Properties

Label 32.32.187...000.1
Degree $32$
Signature $[32, 0]$
Discriminant $1.871\times 10^{74}$
Root discriminant \(209.41\)
Ramified primes $2,5$
Class number $2$ (GRH)
Class group [2] (GRH)
Galois group $C_{32}$ (as 32T33)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 160*x^30 + 11600*x^28 - 504000*x^26 + 14625000*x^24 - 299000000*x^22 + 4427500000*x^20 - 48070000000*x^18 + 383057812500*x^16 - 2220625000000*x^14 + 9185312500000*x^12 - 26243750000000*x^10 + 49207031250000*x^8 - 55781250000000*x^6 + 33203125000000*x^4 - 7812500000000*x^2 + 581850781250)
 
gp: K = bnfinit(y^32 - 160*y^30 + 11600*y^28 - 504000*y^26 + 14625000*y^24 - 299000000*y^22 + 4427500000*y^20 - 48070000000*y^18 + 383057812500*y^16 - 2220625000000*y^14 + 9185312500000*y^12 - 26243750000000*y^10 + 49207031250000*y^8 - 55781250000000*y^6 + 33203125000000*y^4 - 7812500000000*y^2 + 581850781250, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 160*x^30 + 11600*x^28 - 504000*x^26 + 14625000*x^24 - 299000000*x^22 + 4427500000*x^20 - 48070000000*x^18 + 383057812500*x^16 - 2220625000000*x^14 + 9185312500000*x^12 - 26243750000000*x^10 + 49207031250000*x^8 - 55781250000000*x^6 + 33203125000000*x^4 - 7812500000000*x^2 + 581850781250);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - 160*x^30 + 11600*x^28 - 504000*x^26 + 14625000*x^24 - 299000000*x^22 + 4427500000*x^20 - 48070000000*x^18 + 383057812500*x^16 - 2220625000000*x^14 + 9185312500000*x^12 - 26243750000000*x^10 + 49207031250000*x^8 - 55781250000000*x^6 + 33203125000000*x^4 - 7812500000000*x^2 + 581850781250)
 

\( x^{32} - 160 x^{30} + 11600 x^{28} - 504000 x^{26} + 14625000 x^{24} - 299000000 x^{22} + 4427500000 x^{20} - 48070000000 x^{18} + 383057812500 x^{16} + \cdots + 581850781250 \) 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:  $[32, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(187072209578355573530071658587684226515959365500928000000000000000000000000\) \(\medspace = 2^{191}\cdot 5^{24}\) 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:  \(209.41\)
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:  $2^{191/32}5^{3/4}\approx 209.41138535741914$
Ramified primes:   \(2\), \(5\) 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:  \(\Q(\sqrt{2}) \)
$\card{ \Gal(K/\Q) }$:  $32$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(640=2^{7}\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{640}(1,·)$, $\chi_{640}(387,·)$, $\chi_{640}(9,·)$, $\chi_{640}(523,·)$, $\chi_{640}(401,·)$, $\chi_{640}(147,·)$, $\chi_{640}(409,·)$, $\chi_{640}(283,·)$, $\chi_{640}(161,·)$, $\chi_{640}(547,·)$, $\chi_{640}(169,·)$, $\chi_{640}(43,·)$, $\chi_{640}(561,·)$, $\chi_{640}(307,·)$, $\chi_{640}(569,·)$, $\chi_{640}(443,·)$, $\chi_{640}(321,·)$, $\chi_{640}(67,·)$, $\chi_{640}(329,·)$, $\chi_{640}(203,·)$, $\chi_{640}(81,·)$, $\chi_{640}(467,·)$, $\chi_{640}(89,·)$, $\chi_{640}(603,·)$, $\chi_{640}(481,·)$, $\chi_{640}(227,·)$, $\chi_{640}(489,·)$, $\chi_{640}(363,·)$, $\chi_{640}(241,·)$, $\chi_{640}(627,·)$, $\chi_{640}(249,·)$, $\chi_{640}(123,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5}a^{4}$, $\frac{1}{5}a^{5}$, $\frac{1}{5}a^{6}$, $\frac{1}{5}a^{7}$, $\frac{1}{25}a^{8}$, $\frac{1}{25}a^{9}$, $\frac{1}{25}a^{10}$, $\frac{1}{25}a^{11}$, $\frac{1}{125}a^{12}$, $\frac{1}{125}a^{13}$, $\frac{1}{125}a^{14}$, $\frac{1}{125}a^{15}$, $\frac{1}{119375}a^{16}-\frac{16}{23875}a^{14}-\frac{53}{23875}a^{12}-\frac{41}{4775}a^{10}+\frac{74}{4775}a^{8}+\frac{8}{955}a^{6}-\frac{4}{191}a^{4}+\frac{22}{191}a^{2}-\frac{87}{191}$, $\frac{1}{103020625}a^{17}-\frac{9324}{4120825}a^{15}+\frac{34327}{20604125}a^{13}-\frac{39769}{4120825}a^{11}-\frac{16352}{4120825}a^{9}