}-\frac{356}{2065}a^{19}-\frac{271}{2065}a^{18}+\frac{783}{2065}a^{17}-\frac{879}{2065}a^{16}-\frac{72}{295}a^{15}+\frac{729}{2065}a^{14}+\frac{136}{295}a^{13}+\frac{642}{2065}a^{12}+\frac{34}{413}a^{11}-\frac{393}{2065}a^{10}+\frac{272}{2065}a^{9}+\frac{43}{295}a^{8}+\frac{206}{2065}a^{7}-\frac{418}{2065}a^{6}-\frac{111}{295}a^{5}+\frac{37}{413}a^{4}+\frac{334}{2065}a^{3}-\frac{1927}{4130}a^{2}+\frac{593}{2065}a+\frac{667}{4130}$, $\frac{1}{4130}a^{27}+\frac{39}{826}a^{25}+\frac{9}{295}a^{24}+\frac{176}{2065}a^{23}-\frac{57}{2065}a^{22}-\frac{257}{2065}a^{21}-\frac{85}{413}a^{20}-\frac{135}{413}a^{19}-\frac{151}{2065}a^{18}-\frac{951}{2065}a^{17}+\frac{16}{35}a^{16}-\frac{307}{2065}a^{15}+\frac{828}{2065}a^{14}-\frac{338}{2065}a^{13}-\frac{772}{2065}a^{12}+\frac{199}{2065}a^{11}-\frac{58}{413}a^{10}+\frac{257}{2065}a^{9}+\frac{142}{413}a^{8}-\frac{151}{413}a^{7}-\frac{17}{2065}a^{6}-\frac{242}{2065}a^{5}-\frac{528}{2065}a^{4}+\frac{1439}{4130}a^{3}-\frac{396}{2065}a^{2}+\frac{1289}{4130}a+\frac{104}{413}$, $\frac{1}{20650}a^{28}-\frac{1}{20650}a^{27}+\frac{1}{10325}a^{26}+\frac{372}{10325}a^{25}-\frac{563}{20650}a^{24}+\frac{149}{2065}a^{23}+\frac{87}{10325}a^{22}-\frac{71}{175}a^{21}+\frac{154}{1475}a^{20}+\frac{674}{10325}a^{19}+\frac{719}{2065}a^{18}-\frac{131}{10325}a^{17}-\frac{4651}{10325}a^{16}-\frac{886}{2065}a^{15}-\frac{619}{2065}a^{14}+\frac{299}{1475}a^{13}-\frac{44}{413}a^{12}-\frac{37}{1475}a^{11}+\frac{2882}{10325}a^{10}-\frac{1657}{10325}a^{9}+\frac{561}{2065}a^{8}-\frac{2263}{10325}a^{7}-\frac{2151}{10325}a^{6}+\frac{3473}{10325}a^{5}+\frac{539}{2950}a^{4}+\frac{179}{20650}a^{3}-\frac{4636}{10325}a^{2}-\frac{4096}{10325}a-\frac{8349}{20650}$, $\frac{1}{12038950}a^{29}-\frac{7}{1719850}a^{28}-\frac{6}{1203895}a^{27}+\frac{19}{547225}a^{26}-\frac{37146}{1203895}a^{25}-\frac{596681}{12038950}a^{24}-\frac{194448}{6019475}a^{23}+\frac{33601}{1203895}a^{22}+\frac{73373}{171985}a^{21}-\frac{24127}{1203895}a^{20}-\frac{2538542}{6019475}a^{19}-\frac{3137}{14575}a^{18}+\frac{314836}{859925}a^{17}+\frac{1413663}{6019475}a^{16}+\frac{464197}{1203895}a^{15}-\frac{2255697}{6019475}a^{14}-\frac{2891439}{6019475}a^{13}-\frac{408262}{859925}a^{12}-\frac{2317176}{6019475}a^{11}-\frac{136459}{859925}a^{10}-\frac{2441059}{6019475}a^{9}+\frac{1923252}{6019475}a^{8}+\frac{2611883}{6019475}a^{7}+\frac{212991}{6019475}a^{6}-\frac{1379}{6490}a^{5}-\frac{274469}{2407790}a^{4}+\frac{37916}{113575}a^{3}-\frac{2444573}{6019475}a^{2}+\frac{331036}{6019475}a-\frac{860323}{12038950}$, $\frac{1}{34\!\cdots\!50}a^{30}-\frac{39\!\cdots\!22}{17\!\cdots\!25}a^{29}+\frac{72\!\cdots\!79}{49\!\cdots\!50}a^{28}-\frac{15\!\cdots\!13}{68\!\cdots\!90}a^{27}+\frac{30\!\cdots\!91}{34\!\cdots\!50}a^{26}+\frac{13\!\cdots\!83}{55\!\cdots\!75}a^{25}-\frac{33\!\cdots\!98}{68\!\cdots\!49}a^{24}+\frac{60\!\cdots\!71}{98\!\cdots\!07}a^{23}-\frac{31\!\cdots\!24}{17\!\cdots\!25}a^{22}+\frac{37\!\cdots\!19}{79\!\cdots\!25}a^{21}+\frac{60\!\cdots\!47}{17\!\cdots\!25}a^{20}-\frac{84\!\cdots\!22}{24\!\cdots\!75}a^{19}-\frac{10\!\cdots\!61}{32\!\cdots\!25}a^{18}-\frac{12\!\cdots\!79}{34\!\cdots\!45}a^{17}-\frac{62\!\cdots\!28}{17\!\cdots\!25}a^{16}-\frac{15\!\cdots\!77}{17\!\cdots\!25}a^{15}-\frac{64\!\cdots\!09}{17\!\cdots\!25}a^{14}+\frac{12\!\cdots\!58}{34\!\cdots\!45}a^{13}+\frac{79\!\cdots\!59}{17\!\cdots\!25}a^{12}+\frac{62\!\cdots\!49}{31\!\cdots\!95}a^{11}-\frac{18\!\cdots\!03}{17\!\cdots\!25}a^{10}-\frac{21\!\cdots\!69}{17\!\cdots\!25}a^{9}-\frac{23\!\cdots\!12}{17\!\cdots\!25}a^{8}+\frac{17\!\cdots\!59}{84\!\cdots\!75}a^{7}-\frac{27\!\cdots\!51}{31\!\cdots\!50}a^{6}+\frac{77\!\cdots\!24}{17\!\cdots\!25}a^{5}+\frac{39\!\cdots\!87}{16\!\cdots\!30}a^{4}+\frac{19\!\cdots\!29}{58\!\cdots\!50}a^{3}-\frac{59\!\cdots\!27}{49\!\cdots\!50}a^{2}-\frac{83\!\cdots\!87}{17\!\cdots\!25}a+\frac{11\!\cdots\!54}{17\!\cdots\!25}$, $\frac{1}{19\!\cdots\!50}a^{31}+\frac{19\!\cdots\!78}{13\!\cdots\!25}a^{30}-\frac{39\!\cdots\!79}{96\!\cdots\!75}a^{29}-\frac{24\!\cdots\!79}{19\!\cdots\!50}a^{28}-\frac{23\!\cdots\!99}{19\!\cdots\!50}a^{27}+\frac{22\!\cdots\!71}{19\!\cdots\!50}a^{26}+\frac{25\!\cdots\!73}{96\!\cdots\!75}a^{25}-\frac{47\!\cdots\!59}{96\!\cdots\!75}a^{24}-\frac{30\!\cdots\!29}{13\!\cdots\!25}a^{23}+\frac{40\!\cdots\!99}{96\!\cdots\!75}a^{22}+\frac{37\!\cdots\!81}{19\!\cdots\!55}a^{21}-\frac{10\!\cdots\!77}{58\!\cdots\!75}a^{20}-\frac{17\!\cdots\!63}{96\!\cdots\!75}a^{19}-\frac{30\!\cdots\!66}{96\!\cdots\!75}a^{18}-\frac{38\!\cdots\!82}{96\!\cdots\!75}a^{17}+\frac{65\!\cdots\!82}{13\!