LMFDB - Number field 21.13.8830806594437225156672627375869213526379397644288.1

Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

comment:Define the number field

sage:x = polygen(QQ); K.<a> = NumberField(x^21 - 12*x^19 - 8*x^18 - 324*x^17 - 432*x^16 + 2880*x^15 + 6048*x^14 + 37728*x^13 + 90752*x^12 - 96768*x^11 - 582144*x^10 - 1895872*x^9 - 4845312*x^8 - 6218496*x^7 - 1158144*x^6 + 7271424*x^5 + 10764288*x^4 + 7647232*x^3 + 3096576*x^2 + 688128*x + 65536)

gp:K = bnfinit(y^21 - 12*y^19 - 8*y^18 - 324*y^17 - 432*y^16 + 2880*y^15 + 6048*y^14 + 37728*y^13 + 90752*y^12 - 96768*y^11 - 582144*y^10 - 1895872*y^9 - 4845312*y^8 - 6218496*y^7 - 1158144*y^6 + 7271424*y^5 + 10764288*y^4 + 7647232*y^3 + 3096576*y^2 + 688128*y + 65536, 1)

magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 12*x^19 - 8*x^18 - 324*x^17 - 432*x^16 + 2880*x^15 + 6048*x^14 + 37728*x^13 + 90752*x^12 - 96768*x^11 - 582144*x^10 - 1895872*x^9 - 4845312*x^8 - 6218496*x^7 - 1158144*x^6 + 7271424*x^5 + 10764288*x^4 + 7647232*x^3 + 3096576*x^2 + 688128*x + 65536);

oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^21 - 12*x^19 - 8*x^18 - 324*x^17 - 432*x^16 + 2880*x^15 + 6048*x^14 + 37728*x^13 + 90752*x^12 - 96768*x^11 - 582144*x^10 - 1895872*x^9 - 4845312*x^8 - 6218496*x^7 - 1158144*x^6 + 7271424*x^5 + 10764288*x^4 + 7647232*x^3 + 3096576*x^2 + 688128*x + 65536)

$ x^{21} - 12 x^{19} - 8 x^{18} - 324 x^{17} - 432 x^{16} + 2880 x^{15} + 6048 x^{14} + 37728 x^{13} + \cdots + 65536 $ x^21 - 12*x^19 - 8*x^18 - 324*x^17 - 432*x^16 + 2880*x^15 + 6048*x^14 + 37728*x^13 + 90752*x^12 - 96768*x^11 - 582144*x^10 - 1895872*x^9 - 4845312*x^8 - 6218496*x^7 - 1158144*x^6 + 7271424*x^5 + 10764288*x^4 + 7647232*x^3 + 3096576*x^2 + 688128*x + 65536

comment:Defining polynomial

sage:K.defining_polynomial()

gp:K.pol

magma:DefiningPolynomial(K);

oscar:defining_polynomial(K)