+\frac{21973}{824165}a^{7}+\frac{76953}{824165}a^{5}+\frac{15684}{164833}a^{3}-\frac{58533}{164833}a$, $\frac{1}{103020625}a^{18}-\frac{18}{20604125}a^{16}-\frac{67507}{20604125}a^{14}-\frac{21067}{20604125}a^{12}-\frac{9448}{4120825}a^{10}+\frac{9028}{824165}a^{8}-\frac{36963}{824165}a^{6}+\frac{33544}{824165}a^{4}-\frac{42136}{164833}a^{2}+\frac{3}{191}$, $\frac{1}{103020625}a^{19}+\frac{22351}{20604125}a^{15}-\frac{63464}{20604125}a^{13}+\frac{37668}{4120825}a^{11}+\frac{56957}{4120825}a^{9}-\frac{37389}{824165}a^{7}+\frac{36328}{824165}a^{5}+\frac{50760}{164833}a^{3}+\frac{9275}{164833}a$, $\frac{1}{515103125}a^{20}-\frac{87}{103020625}a^{16}-\frac{49284}{20604125}a^{14}+\frac{73051}{20604125}a^{12}-\frac{24682}{4120825}a^{10}-\frac{49471}{4120825}a^{8}-\frac{40372}{824165}a^{6}+\frac{5021}{824165}a^{4}+\frac{2718}{164833}a^{2}-\frac{30}{191}$, $\frac{1}{515103125}a^{21}+\frac{15601}{20604125}a^{15}-\frac{72327}{20604125}a^{13}-\frac{23092}{4120825}a^{11}+\frac{11402}{4120825}a^{9}+\frac{58116}{824165}a^{7}-\frac{58221}{824165}a^{5}+\frac{48562}{164833}a^{3}-\frac{8438}{164833}a$, $\frac{1}{515103125}a^{22}+\frac{67}{20604125}a^{16}+\frac{16562}{20604125}a^{14}+\frac{45058}{20604125}a^{12}+\frac{12809}{824165}a^{10}-\frac{17511}{4120825}a^{8}-\frac{20249}{824165}a^{6}-\frac{16953}{824165}a^{4}-\frac{68848}{164833}a^{2}-\frac{1}{191}$, $\frac{1}{515103125}a^{23}-\frac{24873}{20604125}a^{15}-\frac{16202}{4120825}a^{13}+\frac{35187}{4120825}a^{11}+\frac{4184}{824165}a^{9}+\frac{36281}{824165}a^{7}-\frac{16452}{164833}a^{5}-\frac{48332}{164833}a^{3}-\frac{7435}{164833}a$, $\frac{1}{2575515625}a^{24}+\frac{154}{103020625}a^{16}+\frac{15573}{4120825}a^{14}+\frac{5484}{4120825}a^{12}-\frac{1317}{164833}a^{10}+\frac{75116}{4120825}a^{8}+\frac{18931}{824165}a^{6}-\frac{54373}{824165}a^{4}+\frac{54608}{164833}a^{2}-\frac{40}{191}$, $\frac{1}{2575515625}a^{25}+\frac{4693}{20604125}a^{15}+\frac{15718}{20604125}a^{13}-\frac{1464}{824165}a^{11}-\frac{44004}{4120825}a^{9}-\frac{68251}{824165}a^{7}-\frac{37159}{824165}a^{5}-\frac{53066}{164833}a^{3}+\frac{78580}{164833}a$, $\frac{1}{2575515625}a^{26}+\frac{164}{103020625}a^{16}+\frac{58868}{20604125}a^{14}+\frac{44522}{20604125}a^{12}-\frac{77661}{4120825}a^{10}+\frac{3092}{164833}a^{8}-\frac{58734}{824165}a^{6}+\frac{35857}{824165}a^{4}+\frac{60457}{164833}a^{2}+\frac{57}{191}$, $\frac{1}{2575515625}a^{27}-\frac{42603}{20604125}a^{15}+\frac{19216}{20604125}a^{13}+\frac{15968}{4120825}a^{11}-\frac{43133}{4120825}a^{9}-\frac{7196}{164833}a^{7}-\frac{57127}{824165}a^{5}-\frac{39224}{164833}a^{3}-\frac{76544}{164833}a$, $\frac{1}{12877578125}a^{28}-\frac{316}{103020625}a^{16}-\frac{3174}{824165}a^{14}-\frac{82414}{20604125}a^{12}-\frac{61097}{4120825}a^{10}-\frac{38569}{4120825}a^{8}-\frac{35762}{824165}a^{6}-\frac{60799}{824165}a^{4}+\frac{24907}{164833}a^{2}-\frac{61}{191}$, $\frac{1}{12877578125}a^{29}+\frac{948}{824165}a^{15}+\frac{50773}{20604125}a^{13}+\frac{12808}{824165}a^{11}+\frac{13771}{824165}a^{9}-\frac{16}{863}a^{7}+\frac{25898}{824165}a^{5}+\frac{36061}{164833}a^{3}+\frac{77058}{164833}a$, $\frac{1}{12877578125}a^{30}+\frac{269}{103020625}a^{16}-\frac{35527}{20604125}a^{14}-\frac{6877}{20604125}a^{12}-\frac{28664}{4120825}a^{10}+\frac{75488}{4120825}a^{8}+\frac{69048}{824165}a^{6}+\frac{14486}{164833}a^{4}-\frac{51529}{164833}a^{2}+\frac{77}{191}$, $\frac{1}{12877578125}a^{31}-\frac{4411}{4120825}a^{15}-\frac{10192}{20604125}a^{13}-\frac{44948}{4120825}a^{11}+\frac{4737}{824165}a^{9}-\frac{72534}{824165}a^{7}-\frac{23802}{824165}a^{5}+\frac{15133}{164833}a^{3}-\frac{12140}{164833}a$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