\cdots\!25}a^{16}+\frac{47\!\cdots\!34}{96\!\cdots\!75}a^{15}+\frac{90\!\cdots\!27}{19\!\cdots\!55}a^{14}+\frac{32\!\cdots\!27}{96\!\cdots\!75}a^{13}+\frac{15\!\cdots\!12}{96\!\cdots\!75}a^{12}+\frac{16\!\cdots\!59}{96\!\cdots\!75}a^{11}-\frac{28\!\cdots\!46}{96\!\cdots\!75}a^{10}-\frac{11\!\cdots\!96}{12\!\cdots\!25}a^{9}-\frac{11\!\cdots\!29}{96\!\cdots\!75}a^{8}+\frac{72\!\cdots\!51}{17\!\cdots\!50}a^{7}+\frac{35\!\cdots\!19}{96\!\cdots\!75}a^{6}-\frac{23\!\cdots\!01}{96\!\cdots\!75}a^{5}+\frac{25\!\cdots\!71}{19\!\cdots\!50}a^{4}-\frac{14\!\cdots\!47}{38\!\cdots\!10}a^{3}+\frac{82\!\cdots\!99}{19\!\cdots\!50}a^{2}+\frac{12\!\cdots\!67}{36\!\cdots\!35}a-\frac{41\!\cdots\!93}{96\!\cdots\!75}$ 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

$C_{2}\times C_{12}$, which has order $24$ (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{5813233793250990397555049726219043664958222326}{13843879129084764858961083036080556971441460325} a^{31} + \frac{313058792723162497869348165189543391911301676767}{96907153903593354012727581252563898800090222275} a^{30} - \frac{37054210403733855916135027276944665787622580007}{4507309483888062977336166569886692967446056850} a^{29} + \frac{1363141573795627555646626794476227343974898529287}{96907153903593354012727581252563898800090222275} a^{28} - \frac{13540841942993408602557041640135108047184002083461}{193814307807186708025455162505127797600180444550} a^{27} + \frac{53595764744693003614729576396491644755877091488007}{193814307807186708025455162505127797600180444550} a^{26} - \frac{42768372550827290812157084203186147561824293553582}{96907153903593354012727581252563898800090222275} a^{25} - \frac{49247822236294657827509483759437767783244213564157}{96907153903593354012727581252563898800090222275} a^{24} + \frac{368843992106941123511534181055872872832359940628812}{96907153903593354012727581252563898800090222275} a^{23} - \frac{166566622846549580070499961434818690426282847612934}{19381430780718670802545516250512779760018044455} a^{22} + \frac{1012396760840747374585713091184219639881990414719708}{96907153903593354012727581252563898800090222275} a^{21} - \frac{43133024291071871246068493738469773567943068001007}{8809741263963032182975234659323990800008202025} a^{20} - \frac{395447275940503425117720491653817533475422107485363}{96907153903593354012727581252563898800090222275} a^{19} + \frac{2099354223268852302892452740124769298329985158495238}{96907153903593354012727581252563898800090222275} a^{18} - \frac{11929910309621938335029349421532955824181382382849829}{96907153903593354012727581252563898800090222275} a^{17} + \frac{5798948450096472874577707663504427609376473559138033}{13843879129084764858961083036080556971441460325} a^{16} - \frac{15468404626972161061208414672302436173494312698823118}{19381430780718670802545516250512779760018044455} a^{15} + \frac{93590828169507739053280346457161095951761270523335492}{96907153903593354012727581252563898800090222275} a^{14} - \frac{82446463767698257007722316487106048698558356976218752}{96907153903593354012727581252563898800090222275} a^{13} + \frac{10559940074280426442942612294571259929802601080873907}{19381430780718670802545516250512779760018044455} a^{12} - \frac{24670440691204120787232873477468689131102953593769149}{96907153903593354012727581252563898800090222275} a^{11} + \frac{9255120962340904556387960040955490913520952281597767}{96907153903593354012727581252563898800090222275} a^{10} - \frac{988904651754516272201153935897453050299687714441863}{19381430780718670802545516250512779760018044455} a^{9} + \frac{2098497702604096510018743795660950122343799718049527}{96907153903593354012727581252563898800090222275} a^{8} - \frac{43218920416338525607714988932101176226223579612404}{8809741263963032182975234659323990800008202025} a^{7} + \frac{3048847681603319526363160213496805227585448841102}{2253654741944031488668083284943346483723028425} a^{6} - \frac{44053440887251912559993267187161402082137346322999}{193814307807186708025455162505127797600180444550} a^{5} - \frac{217049248544852947607796794169263992303004737449}{96907153903593354012727581252563898800090222275} a^{4} - \frac{728295092022594877282397190043773960422084001043}{38762861561437341605091032501025559520036088910} a^{3} + \frac{1268108014959344308335996092678611078172330091927}{193814307807186708025455162505127797600180444550} a^{2} - \frac{222631420193665279006894984012361680594655562}{1828436866105534981372218514199318845284721175} a - \frac{246257323728857193349491022855375283225169988}{1167556071127630771237681701838119262651689425} \)  (order $24$) 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:   $\frac{79\!