Invariants

Degree:	$21$	comment:Degree over Q sage:K.degree() gp:poldegree(K.pol) magma:Degree(K); oscar:degree(K)
Signature:	$[13, 4]$	comment:Signature sage:K.signature() gp:K.sign magma:Signature(K); oscar:signature(K)
Discriminant:	$8830806594437225156672627375869213526379397644288$ 8830806594437225156672627375869213526379397644288 $\medspace = 2^{35}\cdot 3^{40}\cdot 7\cdot 13^{15}\cdot 59$	comment:Discriminant sage:K.disc() gp:K.disc magma:OK := Integers(K); Discriminant(OK); oscar:OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$214.17$	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:	not computed
Ramified primes:	$2$, $3$, $7$, $13$, $59$ 2, 3, 7, 13, 59	comment:Ramified primes sage:K.disc().support() gp:factor(abs(K.disc))[,1]~ magma:PrimeDivisors(Discriminant(OK)); oscar:prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{10738}) $
$\Aut(K/\Q)$:	$C_1$	comment:Autmorphisms sage:K.automorphisms() magma:Automorphisms(K); oscar:automorphisms(K)
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{4}a^{5}$, $\frac{1}{8}a^{6}$, $\frac{1}{16}a^{7}-\frac{1}{8}a^{5}-\frac{1}{2}a$, $\frac{1}{16}a^{8}$, $\frac{1}{32}a^{9}-\frac{1}{8}a^{5}-\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{32}a^{10}$, $\frac{1}{64}a^{11}-\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{64}a^{12}$, $\frac{1}{128}a^{13}-\frac{1}{8}a^{5}-\frac{1}{4}a^{3}$, $\frac{1}{256}a^{14}-\frac{1}{64}a^{10}-\frac{1}{4}a^{2}$, $\frac{1}{512}a^{15}-\frac{1}{128}a^{11}-\frac{1}{8}a^{3}$, $\frac{1}{2048}a^{16}-\frac{1}{1024}a^{15}-\frac{1}{256}a^{13}-\frac{1}{512}a^{12}-\frac{1}{256}a^{11}-\frac{1}{128}a^{10}+\frac{1}{64}a^{8}-\frac{1}{32}a^{7}-\frac{1}{16}a^{6}-\frac{1}{8}a^{5}+\frac{1}{32}a^{4}+\frac{1}{16}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{8192}a^{17}-\frac{1}{2048}a^{15}-\frac{1}{1024}a^{14}+\frac{3}{2048}a^{13}+\frac{1}{512}a^{12}-\frac{1}{256}a^{11}-\frac{3}{256}a^{10}-\frac{3}{256}a^{9}+\frac{1}{64}a^{8}-\frac{1}{32}a^{7}-\frac{15}{128}a^{5}+\frac{1}{32}a^{4}-\frac{1}{32}a^{3}-\frac{1}{8}a^{2}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{32768}a^{18}-\frac{1}{16384}a^{17}-\frac{1}{8192}a^{16}+\frac{7}{8192}a^{14}-\frac{1}{4096}a^{13}-\frac{1}{512}a^{12}+\frac{7}{1024}a^{11}-\frac{13}{1024}a^{10}-\frac{3}{512}a^{9}-\frac{1}{64}a