$C_{2}$, which has order $2$ (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:  $31$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) 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:   $\frac{1}{119375}a^{16}-\frac{16}{23875}a^{14}+\frac{104}{4775}a^{12}-\frac{352}{955}a^{10}+\frac{660}{191}a^{8}-\frac{3360}{191}a^{6}+\frac{8400}{191}a^{4}-\frac{8000}{191}a^{2}+\frac{1441}{191}$, $\frac{24}{2575515625}a^{24}-\frac{576}{515103125}a^{22}+\frac{6048}{103020625}a^{20}-\frac{7296}{4120825}a^{18}+\frac{139536}{4120825}a^{16}-\frac{352512}{824165}a^{14}+\frac{594048}{164833}a^{12}-\frac{3294720}{164833}a^{10}+\frac{289586753}{4120825}a^{8}-\frac{120214024}{824165}a^{6}+\frac{25975060}{164833}a^{4}-\frac{11740240}{164833}a^{2}+\frac{1741}{191}$, $\frac{4}{12877578125}a^{28}-\frac{112}{2575515625}a^{26}+\frac{56}{20604125}a^{24}-\frac{10304}{103020625}a^{22}+\frac{49588}{20604125}a^{20}-\frac{163856}{4120825}a^{18}+\frac{379848}{824165}a^{16}-\frac{620160}{164833}a^{14}+\frac{3527160}{164833}a^{12}-\frac{13613600}{164833}a^{10}+\frac{34034000}{164833}a^{8}-\frac{50960000}{164833}a^{6}+\frac{199138943}{824165}a^{4}-\frac{12555772}{164833}a^{2}+\frac{1801}{191}$, $\frac{3}{12877578125}a^{28}-\frac{84}{2575515625}a^{26}+\frac{42}{20604125}a^{24}-\frac{7728}{103020625}a^{22}+\frac{37191}{20604125}a^{20}-\frac{122892}{4120825}a^{18}+\frac{284886}{824165}a^{16}-\frac{465120}{164833}a^{14}+\frac{2645370}{164833}a^{12}-\frac{10210200}{164833}a^{10}+\frac{25525500}{164833}a^{8}-\frac{38220000}{164833}a^{6}+\frac{149312999}{824165}a^{4}-\frac{9251996}{164833}a^{2}+\frac{539}{191}$, $\frac{117}{515103125}a^{20}-\frac{468}{20604125}a^{18}+\frac{3978}{4120825}a^{16}-\frac{3744}{164833}a^{14}+\frac{6657491}{20604125}a^{12}-\frac{11749092}{4120825}a^{10}+\frac{12716514}{824165}a^{8}-\frac{8070992}{164833}a^{6}+\frac{13701525}{164833}a^{4}-\frac{10116900}{164833}a^{2}+\frac{1559}{191}$, $\frac{117}{515103125}a^{20}-\frac{468}{20604125}a^{18}+\frac{3978}{4120825}a^{16}-\frac{3744}{164833}a^{14}+\frac{6657491}{20604125}a^{12}-\frac{11749092}{4120825}a^{10}+\frac{12716514}{824165}a^{8}-\frac{8070992}{164833}a^{6}+\frac{13701525}{164833}a^{4}-\frac{10116900}{164833}a^{2}+\frac{1941}{191}$, $\frac{44}{515103125}a^{20}-\frac{176}{20604125}a^{18}+\frac{1496}{4120825}a^{16}-\frac{1408}{164833}a^{14}+\frac{2502263}{20604125}a^{12}-\frac{4401556}{4120825}a^{10}+\frac{4706202}{824165}a^{8}-\frac{2877456}{164833}a^{6}+\frac{4413075}{164833}a^{4}-\frac{2536700}{164833}a^{2}+\frac{59}{191}$, $\frac{2}{2575515625}a^{26}-\frac{52}{515103125}a^{24}+\frac{598}{103020625}a^{22}-\frac{100056}{515103125}a^{20}+\frac{86274}{20604125}a^{18}-\frac{250444}{4120825}a^{16}+\frac{99368}{164833}a^{14}-\frac{83687737}{20604125}a^{12}+\frac{74605944}{4120825}a^{10}-\frac{41768798}{824165}a^{8}+\frac{66935131}{824165}a^{6}-\frac{10321391}{164833}a^{4}+\frac{2346795}{164833}a^{2}-\frac{173}{191}$, $\frac{3}{12877578125}a^{28}-\frac{84}{2575515625}a^{26}+\frac{42}{20604125}a^{24}-\frac{7728}{103020625}a^{22}+\frac{37191}{20604125}a^{20}-\frac{3072271}{103020625}a^{18}+\frac{7121628}{20604125}a^{16}-\frac{58119707}{20604125}a^{14}+\frac{66045028}{4120825}a^{12}-\frac{50809099}{824165}a^{10}+\frac{25116330}{164833}a^{8}-\frac{36159690}{164833}a^{6}+\frac{121934499}{824165}a^{4}-\frac{3386121}{164833}a^{2}+\frac{53}{191}$, $\frac{38}{515103125}a^{22}-\frac{4063}{515103125}a^{20}+\frac{7474}{20604125}a^{18}-\frac{38658}{4120825}a^{16}+\frac{24680}{164833}a^{14}-\frac{31380509}{20604125}a^{12}+\frac{40559627}{4120825}a^{10}-\frac{32178676}{824165}a^{8}+\frac{14570673}{164833}a^{6}-\frac{16653225}{164833}a^{4}+\frac{8294975}{164833}a^{2}-\frac{927}{191}$, $\frac{2}{12877578125}a^{30}-\frac{304}{12877