\cdots\!67}{19\!\cdots\!50}a^{31}-\frac{31\!\cdots\!07}{96\!\cdots\!75}a^{30}+\frac{39\!\cdots\!09}{45\!\cdots\!50}a^{29}-\frac{30\!\cdots\!64}{19\!\cdots\!55}a^{28}+\frac{23\!\cdots\!99}{32\!\cdots\!50}a^{27}-\frac{55\!\cdots\!91}{19\!\cdots\!50}a^{26}+\frac{96\!\cdots\!43}{19\!\cdots\!50}a^{25}+\frac{14\!\cdots\!13}{38\!\cdots\!10}a^{24}-\frac{37\!\cdots\!04}{96\!\cdots\!75}a^{23}+\frac{12\!\cdots\!56}{13\!\cdots\!25}a^{22}-\frac{11\!\cdots\!18}{96\!\cdots\!75}a^{21}+\frac{65\!\cdots\!28}{88\!\cdots\!25}a^{20}+\frac{27\!\cdots\!83}{96\!\cdots\!75}a^{19}-\frac{21\!\cdots\!84}{96\!\cdots\!75}a^{18}+\frac{12\!\cdots\!57}{96\!\cdots\!75}a^{17}-\frac{42\!\cdots\!36}{96\!\cdots\!75}a^{16}+\frac{85\!\cdots\!84}{96\!\cdots\!75}a^{15}-\frac{22\!\cdots\!03}{19\!\cdots\!55}a^{14}+\frac{10\!\cdots\!28}{96\!\cdots\!75}a^{13}-\frac{71\!\cdots\!52}{96\!\cdots\!75}a^{12}+\frac{36\!\cdots\!83}{96\!\cdots\!75}a^{11}-\frac{14\!\cdots\!04}{96\!\cdots\!75}a^{10}+\frac{12\!\cdots\!27}{19\!\cdots\!55}a^{9}-\frac{60\!\cdots\!69}{19\!\cdots\!55}a^{8}+\frac{69\!\cdots\!05}{70\!\cdots\!62}a^{7}-\frac{44\!\cdots\!66}{22\!\cdots\!25}a^{6}+\frac{52\!\cdots\!73}{19\!\cdots\!50}a^{5}-\frac{10\!\cdots\!06}{19\!\cdots\!55}a^{4}+\frac{45\!\cdots\!41}{27\!\cdots\!50}a^{3}-\frac{22\!\cdots\!73}{19\!\cdots\!50}a^{2}+\frac{17\!\cdots\!47}{19\!\cdots\!50}a+\frac{54\!\cdots\!69}{23\!\cdots\!50}$, $\frac{33\!\cdots\!57}{21\!\cdots\!75}a^{31}-\frac{26\!\cdots\!61}{21\!\cdots\!75}a^{30}+\frac{17\!\cdots\!21}{50\!\cdots\!25}a^{29}-\frac{26\!\cdots\!83}{43\!\cdots\!50}a^{28}+\frac{59\!\cdots\!47}{21\!\cdots\!75}a^{27}-\frac{23\!\cdots\!41}{21\!\cdots\!75}a^{26}+\frac{42\!\cdots\!77}{21\!\cdots\!75}a^{25}+\frac{57\!\cdots\!72}{43\!\cdots\!35}a^{24}-\frac{31\!\cdots\!76}{21\!\cdots\!75}a^{23}+\frac{77\!\cdots\!56}{21\!\cdots\!75}a^{22}-\frac{20\!\cdots\!44}{43\!\cdots\!35}a^{21}+\frac{86\!\cdots\!96}{28\!\cdots\!75}a^{20}+\frac{18\!\cdots\!22}{21\!\cdots\!75}a^{19}-\frac{18\!\cdots\!62}{21\!\cdots\!75}a^{18}+\frac{10\!\cdots\!18}{21\!\cdots\!75}a^{17}-\frac{72\!\cdots\!66}{43\!\cdots\!35}a^{16}+\frac{10\!\cdots\!28}{31\!\cdots\!25}a^{15}-\frac{97\!\cdots\!76}{21\!\cdots\!75}a^{14}+\frac{92\!\cdots\!16}{21\!\cdots\!75}a^{13}-\frac{65\!\cdots\!14}{21\!\cdots\!75}a^{12}+\frac{34\!\cdots\!98}{21\!\cdots\!75}a^{11}-\frac{14\!\cdots\!78}{21\!\cdots\!75}a^{10}+\frac{64\!\cdots\!06}{21\!\cdots\!75}a^{9}-\frac{11\!\cdots\!12}{87\!\cdots\!27}a^{8}+\frac{84\!\cdots\!21}{19\!\cdots\!25}a^{7}-\frac{59\!\cdots\!97}{50\!\cdots\!25}a^{6}+\frac{62\!\cdots\!63}{31\!\cdots\!25}a^{5}+\frac{14\!\cdots\!19}{43\!\cdots\!50}a^{4}+\frac{28\!\cdots\!69}{21\!\cdots\!75}a^{3}-\frac{16\!\cdots\!91}{21\!\cdots\!75}a^{2}+\frac{31\!\cdots\!63}{21\!\cdots\!75}a+\frac{19\!\cdots\!87}{26\!\cdots\!25}$, $\frac{58\!\cdots\!26}{13\!\cdots\!25}a^{31}-\frac{31\!\cdots\!67}{96\!\cdots\!75}a^{30}+\frac{37\!\cdots\!07}{45\!\cdots\!50}a^{29}-\frac{13\!\cdots\!87}{96\!\cdots\!75}a^{28}+\frac{13\!\cdots\!61}{19\!\cdots\!50}a^{27}-\frac{53\!\cdots\!07}{19\!\cdots\!50}a^{26}+\frac{42\!\cdots\!82}{96\!\cdots\!75}a^{25}+\frac{49\!\cdots\!57}{96\!\cdots\!75}a^{24}-\frac{36\!\cdots\!12}{96\!\cdots\!75}a^{23}+\frac{16\!\cdots\!34}{19\!\cdots\!55}a^{22}-\frac{10\!\cdots\!08}{96\!\cdots\!75}a^{21}+\frac{43\!\cdots\!07}{88\!\cdots\!25}a^{20}+\frac{39\!\cdots\!63}{96\!\cdots\!75}a^{19}-\frac{20\!\cdots\!38}{96\!\cdots\!75}a^{18}+\frac{11\!\cdots\!29}{96\!\cdots\!75}a^{17}-\frac{57\!\cdots\!33}{13\!\cdots\!25}a^{16}+\frac{15\!\cdots\!18}{19\!\cdots\!55}a^{15}-\frac{93\!\cdots\!92}{96\!\cdots\!75}a^{14}+\frac{82\!\cdots\!52}{96\!\cdots\!75}a^{13}-\frac{10\!\cdots\!07}{19\!\cdots\!55}a^{12}+\frac{24\!\cdots\!49}{96\!\cdots\!75}a^{11}-\frac{92\!\cdots\!67}{96\!\cdots\!75}a^{10}+\frac{98\!\cdots\!63}{19\!\cdots\!55}a^{9}-\frac{20\!\cdots\!27}{96\!\cdots\!75}a^{8}+\frac{43\!\cdots\!04}{88\!\cdots\!25}a^{7}-\frac{30\!\cdots\!02}{22\!\cdots\!25}a^{6}+\frac{44\!\cdots\!99}{19\!\cdots\!50}a^{5}+\frac{21\!\cdots\!49}{96\!\cdots\!75}a^{4}+\frac{72\!\cdots\!43}{38\!\cdots\!10}a^{3}-\frac{12\!\cdots\!27}{19\!\cdots\!50}a^{2}+\frac{22\!\cdots\!62}{18\!\cdots\!75}a+\frac{14\!\cdots\!13}{11\!\cdots\!25}$, $\frac{60\!\cdots\!03}{19\!\cdots\!50}a^{31}-\frac{23\!\cdots\!74}{96\!\cdots\!75}a^{30}+\frac{11\!\cdots\!79}{19\!\cdots\!55}a^{29}-\frac{93\!\cdots\!04}{96\!\cdots\!75}a^{28}+\frac{98\!\cdots\!34}{19\!\cdots\!55}a^{27}-\frac{27\!