^{7}+\frac{1}{512}a^{6}+\frac{17}{256}a^{5}-\frac{3}{128}a^{4}-\frac{1}{64}a^{3}+\frac{1}{8}a^{2}-\frac{1}{4}$, $\frac{1}{131072}a^{19}-\frac{1}{16384}a^{17}-\frac{1}{16384}a^{16}-\frac{25}{32768}a^{15}+\frac{3}{8192}a^{14}+\frac{27}{8192}a^{13}+\frac{3}{4096}a^{12}-\frac{15}{4096}a^{11}-\frac{1}{128}a^{10}-\frac{3}{1024}a^{9}+\frac{7}{256}a^{8}+\frac{49}{2048}a^{7}+\frac{9}{512}a^{6}-\frac{25}{256}a^{5}-\frac{1}{64}a^{4}+\frac{27}{128}a^{3}-\frac{3}{16}a^{2}-\frac{5}{16}a-\frac{1}{8}$, $\frac{1}{524288}a^{20}-\frac{1}{262144}a^{19}-\frac{1}{65536}a^{18}+\frac{1}{65536}a^{17}-\frac{21}{131072}a^{16}-\frac{33}{65536}a^{15}+\frac{21}{32768}a^{14}-\frac{3}{2048}a^{13}+\frac{43}{16384}a^{12}+\frac{31}{8192}a^{11}-\frac{19}{4096}a^{10}+\frac{17}{2048}a^{9}+\frac{65}{8192}a^{8}-\frac{31}{4096}a^{7}+\frac{15}{512}a^{6}+\frac{55}{512}a^{5}-\frac{33}{512}a^{4}+\frac{9}{256}a^{3}+\frac{9}{64}a^{2}-\frac{3}{8}a+\frac{1}{16}$ 1, a, 1/2*a^2, 1/2*a^3, 1/4*a^4, 1/4*a^5, 1/8*a^6, 1/16*a^7 - 1/8*a^5 - 1/2*a, 1/16*a^8, 1/32*a^9 - 1/8*a^5 - 1/4*a^3 - 1/2*a, 1/32*a^10, 1/64*a^11 - 1/4*a^3 - 1/2*a, 1/64*a^12, 1/128*a^13 - 1/8*a^5 - 1/4*a^3, 1/256*a^14 - 1/64*a^10 - 1/4*a^2, 1/512*a^15 - 1/128*a^11 - 1/8*a^3, 1/2048*a^16 - 1/1024*a^15 - 1/256*a^13 - 1/512*a^12 - 1/256*a^11 - 1/128*a^10 + 1/64*a^8 - 1/32*a^7 - 1/16*a^6 - 1/8*a^5 + 1/32*a^4 + 1/16*a^3 - 1/4*a^2 - 1/2*a, 1/8192*a^17 - 1/2048*a^15 - 1/1024*a^14 + 3/2048*a^13 + 1/512*a^12 - 1/256*a^11 - 3/256*a^10 - 3/256*a^9 + 1/64*a^8 - 1/32*a^7 - 15/128*a^5 + 1/32*a^4 - 1/32*a^3 - 1/8*a^2 - 1/4*a - 1/2, 1/32768*a^18 - 1/16384*a^17 - 1/8192*a^16 + 7/8192*a^14 - 1/4096*a^13 - 1/512*a^12 + 7/1024*a^11 - 13/1024*a^10 - 3/512*a^9 - 1/64*a^7 + 1/512*a^6 + 17/256*a^5 - 3/128*a^4 - 1/64*a^3 + 1/8*a^2 - 1/4, 1/131072*a^19 - 1/16384*a^17 - 1/16384*a^16 - 25/32768*a^15 + 3/8192*a^14 + 27/8192*a^13 + 3/4096*a^12 - 15/4096*a^11 - 1/128*a^10 - 3/1024*a^9 + 7/256*a^8 + 49/2048*a^7 + 9/512*a^6 - 25/256*a^5 - 1/64*a^4 + 27/128*a^3 - 3/16*a^2 - 5/16*a - 1/8, 1/524288*a^20 - 1/262144*a^19 - 1/65536*a^18 + 1/65536*a^17 - 21/131072*a^16 - 33/65536*a^15 + 21/32768*a^14 - 3/2048*a^13 + 43/16384*a^12 + 31/8192*a^11 - 19/4096*a^10 + 17/2048*a^9 + 65/8192*a^8 - 31/4096*a^7 + 15/512*a^6 + 55/512*a^5 - 33/512*a^4 + 9/256*a^3 + 9/64*a^2 - 3/8*a + 1/16