578125}a^{28}+\frac{4162}{2575515625}a^{26}-\frac{1356}{20604125}a^{24}+\frac{182804}{103020625}a^{22}-\frac{687148}{20604125}a^{20}+\frac{1846306}{4120825}a^{18}-\frac{3577548}{824165}a^{16}+\frac{4980660}{164833}a^{14}-\frac{24522160}{164833}a^{12}+\frac{82897100}{164833}a^{10}-\frac{183209000}{164833}a^{8}+\frac{244335000}{164833}a^{6}-\frac{855388943}{824165}a^{4}+\frac{47820713}{164833}a^{2}-\frac{4107}{191}$, $\frac{41}{515103125}a^{22}-\frac{4627}{515103125}a^{20}+\frac{9037}{20604125}a^{18}-\frac{9996}{824165}a^{16}+\frac{34412}{164833}a^{14}-\frac{47698491}{20604125}a^{12}+\frac{68183109}{4120825}a^{10}-\frac{61112684}{824165}a^{8}+\frac{32348337}{164833}a^{6}-\frac{45368275}{164833}a^{4}+\frac{27271275}{164833}a^{2}-\frac{3427}{191}$, $\frac{3}{12877578125}a^{28}-\frac{84}{2575515625}a^{26}+\frac{42}{20604125}a^{24}-\frac{7728}{103020625}a^{22}+\frac{37191}{20604125}a^{20}-\frac{3072022}{103020625}a^{18}+\frac{7117146}{20604125}a^{16}-\frac{57951151}{20604125}a^{14}+\frac{65358524}{4120825}a^{12}-\frac{49169747}{824165}a^{10}+\frac{22796730}{164833}a^{8}-\frac{26824770}{164833}a^{6}+\frac{26112499}{824165}a^{4}+\frac{12165629}{164833}a^{2}-\frac{3593}{191}$, $\frac{11}{2575515625}a^{26}-\frac{286}{515103125}a^{24}+\frac{3289}{103020625}a^{22}-\frac{638}{596875}a^{20}+\frac{475651}{20604125}a^{18}-\frac{1387166}{4120825}a^{16}+\frac{555676}{164833}a^{14}-\frac{476547263}{20604125}a^{12}+\frac{438942806}{4120825}a^{10}-\frac{260318702}{824165}a^{8}+\frac{461742957}{824165}a^{6}-\frac{85947137}{164833}a^{4}+\frac{33104665}{164833}a^{2}-\frac{4773}{191}$, $\frac{1}{12877578125}a^{30}-\frac{146}{12877578125}a^{28}+\frac{1913}{2575515625}a^{26}-\frac{594}{20604125}a^{24}+\frac{75946}{103020625}a^{22}-\frac{269192}{20604125}a^{20}+\frac{677369}{4120825}a^{18}-\frac{1219002}{824165}a^{16}+\frac{1560090}{164833}a^{14}-\frac{6970340}{164833}a^{12}+\frac{21028150}{164833}a^{10}-\frac{40553500}{164833}a^{8}+\frac{45727500}{164833}a^{6}-\frac{128986057}{824165}a^{4}+\frac{5159115}{164833}a^{2}-\frac{307}{191}$, $\frac{3}{12877578125}a^{30}+\frac{3}{12877578125}a^{29}-\frac{454}{12877578125}a^{28}-\frac{87}{2575515625}a^{27}+\frac{6187}{2575515625}a^{26}+\frac{1131}{515103125}a^{25}-\frac{2006}{20604125}a^{24}-\frac{348}{4120825}a^{23}+\frac{269054}{103020625}a^{22}+\frac{44022}{20604125}a^{21}-\frac{1005928}{20604125}a^{20}-\frac{154077}{4120825}a^{19}+\frac{2687531}{4120825}a^{18}+\frac{381843}{824165}a^{17}-\frac{5176398}{824165}a^{16}-\frac{674424}{164833}a^{15}+\frac{7160910}{164833}a^{14}+\frac{4215150}{164833}a^{13}-\frac{35019660}{164833}a^{12}-\frac{18265650}{164833}a^{11}+\frac{117538850}{164833}a^{10}+\frac{52874250}{164833}a^{9}-\frac{257796500}{164833}a^{8}-\frac{96135000}{164833}a^{7}+\frac{341022500}{164833}a^{6}+\frac{98962500}{164833}a^{5}-\frac{1183513943}{824165}a^{4}-\frac{47562001}{164833}a^{3}+\frac{65535600}{164833}a^{2}+\frac{6719848}{164833}a-\frac{6979}{191}$, $\frac{3}{12877578125}a^{28}-\frac{3}{2575515625}a^{27}-\frac{84}{2575515625}a^{26}+\frac{81}{515103125}a^{25}+\frac{42}{20604125}a^{24}-\frac{4853}{515103125}a^{23}-\frac{7728}{103020625}a^{22}+\frac{33994}{103020625}a^{21}+\frac{37191}{20604125}a^{20}-\frac{30863}{4120825}a^{19}-\frac{3071993}{103020625}a^{18}+\frac{475608}{4120825}a^{17}+\frac{7116624}{20604125}a^{16}-\frac{1013676}{824165}a^{15}-\frac{57930858}{20604125}a^{14}+\frac{1493144}{164833}a^{13}+\frac{65269302}{4120825}a^{12}-\frac{7469410}{164833}a^{11}-\frac{48927846}{824165}a^{10}+\frac{611782171}{4120825}a^{9}+\frac{22387560}{164833}a^{8}-\frac{246499539}{824165}a^{7}-\frac{24764460}{164833}a^{6}+\frac{274942586}{824165}a^{5}-\frac{1266001}{824165}a^{4}-\frac{28785655}{164