\cdots\!79}{13\!\cdots\!25}a^{26}+\frac{58\!\cdots\!91}{19\!\cdots\!50}a^{25}+\frac{41\!\cdots\!93}{95\!\cdots\!50}a^{24}-\frac{27\!\cdots\!53}{96\!\cdots\!75}a^{23}+\frac{58\!\cdots\!79}{96\!\cdots\!75}a^{22}-\frac{66\!\cdots\!94}{96\!\cdots\!75}a^{21}+\frac{26\!\cdots\!71}{12\!\cdots\!89}a^{20}+\frac{76\!\cdots\!01}{18\!\cdots\!75}a^{19}-\frac{15\!\cdots\!09}{96\!\cdots\!75}a^{18}+\frac{12\!\cdots\!91}{13\!\cdots\!25}a^{17}-\frac{58\!\cdots\!32}{19\!\cdots\!55}a^{16}+\frac{21\!\cdots\!65}{38\!\cdots\!91}a^{15}-\frac{12\!\cdots\!71}{19\!\cdots\!55}a^{14}+\frac{15\!\cdots\!13}{31\!\cdots\!25}a^{13}-\frac{26\!\cdots\!12}{96\!\cdots\!75}a^{12}+\frac{27\!\cdots\!47}{31\!\cdots\!25}a^{11}-\frac{15\!\cdots\!59}{96\!\cdots\!75}a^{10}+\frac{14\!\cdots\!39}{96\!\cdots\!75}a^{9}-\frac{60\!\cdots\!23}{96\!\cdots\!75}a^{8}-\frac{25\!\cdots\!43}{17\!\cdots\!50}a^{7}+\frac{95\!\cdots\!89}{96\!\cdots\!75}a^{6}-\frac{42\!\cdots\!66}{19\!\cdots\!55}a^{5}+\frac{11\!\cdots\!03}{13\!\cdots\!25}a^{4}-\frac{66\!\cdots\!12}{96\!\cdots\!75}a^{3}-\frac{16\!\cdots\!13}{96\!\cdots\!75}a^{2}-\frac{43\!\cdots\!29}{19\!\cdots\!50}a+\frac{24\!\cdots\!89}{19\!\cdots\!50}$, $\frac{41\!\cdots\!79}{64\!\cdots\!25}a^{31}-\frac{32\!\cdots\!36}{64\!\cdots\!25}a^{30}+\frac{11\!\cdots\!42}{91\!\cdots\!75}a^{29}-\frac{26\!\cdots\!01}{12\!\cdots\!50}a^{28}+\frac{68\!\cdots\!09}{64\!\cdots\!25}a^{27}-\frac{10\!\cdots\!59}{25\!\cdots\!10}a^{26}+\frac{17\!\cdots\!37}{25\!\cdots\!10}a^{25}+\frac{19\!\cdots\!22}{22\!\cdots\!25}a^{24}-\frac{38\!\cdots\!34}{64\!\cdots\!25}a^{23}+\frac{84\!\cdots\!74}{64\!\cdots\!25}a^{22}-\frac{96\!\cdots\!19}{64\!\cdots\!25}a^{21}+\frac{14\!\cdots\!71}{28\!\cdots\!25}a^{20}+\frac{59\!\cdots\!88}{64\!\cdots\!25}a^{19}-\frac{31\!\cdots\!67}{91\!\cdots\!75}a^{18}+\frac{12\!\cdots\!59}{64\!\cdots\!25}a^{17}-\frac{41\!\cdots\!49}{64\!\cdots\!25}a^{16}+\frac{77\!\cdots\!43}{64\!\cdots\!25}a^{15}-\frac{87\!\cdots\!33}{64\!\cdots\!25}a^{14}+\frac{31\!\cdots\!53}{29\!\cdots\!25}a^{13}-\frac{36\!\cdots\!86}{64\!\cdots\!25}a^{12}+\frac{34\!\cdots\!77}{20\!\cdots\!75}a^{11}-\frac{12\!\cdots\!72}{91\!\cdots\!75}a^{10}+\frac{15\!\cdots\!28}{64\!\cdots\!25}a^{9}-\frac{93\!\cdots\!96}{64\!\cdots\!25}a^{8}-\frac{29\!\cdots\!37}{58\!\cdots\!75}a^{7}+\frac{19\!\cdots\!21}{64\!\cdots\!25}a^{6}-\frac{37\!\cdots\!47}{12\!\cdots\!05}a^{5}+\frac{31\!\cdots\!83}{12\!\cdots\!50}a^{4}-\frac{23\!\cdots\!92}{91\!\cdots\!75}a^{3}-\frac{15\!\cdots\!91}{12\!\cdots\!50}a^{2}-\frac{76\!\cdots\!83}{12\!\cdots\!50}a+\frac{57\!\cdots\!49}{64\!\cdots\!25}$, $\frac{12\!\cdots\!76}{11\!\cdots\!25}a^{31}-\frac{78\!\cdots\!66}{96\!\cdots\!75}a^{30}+\frac{18\!\cdots\!64}{96\!\cdots\!75}a^{29}-\frac{30\!\cdots\!07}{96\!\cdots\!75}a^{28}+\frac{34\!\cdots\!19}{19\!\cdots\!50}a^{27}-\frac{65\!\cdots\!77}{96\!\cdots\!75}a^{26}+\frac{18\!\cdots\!62}{19\!\cdots\!55}a^{25}+\frac{21\!\cdots\!84}{13\!\cdots\!25}a^{24}-\frac{12\!\cdots\!84}{13\!\cdots\!25}a^{23}+\frac{18\!\cdots\!84}{96\!\cdots\!75}a^{22}-\frac{60\!\cdots\!98}{27\!\cdots\!65}a^{21}+\frac{63\!\cdots\!56}{80\!\cdots\!75}a^{20}+\frac{14\!\cdots\!48}{13\!\cdots\!25}a^{19}-\frac{50\!\cdots\!69}{96\!\cdots\!75}a^{18}+\frac{10\!\cdots\!24}{33\!\cdots\!75}a^{17}-\frac{97\!\cdots\!26}{96\!\cdots\!75}a^{16}+\frac{17\!\cdots\!32}{96\!\cdots\!75}a^{15}-\frac{19\!\cdots\!04}{96\!\cdots\!75}a^{14}+\frac{17\!\cdots\!68}{96\!\cdots\!75}a^{13}-\frac{15\!\cdots\!37}{13\!\cdots\!25}a^{12}+\frac{55\!\cdots\!56}{96\!\cdots\!75}a^{11}-\frac{51\!\cdots\!34}{19\!\cdots\!55}a^{10}+\frac{15\!\cdots\!44}{96\!\cdots\!75}a^{9}-\frac{51\!\cdots\!86}{96\!\cdots\!75}a^{8}+\frac{15\!\cdots\!64}{88\!\cdots\!25}a^{7}-\frac{73\!\cdots\!11}{96\!\cdots\!75}a^{6}+\frac{12\!\cdots\!44}{96\!\cdots\!75}a^{5}-\frac{31\!\cdots\!29}{96\!\cdots\!75}a^{4}+\frac{20\!\cdots\!09}{19\!\cdots\!50}a^{3}-\frac{27\!\cdots\!87}{96\!\cdots\!75}a^{2}+\frac{51\!\cdots\!26}{96\!\cdots\!75}a+\frac{27\!\cdots\!77}{96\!\cdots\!75}$, $\frac{65\!\cdots\!78}{96\!\cdots\!75}a^{31}-\frac{14\!\cdots\!53}{27\!\cdots\!50}a^{30}+\frac{24\!\cdots\!13}{19\!\cdots\!50}a^{29}-\frac{20\!\cdots\!62}{96\!\cdots\!75}a^{28}+\frac{40\!\cdots\!57}{36\!\cdots\!50}a^{27}-\frac{83\!\cdots\!57}{19\!\cdots\!55}a^{26}+\frac{62\!\cdots\!54}{96\!\cdots\!75}a^{25}+\frac{30\!\cdots\!71}{33\!\cdots\!75}a^{24}-\frac{11\!\cdots\!41}{19\!\cdots\!55}a^{23}+\frac{21\!\cdots\!39}{16\!\cdots\!25}a^{22}-\frac{14\!\cdots\!14}{96\!\cdots\!75}a^{21}+\frac{19\!\cdots\!53}{30\!\cdots\!25}a^{20}+\frac{58\!\cdots\!81}{96\!\cdots\!75}a^{19}-\frac{32\!\cdots\!66}{96\!\cdots\!75}a^{18}+\frac{18\!\cdots\!46}{96\!\cdots\!75}a^{17}-\frac{17\!\cdots\!29}{27\!\cdots\!65}a^{16}+\frac{46\!\cdots\!77}{38\!\cdots\!91}a^{15}-\frac{13\!\cdots\!11}{96\!\cdots\!75}a^{14}+\frac{12\!\cdots\!56}{96\!\cdots\!75}a^{13}-\frac{79\!\cdots\!91}{96\!\cdots\!75}a^{12}+\frac{79\!\cdots\!42}{19\!\cdots\!55}a^{11}-\frac{17\!\cdots\!04}{96\!\cdots\!75}a^{10}+\frac{96\!\cdots\!03}{96\!\cdots\!75}a^{9}-\frac{14\!\cdots\!67}{38\!\cdots\!91}a^{8}+\frac{99\!\cdots\!39}{88\!\cdots\!25}a^{7}-\frac{99\!\cdots\!21}{19\!\cdots\!50}a^{6}+\frac{15\!\cdots\!69}{19\!\cdots\!50}a^{5}-\frac{87\!\cdots\!46}{44\!\cdots\!75}a^{4}+\frac{77\!\cdots\!81}{62\!\cdots\!50}a^{3}-\frac{12\!\cdots\!44}{96\!\cdots\!75}a^{2}+\frac{17\!\cdots\!23}{19\!\cdots\!55}a-\frac{11\!\cdots\!76}{96\!\cdots\!75}$, $\frac{51\!\cdots\!17}{19\!\cdots\!50}a^{31}-\frac{61\!\cdots\!51}{19\!\cdots\!50}a^{30}+\frac{27\!\cdots\!43}{19\!\cdots\!55}a^{29}-\frac{46\!\cdots\!71}{13\!\cdots\!25}a^{28}+\frac{14\!\cdots\!03}{16\!\cdots\!25}a^{27}-\frac{36\!\cdots\!57}{96\!\cdots\!75}a^{26}+\frac{10\!\cdots\!87}{96\!\cdots\!75}a^{25}-\frac{21\!\cdots\!21}{19\!\cdots\!50}a^{24}-\frac{32\!\cdots\!89}{96\!\cdots\!75}a^{23}+\frac{22\!\cdots\!42}{13\!\cdots\!25}a^{22}-\frac{31\!\cdots\!12}{96\!\cdots\!75}a^{21}+\frac{11\!\cdots\!81}{29\!\cdots\!95}a^{20}-\frac{26\!\cdots\!53}{13\!\cdots\!25}a^{19}-\frac{28\!\cdots\!24}{13\!\cdots\!25}a^{18}+\frac{38\!\cdots\!07}{27\!\cdots\!65}a^{17}-\frac{59\!\cdots\!07}{96\!\cdots\!75}a^{16}+\frac{16\!\cdots\!43}{96\!\cdots\!75}a^{15}-\frac{30\!\cdots\!12}{96\!\cdots\!75}a^{14}+\frac{36\!\cdots\!24}{96\!\cdots\!75}a^{13}-\frac{66\!\cdots\!76}{19\!\cdots\!55}a^{12}+\frac{45\!\cdots\!28}{19\!\cdots\!55}a^{11}-\frac{11\!\cdots\!19}{96\!\cdots\!75}a^{10}+\frac{66\!\cdots\!37}{13\!\cdots\!25}a^{9}-\frac{20\!\cdots\!36}{96\!\cdots\!75}a^{8}+\frac{34\!\cdots\!19}{35\!\cdots\!10}a^{7}-\frac{64\!\cdots\!63}{19\!\cdots\!50}a^{6}+\frac{86\!\cdots\!87}{96\!\cdots\!75}a^{5}-\frac{17\!\cdots\!22}{96\!\cdots\!75}a^{4}+\frac{33\!\cdots\!19}{96\!\cdots\!75}a^{3}-\frac{14\!\cdots\!92}{96\!\cdots\!75}a^{2}+\frac{49\!\cdots\!28}{96\!\cdots\!75}a-\frac{23\!\cdots\!97}{19\!\cdots\!50}$, $\frac{26\!\cdots\!11}{77\!\cdots\!75}a^{31}-\frac{73\!\cdots\!60}{25\!\cdots\!41}a^{30}+\frac{43\!\cdots\!59}{51\!\cdots\!82}a^{29}-\frac{20\!\cdots\!89}{12\!\cdots\!50}a^{28}+\frac{16\!\cdots\!07}{25\!\cdots\!10}a^{27}-\frac{33\!\cdots\!19}{12\!\cdots\!50}a^{26}+\frac{32\!\cdots\!14}{64\!\cdots\!25}a^{25}+\frac{21\!\cdots\!99}{12\!\cdots\!50}a^{24}-\frac{21\!\cdots\!18}{64\!\cdots\!25}a^{23}+\frac{58\!\cdots\!14}{64\!\cdots\!25}a^{22}-\frac{17\!\cdots\!99}{12\!\cdots\!05}a^{21}+\frac{59\!\cdots\!52}{58\!\cdots\!75}a^{20}-\frac{55\!\cdots\!71}{64\!\cdots\!25}a^{19}-\frac{24\!\cdots\!22}{12\!\cdots\!05}a^{18}+\frac{24\!\cdots\!03}{22\!\cdots\!25}a^{17}-\frac{26\!\cdots\!01}{64\!\cdots\!25}a^{16}+\frac{56\!\cdots\!83}{64\!\cdots\!25}a^{15}-\frac{79\!\cdots\!52}{64\!\cdots\!25}a^{14}+\frac{16\!\cdots\!54}{12\!\cdots\!05}a^{13}-\frac{62\!\cdots\!71}{64\!\cdots\!25}a^{12}+\frac{72\!\cdots\!76}{12\!\cdots\!05}a^{11}-\frac{16\!\cdots\!93}{64\!\cdots\!25}a^{10}+\frac{73\!\cdots\!28}{64\!\cdots\!25}a^{9}-\frac{13\!\cdots\!25}{25\!\cdots\!41}a^{8}+\frac{12\!\cdots\!66}{58\!\cdots\!75}a^{7}-\frac{38\!\cdots\!86}{64\!\cdots\!25}a^{6}+\frac{26\!\cdots\!89}{18\!\cdots\!50}a^{5}-\frac{39\!\cdots\!31}{12\!\cdots\!50}a^{4}+\frac{99\!\cdots\!27}{12\!\cdots\!50}a^{3}-\frac{13\!\cdots\!73}{51\!\cdots\!82}a^{2}+\frac{63\!\cdots\!69}{91\!\cdots\!75}a-\frac{86\!\cdots\!71}{12\!\cdots\!50}$, $\frac{58\!\cdots\!02}{19\!\cdots\!55}a^{31}-\frac{18\!\cdots\!83}{77\!\cdots\!82}a^{30}+\frac{60\!\cdots\!33}{96\!\cdots\!75}a^{29}-\frac{21\!\cdots\!31}{19\!\cdots\!50}a^{28}+\frac{86\!\cdots\!01}{16\!\cdots\!75}a^{27}-\frac{19\!\cdots\!63}{96\!\cdots\!75}a^{26}+\frac{33\!\cdots\!71}{96\!\cdots\!75}a^{25}+\frac{11\!\cdots\!11}{38\!\cdots\!94}a^{24}-\frac{12\!\cdots\!89}{44\!\cdots\!75}a^{23}+\frac{63\!\cdots\!11}{96\!\cdots\!75}a^{22}-\frac{82\!\cdots\!52}{96\!\cdots\!75}a^{21}+\frac{14\!\cdots\!71}{30\!\cdots\!25}a^{20}+\frac{42\!\cdots\!42}{19\!\cdots\!55}a^{19}-\frac{35\!\cdots\!07}{22\!\cdots\!25}a^{18}+\frac{87\!\cdots\!29}{96\!\cdots\!75}a^{17}-\frac{60\!\cdots\!41}{19\!\cdots\!55}a^{16}+\frac{22\!\cdots\!44}{36\!\cdots\!35}a^{15}-\frac{76\!\cdots\!97}{96\!\cdots\!75}a^{14}+\frac{14\!\cdots\!24}{19\!\cdots\!55}a^{13}-\frac{48\!\cdots\!49}{96\!\cdots\!75}a^{12}+\frac{24\!\cdots\!52}{96\!\cdots\!75}a^{11}-\frac{10\!