833}a^{3}+\frac{18031504}{164833}a^{2}+\frac{5726000}{164833}a-\frac{4079}{191}$, $\frac{2}{12877578125}a^{31}+\frac{1}{12877578125}a^{30}-\frac{306}{12877578125}a^{29}-\frac{33}{2575515625}a^{28}+\frac{4213}{2575515625}a^{27}+\frac{2438}{2575515625}a^{26}-\frac{172301}{2575515625}a^{25}-\frac{106566}{2575515625}a^{24}+\frac{931087}{515103125}a^{23}+\frac{613368}{515103125}a^{22}-\frac{17481847}{515103125}a^{21}-\frac{12233231}{515103125}a^{20}+\frac{46706982}{103020625}a^{19}+\frac{55445}{164833}a^{18}-\frac{467853}{107875}a^{17}-\frac{14047576}{4120825}a^{16}+\frac{121497504}{4120825}a^{15}+\frac{20252778}{824165}a^{14}-\frac{22958072}{164833}a^{13}-\frac{2546079487}{20604125}a^{12}+\frac{1805770659}{4120825}a^{11}+\frac{1722588927}{4120825}a^{10}-\frac{3504111324}{4120825}a^{9}-\frac{3686963887}{4120825}a^{8}+\frac{716016194}{824165}a^{7}+\frac{898238616}{824165}a^{6}-\frac{44155046}{164833}a^{5}-\frac{100817984}{164833}a^{4}-\frac{24065947}{164833}a^{3}+\frac{11742577}{164833}a^{2}+\frac{6944010}{164833}a+\frac{1205}{191}$, $\frac{79}{515103125}a^{22}+\frac{44}{515103125}a^{21}-\frac{8573}{515103125}a^{20}-\frac{924}{103020625}a^{19}+\frac{16043}{20604125}a^{18}+\frac{8316}{20604125}a^{17}-\frac{16932}{824165}a^{16}-\frac{41888}{4120825}a^{15}+\frac{55348}{164833}a^{14}+\frac{25872}{164833}a^{13}-\frac{72421509}{20604125}a^{12}-\frac{6306537}{4120825}a^{11}+\frac{96993644}{4120825}a^{10}+\frac{7710307}{824165}a^{9}-\frac{80574846}{824165}a^{8}-\frac{5673228}{164833}a^{7}+\frac{38848018}{164833}a^{6}+\frac{11525745}{164833}a^{5}-\frac{48319975}{164833}a^{4}-\frac{10913375}{164833}a^{3}+\frac{25449350}{164833}a^{2}+\frac{3048542}{164833}a-\frac{3941}{191}$, $\frac{1}{12877578125}a^{31}-\frac{159}{12877578125}a^{29}+\frac{3}{12877578125}a^{28}+\frac{2297}{2575515625}a^{27}-\frac{86}{2575515625}a^{26}-\frac{3988}{103020625}a^{25}+\frac{1126}{515103125}a^{24}+\frac{579577}{515103125}a^{23}-\frac{44551}{515103125}a^{22}-\frac{11898247}{515103125}a^{21}+\frac{237117}{103020625}a^{20}+\frac{35455561}{103020625}a^{19}-\frac{891996}{20604125}a^{18}-\frac{77565749}{20604125}a^{17}+\frac{60554813}{103020625}a^{16}+\frac{124554262}{4120825}a^{15}-\frac{119046299}{20604125}a^{14}-\frac{726201757}{4120825}a^{13}+\frac{167566876}{4120825}a^{12}+\frac{3007007514}{4120825}a^{11}-\frac{824077432}{4120825}a^{10}-\frac{8520149949}{4120825}a^{9}+\frac{544266873}{824165}a^{8}+\frac{3111208241}{824165}a^{7}-\frac{1133631506}{824165}a^{6}-\frac{660452771}{164833}a^{5}+\frac{1340721079}{824165}a^{4}+\frac{330547982}{164833}a^{3}-\frac{145596191}{164833}a^{2}-\frac{38129736}{164833}a+\frac{19625}{191}$, $\frac{2}{12877578125}a^{31}+\frac{1}{12877578125}a^{30}-\frac{62}{2575515625}a^{29}-\frac{6}{515103125}a^{28}+\frac{4336}{2575515625}a^{27}+\frac{408}{515103125}a^{26}-\frac{36151}{515103125}a^{25}-\frac{16633}{515103125}a^{24}+\frac{999621}{515103125}a^{23}+\frac{90567}{103020625}a^{22}-\frac{3860283}{103020625}a^{21}-\frac{8676106}{515103125}a^{20}+\frac{53352767}{103020625}a^{19}+\frac{24005649}{103020625}a^{18}-\frac{532082417}{103020625}a^{17}-\frac{241973823}{103020625}a^{16}+\frac{761823343}{20604125}a^{15}+\frac{354014851}{20604125}a^{14}-\frac{3843006374}{20604125}a^{13}-\frac{1848715107}{20604125}a^{12}+\frac{2637633287}{4120825}a^{11}+\frac{1336806829}{4120825}a^{10}-\frac{5797100053}{4120825}a^{9}-\frac{637217877}{824165}a^{8}+\frac{1465957824}{824165}a^{7}+\frac{928263829}{824165}a^{6}-\frac{849553993}{824165}a^{5}-\frac{146372719}{164833}a^{4}+\frac{18715280}{164833}a^{3}+\frac{48414617}{164833}a^{2}+\frac{67980}{164833}a-\frac{5095}{191}$, $\frac{1}{12877578125}a^{31}+\frac{2}{12877578125}a^{30}-\frac{159}{12877578125}a^{29}-