\cdots\!32}{96\!\cdots\!75}a^{10}+\frac{10\!\cdots\!74}{19\!\cdots\!55}a^{9}-\frac{24\!\cdots\!18}{96\!\cdots\!75}a^{8}+\frac{17\!\cdots\!38}{20\!\cdots\!75}a^{7}-\frac{45\!\cdots\!29}{19\!\cdots\!50}a^{6}+\frac{54\!\cdots\!59}{96\!\cdots\!75}a^{5}-\frac{50\!\cdots\!21}{19\!\cdots\!50}a^{4}+\frac{69\!\cdots\!69}{96\!\cdots\!75}a^{3}-\frac{11\!\cdots\!36}{96\!\cdots\!75}a^{2}+\frac{67\!\cdots\!59}{13\!\cdots\!25}a-\frac{60\!\cdots\!23}{38\!\cdots\!10}$, $\frac{30\!\cdots\!17}{11\!\cdots\!26}a^{31}+\frac{20\!\cdots\!29}{19\!\cdots\!50}a^{30}-\frac{11\!\cdots\!01}{13\!\cdots\!25}a^{29}+\frac{44\!\cdots\!53}{19\!\cdots\!50}a^{28}-\frac{38\!\cdots\!57}{96\!\cdots\!75}a^{27}+\frac{17\!\cdots\!94}{96\!\cdots\!75}a^{26}-\frac{41\!\cdots\!87}{55\!\cdots\!30}a^{25}+\frac{30\!\cdots\!87}{23\!\cdots\!50}a^{24}+\frac{11\!\cdots\!19}{96\!\cdots\!75}a^{23}-\frac{99\!\cdots\!93}{96\!\cdots\!75}a^{22}+\frac{23\!\cdots\!46}{96\!\cdots\!75}a^{21}-\frac{54\!\cdots\!02}{17\!\cdots\!05}a^{20}+\frac{15\!\cdots\!11}{96\!\cdots\!75}a^{19}+\frac{10\!\cdots\!31}{96\!\cdots\!75}a^{18}-\frac{11\!\cdots\!91}{19\!\cdots\!55}a^{17}+\frac{63\!\cdots\!83}{19\!\cdots\!55}a^{16}-\frac{11\!\cdots\!68}{96\!\cdots\!75}a^{15}+\frac{62\!\cdots\!42}{27\!\cdots\!65}a^{14}-\frac{27\!\cdots\!61}{96\!\cdots\!75}a^{13}+\frac{56\!\cdots\!41}{22\!\cdots\!25}a^{12}-\frac{15\!\cdots\!73}{96\!\cdots\!75}a^{11}+\frac{23\!\cdots\!69}{32\!\cdots\!45}a^{10}-\frac{36\!\cdots\!76}{13\!\cdots\!25}a^{9}+\frac{13\!\cdots\!41}{96\!\cdots\!75}a^{8}-\frac{13\!\cdots\!11}{17\!\cdots\!50}a^{7}+\frac{30\!\cdots\!17}{19\!\cdots\!50}a^{6}-\frac{22\!\cdots\!22}{96\!\cdots\!75}a^{5}+\frac{28\!\cdots\!41}{27\!\cdots\!50}a^{4}-\frac{14\!\cdots\!77}{13\!\cdots\!25}a^{3}+\frac{11\!\cdots\!67}{96\!\cdots\!75}a^{2}-\frac{27\!\cdots\!19}{55\!\cdots\!30}a-\frac{27\!\cdots\!71}{38\!\cdots\!10}$, $\frac{14\!\cdots\!57}{19\!\cdots\!50}a^{31}-\frac{11\!\cdots\!79}{19\!\cdots\!50}a^{30}+\frac{48\!\cdots\!91}{27\!\cdots\!50}a^{29}-\frac{45\!\cdots\!02}{13\!\cdots\!25}a^{28}+\frac{44\!\cdots\!12}{32\!\cdots\!75}a^{27}-\frac{10\!\cdots\!77}{19\!\cdots\!50}a^{26}+\frac{10\!\cdots\!58}{96\!\cdots\!75}a^{25}+\frac{97\!\cdots\!69}{19\!\cdots\!50}a^{24}-\frac{70\!\cdots\!79}{96\!\cdots\!75}a^{23}+\frac{26\!\cdots\!08}{13\!\cdots\!25}a^{22}-\frac{25\!\cdots\!11}{96\!\cdots\!75}a^{21}+\frac{16\!\cdots\!69}{88\!\cdots\!25}a^{20}+\frac{37\!\cdots\!08}{13\!\cdots\!25}a^{19}-\frac{41\!\cdots\!29}{96\!\cdots\!75}a^{18}+\frac{13\!\cdots\!86}{55\!\cdots\!13}a^{17}-\frac{11\!\cdots\!44}{13\!\cdots\!25}a^{16}+\frac{25\!\cdots\!62}{13\!\cdots\!25}a^{15}-\frac{47\!\cdots\!18}{19\!\cdots\!55}a^{14}+\frac{23\!\cdots\!99}{96\!\cdots\!75}a^{13}-\frac{24\!\cdots\!11}{13\!\cdots\!25}a^{12}+\frac{18\!\cdots\!87}{19\!\cdots\!55}a^{11}-\frac{11\!\cdots\!64}{27\!\cdots\!65}a^{10}+\frac{17\!\cdots\!81}{96\!\cdots\!75}a^{9}-\frac{33\!\cdots\!47}{38\!\cdots\!91}a^{8}+\frac{54\!\cdots\!23}{17\!\cdots\!50}a^{7}-\frac{29\!\cdots\!03}{38\!\cdots\!10}a^{6}+\frac{66\!\cdots\!87}{38\!\cdots\!10}a^{5}-\frac{11\!\cdots\!82}{26\!\cdots\!25}a^{4}+\frac{20\!\cdots\!13}{19\!\cdots\!55}a^{3}-\frac{13\!\cdots\!77}{19\!\cdots\!50}a^{2}+\frac{12\!\cdots\!22}{13\!\cdots\!25}a-\frac{17\!\cdots\!51}{19\!\cdots\!50}$, $\frac{42\!\cdots\!17}{23\!\cdots\!50}a^{31}-\frac{26\!\cdots\!83}{19\!\cdots\!50}a^{30}+\frac{66\!\cdots\!37}{19\!\cdots\!50}a^{29}-\frac{54\!\cdots\!46}{96\!\cdots\!75}a^{28}+\frac{81\!\cdots\!97}{27\!\cdots\!50}a^{27}-\frac{45\!\cdots\!44}{38\!\cdots\!91}a^{26}+\frac{17\!\cdots\!62}{96\!\cdots\!75}a^{25}+\frac{48\!\cdots\!11}{19\!\cdots\!50}a^{24}-\frac{16\!\cdots\!48}{96\!\cdots\!75}a^{23}+\frac{34\!\cdots\!78}{96\!\cdots\!75}a^{22}-\frac{38\!\cdots\!68}{96\!\cdots\!75}a^{21}+\frac{10\!\cdots\!14}{80\!\cdots\!75}a^{20}+\frac{46\!\cdots\!51}{19\!\cdots\!55}a^{19}-\frac{91\!\cdots\!06}{96\!\cdots\!75}a^{18}+\frac{17\!\cdots\!14}{33\!\cdots\!75}a^{17}-\frac{17\!\cdots\!18}{96\!\cdots\!75}a^{16}+\frac{31\!\cdots\!49}{96\!\cdots\!75}a^{15}-\frac{71\!\cdots\!76}{19\!\cdots\!55}a^{14}+\frac{28\!\cdots\!53}{96\!\cdots\!75}a^{13}-\frac{15\!\cdots\!28}{96\!\cdots\!75}a^{12}+\frac{52\!\cdots\!51}{96\!\cdots\!75}a^{11}-\frac{10\!\cdots\!22}{96\!\cdots\!75}a^{10}+\frac{10\!\cdots\!53}{96\!\cdots\!75}a^{9}-\frac{57\!\cdots\!84}{96\!\cdots\!75}a^{8}+\frac{10\!\cdots\!99}{17\!\cdots\!50}a^{7}+\frac{30\!\cdots\!39}{19\!\cdots\!50}a^{6}+\frac{32\!\cdots\!19}{77\!\cdots\!82}a^{5}-\frac{53\!\cdots\!44}{96\!\cdots\!75}a^{4}+\frac{31\!\cdots\!73}{19\!