\frac{12}{515103125}a^{28}+\frac{2286}{2575515625}a^{27}+\frac{814}{515103125}a^{26}-\frac{98222}{2575515625}a^{25}-\frac{33027}{515103125}a^{24}+\frac{561884}{515103125}a^{23}+\frac{178648}{103020625}a^{22}-\frac{11275943}{515103125}a^{21}-\frac{16984056}{515103125}a^{20}+\frac{32598043}{103020625}a^{19}+\frac{46648592}{103020625}a^{18}-\frac{68567953}{20604125}a^{17}-\frac{467721467}{103020625}a^{16}+\frac{523463783}{20604125}a^{15}+\frac{683505843}{20604125}a^{14}-\frac{114374633}{824165}a^{13}-\frac{3587735347}{20604125}a^{12}+\frac{2172066241}{4120825}a^{11}+\frac{2626955799}{4120825}a^{10}-\frac{1092902693}{824165}a^{9}-\frac{1273896719}{824165}a^{8}+\frac{1680338614}{824165}a^{7}+\frac{1869988953}{824165}a^{6}-\frac{275404483}{164833}a^{5}-\frac{281848383}{164833}a^{4}+\frac{95883212}{164833}a^{3}+\frac{80527451}{164833}a^{2}-\frac{10638536}{164833}a-\frac{9165}{191}$, $\frac{3}{2575515625}a^{27}-\frac{81}{515103125}a^{25}+\frac{972}{103020625}a^{23}+\frac{3}{515103125}a^{22}-\frac{6831}{20604125}a^{21}-\frac{447}{515103125}a^{20}+\frac{6237}{824165}a^{19}+\frac{219}{4120825}a^{18}-\frac{12119961}{103020625}a^{17}-\frac{7344}{4120825}a^{16}+\frac{5233637}{4120825}a^{15}+\frac{5988}{164833}a^{14}-\frac{39276579}{4120825}a^{13}-\frac{9660491}{20604125}a^{12}+\frac{40378332}{824165}a^{11}+\frac{3174878}{824165}a^{10}-\frac{27517985}{164833}a^{9}-\frac{16217494}{824165}a^{8}+\frac{58614210}{164833}a^{7}+\frac{9706672}{164833}a^{6}-\frac{350586751}{824165}a^{5}-\frac{15013525}{164833}a^{4}+\frac{37655505}{164833}a^{3}+\frac{8859400}{164833}a^{2}-\frac{4554400}{164833}a-\frac{941}{191}$, $\frac{9}{12877578125}a^{29}-\frac{261}{2575515625}a^{27}-\frac{7}{515103125}a^{26}+\frac{3393}{515103125}a^{25}+\frac{4557}{2575515625}a^{24}-\frac{130459}{515103125}a^{23}-\frac{52493}{515103125}a^{22}+\frac{3296818}{515103125}a^{21}+\frac{1760306}{515103125}a^{20}-\frac{11506168}{103020625}a^{19}-\frac{7612017}{103020625}a^{18}+\frac{141736113}{103020625}a^{17}+\frac{22198356}{20604125}a^{16}-\frac{49506482}{4120825}a^{15}-\frac{221773051}{20604125}a^{14}+\frac{303130492}{4120825}a^{13}+\frac{1511090792}{20604125}a^{12}-\frac{1266334021}{4120825}a^{11}-\frac{1370839649}{4120825}a^{10}+\frac{686682218}{824165}a^{9}+\frac{3941626861}{4120825}a^{8}-\frac{221541187}{164833}a^{7}-\frac{263041396}{164833}a^{6}+\frac{181722930}{164833}a^{5}+\frac{215931522}{164833}a^{4}-\frac{57111795}{164833}a^{3}-\frac{63923245}{164833}a^{2}+\frac{4778050}{164833}a+\frac{6345}{191}$, $\frac{1}{12877578125}a^{31}+\frac{1}{12877578125}a^{30}-\frac{174}{12877578125}a^{29}-\frac{153}{12877578125}a^{28}+\frac{2729}{2575515625}a^{27}+\frac{2137}{2575515625}a^{26}-\frac{25514}{515103125}a^{25}-\frac{90253}{2575515625}a^{24}+\frac{158451}{103020625}a^{23}+\frac{514462}{515103125}a^{22}-\frac{17234504}{515103125}a^{21}-\frac{10443816}{515103125}a^{20}+\frac{53977384}{103020625}a^{19}+\frac{6205806}{20604125}a^{18}-\frac{24620827}{4120825}a^{17}-\frac{13630802}{4120825}a^{16}+\frac{1022141537}{20604125}a^{15}+\frac{551964841}{20604125}a^{14}-\frac{244573757}{824165}a^{13}-\frac{3250640627}{20604125}a^{12}+\frac{1030907787}{824165}a^{11}+\frac{2711593469}{4120825}a^{10}-\frac{14748916534}{4120825}a^{9}-\frac{7678243734}{4120825}a^{8}+\frac{5392024556}{824165}a^{7}+\frac{2762746609}{824165}a^{6}-\frac{5674393079}{824165}a^{5}-\frac{2829184134}{824165}a^{4}+\frac{557426274}{164833}a^{3}+\frac{266748410}{164833}a^{2}-\frac{61380416}{164833}a-\frac{32441}{191}$, $\frac{3}{12877578125}a^{29}-\frac{87}{2575515625}a^{27}-\frac{9}{2575515625}a^{26}+\frac{1131}{515103125}a^{25}+\frac{234}{515103125}a^{24}-\frac{43524}{515103125}a^{23}-\frac{2691}{103020625}a^{22}+\frac{220662}{103020625}a^{21}+\frac{450406}{515103125}a^{20}-\frac{155181}{4120825}a^{19}-\frac{388849}{20604125}a^{18}+\frac{1940679}{4120825}a^{17}+\frac{1132234}{4120825}a^{16}-\frac{3484728}{824165}a^{15}-\frac{452084}{164833}a^{14}+\frac{4477902}{164833}a^{13}+\frac{385352737}{20604125}a^{12}-\frac{20274930}{164833}a^{11}-\frac{351132194}{4120825}a^{10}+\frac{1568519497}{4120825}a^{9}+\frac{204431298}{824165}a^{8}-\frac{628574223}{824165}a^{7}-\frac{351645986}{824165}a^{6}+\frac{147980169}{164833}a^{5}+\frac{326631}{863}a^{4}-\frac{83749051}{164833}a^{3}-\frac{23147770}{164833}a^{2}+\frac{12535590}{164833}a+\frac{4041}{191}$, $\frac{12}{12877578125}a^{28}-\frac{2}{2575515625}a^{27}-\frac{336}{2575515625}a^{26}+\frac{54}{515103125}a^{25}+\frac{20976}{2575515625}a^{24}-\frac{3223}{515103125}a^{23}-\frac{6161}{20604125}a^{22}+\frac{22379}{103020625}a^{21}+\frac{738674}{103020625}a^{20}-\frac{20008}{4120825}a^{19}-\frac{12149396}{103020625}a^{18}+\frac{300903}{4120825}a^{17}+\frac{28016448}{20604125}a^{16}-\frac{617916}{824165}a^{15}-\frac{227412144}{20604125}a^{14}+\frac{860404}{164833}a^{13}+\frac{257207496}{4120825}a^{12}-\frac{3947060}{164833}a^{11}-\frac{986889257}{4120825}a^{10}+\frac{281085832}{4120825}a^{9}+\frac{2446954097}{4120825}a^{8}-\frac{88226238}{824165}a^{7}-\frac{718898101}{824165}a^{6}+\frac{11171303}{164833}a^{5}+\frac{529888446}{824165}a^{4}+\frac{847975}{164833}a^{3}-\frac{26870369}{164833}a^{2}-\frac{622758}{164833}a+\frac{2075}{191}$, $\frac{1}{12877578125}a^{30}-\frac{146}{12877578125}a^{28}+\frac{1913}{2575515625}a^{26}-\frac{594}{20604125}a^{24}+\frac{75946}{103020625}a^{22}-\frac{269192}{20604125}a^{20}+\frac{44}{103020625}a^{19}+\frac{677369}{4120825}a^{18}-\frac{3844}{103020625}a^{17}-\frac{1219002}{824165}a^{16}+\frac{5651}{4120825}a^{15}+\frac{1560090}{164833}a^{14}-\frac{571342}{20604125}a^{13}-\frac{6970340}{164833}a^{12}+\frac{1399571}{4120825}a^{11}+\frac{21028150}{164833}a^{10}-\frac{431586}{164833}a^{9}-\frac{40553500}{164833}a^{8}+\frac{2096562}{164833}a^{7}+\frac{45727500}{164833}a^{6}-\frac{6095220}{164833}a^{5}-\frac{128986057}{824165}a^{4}+\frac{9031175}{164833}a^{3}+\frac{4994282}{164833}a^{2}-\frac{4368250}{164833}a+\frac{1221}{191}$, $\frac{3}{12877578125}a^{28}-\frac{84}{2575515625}a^{26}+\frac{24}{2575515625}a^{25}+\frac{42}{20604125}a^{24}-\frac{24}{20604125}a^{23}-\frac{7728}{103020625}a^{22}+\frac{32956}{515103125}a^{21}+\frac{37191}{20604125}a^{20}-\frac{209076}{103020625}a^{19}-\frac{3072549}{103020625}a^{18}+\frac{846684}{20604125}a^{17}+\frac{7126632}{20604125}a^{16}-\frac{2283712}{4120825}a^{15}-\frac{58308556}{20604125}a^{14}+\frac{830928}{164833}a^{13}+\frac{66820754}{4120825}a^{12}-\frac{126293463}{4120825}a^{11}-\frac{52690352}{824165}a^{10}+\frac{99539693}{824165}a^{9}+\frac{27845100}{164833}a^{8}-\frac{239747107}{824165}a^{7}-\frac{47554920}{164833}a^{6}+\frac{63466984}{164833}a^{5}+\frac{245134999}{824165}a^{4}-\frac{37013915}{164833}a^{3}-\frac{24803746}{164833}a^{2}+\frac{4104850}{164833}a+\frac{3039}{191}$, $\frac{1}{12877578125}a^{31}+\frac{1}{12877578125}a^{30}-\frac{32}{2575515625}a^{29}-\frac{166}{12877578125}a^{28}+\frac{463}{515103125}a^{27}+\frac{2484}{2575515625}a^{26}-\frac{100076}{2575515625}a^{25}-\frac{110656}{2575515625}a^{24}+\frac{23026}{20604125}a^{23}+\frac{653199}{515103125}a^{22}-\frac{18568}{824165}a^{21}-\frac{2689722}{103020625}a^{20}+\frac{269283}{824165}a^{19}+\frac{1584448}{4120825}a^{18}-\frac{567492}{164833}a^{17}-\frac{16837344}{4120825}a^{16}+\frac{4335306}{164833}a^{15}+\frac{25723278}{824165}a^{14}-\frac{23691200}{164833}a^{13}-\frac{27804452}{164833}a^{12}+\frac{90223250}{164833}a^{11}+\frac{103183080}{164833}a^{10}-\frac{229385000