\cdots\!50}a^{3}-\frac{14\!\cdots\!18}{19\!\cdots\!55}a^{2}-\frac{67\!\cdots\!72}{96\!\cdots\!75}a+\frac{49\!\cdots\!03}{27\!\cdots\!50}$, $\frac{17\!\cdots\!14}{61\!\cdots\!65}a^{31}-\frac{19\!\cdots\!56}{96\!\cdots\!75}a^{30}+\frac{42\!\cdots\!16}{96\!\cdots\!75}a^{29}-\frac{60\!\cdots\!31}{96\!\cdots\!75}a^{28}+\frac{14\!\cdots\!37}{36\!\cdots\!35}a^{27}-\frac{61\!\cdots\!69}{38\!\cdots\!10}a^{26}+\frac{37\!\cdots\!73}{19\!\cdots\!50}a^{25}+\frac{49\!\cdots\!77}{95\!\cdots\!50}a^{24}-\frac{23\!\cdots\!57}{96\!\cdots\!75}a^{23}+\frac{43\!\cdots\!83}{96\!\cdots\!75}a^{22}-\frac{35\!\cdots\!21}{96\!\cdots\!75}a^{21}-\frac{38\!\cdots\!06}{30\!\cdots\!25}a^{20}+\frac{11\!\cdots\!98}{19\!\cdots\!55}a^{19}-\frac{13\!\cdots\!23}{96\!\cdots\!75}a^{18}+\frac{72\!\cdots\!88}{96\!\cdots\!75}a^{17}-\frac{22\!\cdots\!52}{96\!\cdots\!75}a^{16}+\frac{36\!\cdots\!49}{96\!\cdots\!75}a^{15}-\frac{63\!\cdots\!33}{19\!\cdots\!55}a^{14}+\frac{45\!\cdots\!59}{31\!\cdots\!25}a^{13}+\frac{43\!\cdots\!06}{96\!\cdots\!75}a^{12}-\frac{38\!\cdots\!19}{31\!\cdots\!25}a^{11}+\frac{91\!\cdots\!09}{96\!\cdots\!75}a^{10}-\frac{31\!\cdots\!18}{96\!\cdots\!75}a^{9}+\frac{15\!\cdots\!97}{96\!\cdots\!75}a^{8}-\frac{94\!\cdots\!42}{80\!\cdots\!75}a^{7}+\frac{86\!\cdots\!47}{19\!\cdots\!55}a^{6}-\frac{11\!\cdots\!73}{96\!\cdots\!75}a^{5}+\frac{64\!\cdots\!68}{19\!\cdots\!55}a^{4}-\frac{70\!\cdots\!44}{96\!\cdots\!75}a^{3}+\frac{45\!\cdots\!29}{19\!\cdots\!50}a^{2}-\frac{23\!\cdots\!97}{27\!\cdots\!50}a+\frac{23\!\cdots\!87}{19\!\cdots\!50}$, $\frac{23\!\cdots\!28}{27\!\cdots\!65}a^{31}-\frac{18\!\cdots\!13}{27\!\cdots\!50}a^{30}+\frac{35\!\cdots\!63}{19\!\cdots\!50}a^{29}-\frac{29\!\cdots\!77}{95\!\cdots\!50}a^{28}+\frac{14\!\cdots\!91}{96\!\cdots\!75}a^{27}-\frac{57\!\cdots\!39}{96\!\cdots\!75}a^{26}+\frac{33\!\cdots\!42}{33\!\cdots\!75}a^{25}+\frac{92\!\cdots\!06}{96\!\cdots\!75}a^{24}-\frac{80\!\cdots\!72}{96\!\cdots\!75}a^{23}+\frac{53\!\cdots\!34}{27\!\cdots\!65}a^{22}-\frac{65\!\cdots\!06}{27\!\cdots\!65}a^{21}+\frac{92\!\cdots\!97}{88\!\cdots\!25}a^{20}+\frac{11\!\cdots\!28}{96\!\cdots\!75}a^{19}-\frac{68\!\cdots\!68}{13\!\cdots\!25}a^{18}+\frac{50\!\cdots\!82}{19\!\cdots\!55}a^{17}-\frac{87\!\cdots\!43}{96\!\cdots\!75}a^{16}+\frac{24\!\cdots\!61}{13\!\cdots\!25}a^{15}-\frac{20\!\cdots\!94}{96\!\cdots\!75}a^{14}+\frac{55\!\cdots\!23}{31\!\cdots\!25}a^{13}-\frac{18\!\cdots\!24}{19\!\cdots\!55}a^{12}+\frac{31\!\cdots\!81}{10\!\cdots\!25}a^{11}+\frac{23\!\cdots\!87}{96\!\cdots\!75}a^{10}+\frac{41\!\cdots\!77}{96\!\cdots\!75}a^{9}-\frac{36\!\cdots\!79}{19\!\cdots\!55}a^{8}-\frac{41\!\cdots\!44}{88\!\cdots\!25}a^{7}+\frac{26\!\cdots\!79}{27\!\cdots\!50}a^{6}-\frac{13\!\cdots\!87}{77\!\cdots\!82}a^{5}+\frac{13\!\cdots\!67}{27\!\cdots\!50}a^{4}-\frac{38\!\cdots\!06}{19\!\cdots\!55}a^{3}-\frac{50\!\cdots\!34}{13\!\cdots\!25}a^{2}-\frac{42\!\cdots\!77}{96\!\cdots\!75}a+\frac{13\!\cdots\!13}{18\!\cdots\!75}$ 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:  \( 184122377962.3511 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 184122377962.3511 \cdot 24}{24\cdot\sqrt{6649076259173893054484111016591360000000000000000}}\cr\approx \mathstrut & 0.421311983406428 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^32 - 8*x^31 + 22*x^30 - 40*x^29 + 178*x^28 - 712*x^27 + 1264*x^26 + 844*x^25 - 9359*x^24 + 23304*x^23 - 31738*x^22 + 20712*x^21 + 4242*x^20 - 53384*x^19 + 309318*x^18 - 1090772*x^17 + 2225756*x^16 - 2949980*x^15 + 2866688*x^14 - 2095500*x^13 + 1173576*x^12 - 535544*x^11 + 258862*x^10 - 119572*x^9 + 42617*x^8 - 13676*x^7 + 3860*x^6 - 916*x^5 + 264*x^4 - 80*x^3 + 20*x^2 - 4*x + 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 - 8*x^31 + 22*x^30 - 40*x^29 + 178*x^28 - 712*x^27 + 1264*x^26 + 844*x^25 - 9359*x^24 + 23304*x^23 - 31738*x^22 + 20712*x^21 + 4242*x^20 - 53384*x^19 + 309318*x^18 - 1090772*x^17 + 2225756*x^16 - 2949980*x^15 + 2866688*x^14 - 2095500*x^13 + 1173576*x^12 - 535544*x^11 + 258862*x^10 - 119572*x^9 + 42617*x^8 - 13676*x^7 + 3860*x^6 - 916*x^5 + 264*x^4 - 80*x^3 + 20*x^2 - 4*x + 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 - 8*x^31 + 22*x^30 - 40*x^29 + 178*x^28 - 712*x^27 + 1264*x^26 + 844*x^25 - 9359*x^24 + 23304*x^23 - 31738*x^22 + 20712*x^21 + 4242*x^20 - 53384*x^19 + 309318*x^18 - 1090772*x^17 + 2225756*x^16 - 2949980*x^15 + 2866688*x^14 - 2095500*x^13 + 1173576*x^12 - 535544*x^11 + 258862*x^10 - 119572*x^9 + 42617*x^8 - 13676*x^7 + 3860*x^6 - 916*x^5 + 264*x^4 - 80*x^3 + 20*x^2 - 4*x + 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 - 8*x^31 + 22*x^30 - 40*x^29 + 178*x^28 - 712*x^27 + 1264*x^26 + 844*x^25 - 9359*x^24 + 23304*x^23 - 31738*x^22 + 20712*x^21 + 4242*x^20 - 53384*x^19 + 309318*x^18 - 1090772*x^17 + 2225756*x^16 - 2949980*x^15 + 2866688*x^14 - 2095500*x^13 + 1173576*x^12 - 535544*x^11 + 258862*x^10 - 119572*x^9 + 42617*x^8 - 13676*x^7 + 3860*x^6 - 916*x^5 + 264*x^4 - 80*x^3 + 20*x^2 - 4*x + 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