}{164833}a^{9}-\frac{6256563247}{4120825}a^{8}+\frac{1817574253}{824165}a^{7}+\frac{1829779153}{824165}a^{6}-\frac{320507271}{164833}a^{5}-\frac{1402475782}{824165}a^{4}+\frac{128107823}{164833}a^{3}+\frac{86600867}{164833}a^{2}-\frac{13548868}{164833}a-\frac{8465}{191}$, $\frac{4}{12877578125}a^{29}-\frac{116}{2575515625}a^{27}-\frac{9}{2575515625}a^{26}+\frac{1508}{515103125}a^{25}+\frac{234}{515103125}a^{24}-\frac{57993}{515103125}a^{23}-\frac{2691}{103020625}a^{22}+\frac{293319}{103020625}a^{21}+\frac{450494}{515103125}a^{20}-\frac{205114}{4120825}a^{19}-\frac{389201}{20604125}a^{18}+\frac{2536443}{4120825}a^{17}+\frac{1135226}{4120825}a^{16}-\frac{4463316}{824165}a^{15}-\frac{454900}{164833}a^{14}+\frac{5543564}{164833}a^{13}+\frac{390357263}{20604125}a^{12}-\frac{23768160}{164833}a^{11}-\frac{359935306}{4120825}a^{10}+\frac{1690538421}{4120825}a^{9}+\frac{213843702}{824165}a^{8}-\frac{597824539}{824165}a^{7}-\frac{380420546}{824165}a^{6}+\frac{117838617}{164833}a^{5}+\frac{71212671}{164833}a^{4}-\frac{53836707}{164833}a^{3}-\frac{28221170}{164833}a^{2}+\frac{7716830}{164833}a+\frac{4159}{191}$ 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:  \( 1035918496783043300000000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{32}\cdot(2\pi)^{0}\cdot 1035918496783043300000000000 \cdot 2}{2\cdot\sqrt{187072209578355573530071658587684226515959365500928000000000000000000000000}}\cr\approx \mathstrut & 0.325297563922018 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^32 - 160*x^30 + 11600*x^28 - 504000*x^26 + 14625000*x^24 - 299000000*x^22 + 4427500000*x^20 - 48070000000*x^18 + 383057812500*x^16 - 2220625000000*x^14 + 9185312500000*x^12 - 26243750000000*x^10 + 49207031250000*x^8 - 55781250000000*x^6 + 33203125000000*x^4 - 7812500000000*x^2 + 581850781250)
 
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 - 160*x^30 + 11600*x^28 - 504000*x^26 + 14625000*x^24 - 299000000*x^22 + 4427500000*x^20 - 48070000000*x^18 + 383057812500*x^16 - 2220625000000*x^14 + 9185312500000*x^12 - 26243750000000*x^10 + 49207031250000*x^8 - 55781250000000*x^6 + 33203125000000*x^4 - 7812500000000*x^2 + 581850781250, 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 - 160*x^30 + 11600*x^28 - 504000*x^26 + 14625000*x^24 - 299000000*x^22 + 4427500000*x^20 - 48070000000*x^18 + 383057812500*x^16 - 2220625000000*x^14 + 9185312500000*x^12 - 26243750000000*x^10 + 49207031250000*x^8 - 55781250000000*x^6 + 33203125000000*x^4 - 7812500000000*x^2 + 581850781250);
 
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 - 160*x^30 + 11600*x^28 - 504000*x^26 + 14625000*x^24 - 299000000*x^22 + 4427500000*x^20 - 48070000000*x^18 + 383057812500*x^16 - 2220625000000*x^14 + 9185312500000*x^12 - 26243750000000*x^10 + 49207031250000*x^8 - 55781250000000*x^6 + 33203125000000*x^4 - 7812500000000*x^2 + 581850781250);
 
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

$C_{32}$ (as 32T33):

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 cyclic group of order 32
The 32 conjugacy class representatives for $C_{32}$
Character table for $C_{32}$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), \(\Q(\zeta_{32})^+\), 16.16.236118324143482260684800000000.1

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

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

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R $32$ R $16^{2}$ $32$ $32$ ${\href{/padicField/17.8.0.1}{8} }^{4}$ $32$ $16^{2}$ $32$ ${\href{/padicField/31.4.0.1}{4} }^{8}$ $32$ $16^{2}$ $32$ ${\href{/padicField/47.8.0.1}{8} }^{4}$ $32$ $32$

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 Deg $32$$32$$1$$191$
\(5\) Copy content Toggle raw display Deg $32$$4$$8$$24$