$D_4\times C_2^3$ (as 32T273):

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 64
The 40 conjugacy class representatives for $D_4\times C_2^3$
Character table for $D_4\times C_2^3$

Intermediate fields

\(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{30}) \), 4.4.17600.1, 4.4.4400.1, 4.4.158400.1, 4.4.39600.1, 4.0.4400.1, 4.0.17600.1, 4.0.39600.1, 4.0.158400.1, \(\Q(\sqrt{3}, \sqrt{10})\), \(\Q(i, \sqrt{10})\), \(\Q(\sqrt{3}, \sqrt{-10})\), \(\Q(\zeta_{12})\), \(\Q(\sqrt{-3}, \sqrt{10})\), \(\Q(\sqrt{-3}, \sqrt{-10})\), \(\Q(i, \sqrt{30})\), \(\Q(\sqrt{3}, \sqrt{-5})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\sqrt{-2}, \sqrt{-5})\), \(\Q(\sqrt{-6}, \sqrt{10})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{6}, \sqrt{10})\), \(\Q(\sqrt{2}, \sqrt{-5})\), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(\sqrt{6}, \sqrt{-10})\), \(\Q(\sqrt{-6}, \sqrt{-10})\), \(\Q(\sqrt{-3}, \sqrt{-5})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(i, \sqrt{5})\), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{15})\), \(\Q(i, \sqrt{6})\), \(\Q(\sqrt{-5}, \sqrt{6})\), \(\Q(\sqrt{2}, \sqrt{-15})\), \(\Q(\sqrt{-2}, \sqrt{15})\), \(\Q(\sqrt{5}, \sqrt{-6})\), \(\Q(\sqrt{-5}, \sqrt{-6})\), \(\Q(\sqrt{-2}, \sqrt{-15})\), \(\Q(\sqrt{2}, \sqrt{15})\), \(\Q(\sqrt{5}, \sqrt{6})\), 8.0.3317760000.2, 8.0.3317760000.1, 8.8.3317760000.1, 8.0.40960000.1, 8.0.3317760000.5, 8.0.3317760000.6, 8.0.3317760000.7, 8.0.12960000.1, \(\Q(\zeta_{24})\), 8.0.3317760000.8, 8.0.207360000.1, 8.0.3317760000.3, 8.0.207360000.2, 8.0.3317760000.9, 8.0.3317760000.4, 8.8.401448960000.5, 8.8.25090560000.1, 8.0.25090560000.24, 8.0.401448960000.60, 8.8.4956160000.1, 8.8.401448960000.1, 8.0.4956160000.5, 8.0.401448960000.14, 8.0.4956160000.2, 8.0.4956160000.6, 8.0.401448960000.11, 8.0.401448960000.6, 8.0.25090560000.13, 8.0.25090560000.10, 8.0.1568160000.8, 8.0.1568160000.5, 8.0.4956160000.13, 8.0.309760000.3, 8.0.401448960000.31, 8.0.25090560000.4, 8.0.401448960000.55, 8.0.401448960000.19, 8.0.401448960000.47, 8.0.401448960000.43, 8.8.401448960000.2, 8.8.401448960000.3, 8.0.401448960000.22, 8.0.401448960000.53, 16.0.11007531417600000000.1, 16.16.2578580279761305600000000.1, 16.0.2578580279761305600000000.1, 16.0.393016351129600000000.2, 16.0.2578580279761305600000000.7, 16.0.2578580279761305600000000.6, 16.0.2578580279761305600000000.2, 16.0.161161267485081600000000.3, 16.0.629536201113600000000.4, 16.0.161161267485081600000000.5, 16.0.161161267485081600000000.6, 16.0.161161267485081600000000.7, 16.0.161161267485081600000000.4, 16.0.2578580279761305600000000.4, 16.0.2578580279761305600000000.3

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

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

Sibling fields

Degree 32 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 R ${\href{/padicField/7.2.0.1}{2} }^{16}$ R ${\href{/padicField/13.4.0.1}{4} }^{8}$ ${\href{/padicField/17.4.0.1}{4} }^{8}$ ${\href{/padicField/19.2.0.1}{2} }^{16}$ ${\href{/padicField/23.4.0.1}{4} }^{8}$ ${\href{/padicField/29.2.0.1}{2} }^{16}$ ${\href{/padicField/31.2.0.1}{2} }^{16}$ ${\href{/padicField/37.2.0.1}{2} }^{16}$ ${\href{/padicField/41.2.0.1}{2} }^{16}$ ${\href{/padicField/43.2.0.1}{2} }^{16}$ ${\href{/padicField/47.4.0.1}{4} }^{8}$ ${\href{/padicField/53.2.0.1}{2} }^{16}$ ${\href{/padicField/59.2.0.1}{2} }^{16}$

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 2.16.36.1$x^{16} + 8 x^{12} + 8 x^{11} + 24 x^{10} + 44 x^{8} - 16 x^{7} + 16 x^{6} - 56 x^{4} + 112 x^{3} + 196$$8$$2$$36$$D_4\times C_2$$[2, 2, 3]^{2}$
2.16.36.1$x^{16} + 8 x^{12} + 8 x^{11} + 24 x^{10} + 44 x^{8} - 16 x^{7} + 16 x^{6} - 56 x^{4} + 112 x^{3} + 196$$8$$2$$36$$D_4\times C_2$$[2, 2, 3]^{2}$
\(3\) Copy content Toggle raw display 3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(5\) 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}$
5.4.2.1$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}$
\(11\) Copy content Toggle raw display 11.2.0.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.1$x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$