comment:Integral basis

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

Ideal class group:		Trivial group, which has order $1$ (assuming GRH)	comment:Class group sage:K.class_group().invariants() gp:K.clgp magma:ClassGroup(K); oscar:class_group(K)
Narrow class group:		Trivial group, which has order $1$ (assuming GRH)	comment:Narrow class group sage:K.narrow_class_group().invariants() gp:bnfnarrow(K) magma:NarrowClassGroup(K);

Unit group

comment:Unit group

sage:UK = K.unit_group()

magma:UK, fUK := UnitGroup(K);

oscar:UK, fUK = unit_group(OK)

Rank:	$16$	comment:Unit rank sage:UK.rank() gp:K.fu magma:UnitRank(K); oscar:rank(UK)
Torsion generator:	$ -1 $ -1 (order $2$)	comment:Generator for roots of unity 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:	$\frac{18206775}{16384}a^{20}-\frac{11291481}{16384}a^{19}-\frac{211511331}{16384}a^{18}-\frac{3608793}{4096}a^{17}-\frac{736211673}{2048}a^{16}-\frac{1053220185}{4096}a^{15}+\frac{13764756015}{4096}a^{14}+\frac{4747994229}{1024}a^{13}+\frac{19990387037}{512}a^{12}+\frac{4904340183}{64}a^{11}-\frac{79424405991}{512}a^{10}-\frac{70501511461}{128}a^{9}-\frac{451804038165}{256}a^{8}-\frac{1098007945005}{256}a^{7}-\frac{1087403811047}{256}a^{6}+\frac{43300927053}{32}a^{5}+\frac{463658121315}{64}a^{4}+\frac{119422921803}{16}a^{3}+\frac{30876190659}{8}a^{2}+\frac{4163485941}{4}a+116956393$, $\frac{415055421}{262144}a^{20}-\frac{136934631}{131072}a^{19}-\frac{299988603}{16384}a^{18}-\frac{19190925}{32768}a^{17}-\frac{33594531057}{65536}a^{16}-\frac{11329402851}{32768}a^{15}+\frac{78444924903}{16384}a^{14}+\frac{13141258479}{2048}a^{13}+\frac{454678980291}{8192}a^{12}+\frac{438545699181}{4096}a^{11}-\frac{458430060717}{2048}a^{10}-\frac{792567185551}{1024}a^{9}-\frac{10203694323363}{4096}a^{8}-\frac{12345517380621}{2048}a^{7}-\frac{3004901869805}{512}a^{6}+\frac{130388518017}{64}a^{5}+\frac{2602934776233}{256}a^{4}+\frac{1322939996241}{128}a^{3}+\frac{169302753693}{32}a^{2}+\frac{11308533039}{8}a+\frac{1258983689}{8}$, $\frac{238538277}{262144}a^{20}-\frac{88055181}{131072}a^{19}-\frac{170850627}{16384}a^{18}+\frac{13930947}{32768}a^{17}-\frac{19339045569}{65536}a^{16}-\frac{5743150317}{32768}a^{15}+\frac{45080975835}{16384}a^{14}+\frac{14221132107}{4096}a^{13}+\frac{260127361975}{8192}a^{12}+\frac{242185499127}{4096}a^{11}-\frac{270044793117}{2048}a^{10}-\frac{442956460529}{1024}a^{9}-\frac{5754959145243}{4096}a^{8}-\frac{6901793275971}{2048}a^{7}-\frac{1620259858451}{512}a^{6}+\frac{20729033325}{16}a^{5}+\frac{1450996964577}{256}a^{4}+\frac{716722924755}{128}a^{3}+\frac{89705646477}{32}a^{2}+733211433a+\frac{639008555}{8}$, $\frac{1761820119}{262144}a^{20}-\frac{613261809}{131072}a^{19}-\frac{2536052795}{32768}a^{18}+\frac{3774519}{32768}a^{17}-\frac{142711490331}{65536}a^{16}-\frac{45463047741}{32768}a^{15}+\frac{332961785343}{16384}a^{14}+\frac{108542608647}{4096}a^{13}+\frac{1926015726289}{8192}a^{12}+\frac{1827831086415}{4096}a^{11}-\frac{1968293136069}{2048}a^{10}-\frac{3321325789631}{1024}a^{9}-\frac{42940386870921}{4096}a^{8}-\frac{51743760379203}{2048}a^{7}-\frac{3097858934183}{128}a^{6}+\frac{2321921278089}{256}a^{5}+\frac{10894999038801}{256}a^{4}+\frac{5467211076757}{128}a^{3}+\frac{692873430327}{32}a^{2}+\frac{45857445057}{8}a+\frac{5058574613}{8}$, $\frac{142988247}{32768}a^{20}-\frac{46949625}{16384}a^{19}-\frac{103383081}{2048}a^{18}-\frac{7217721}{4096}a^{17}-\frac{11572746231}{8192}a^{16}-\frac{3921442609}{4096}a^{15}+\frac{27024103989}{2048}a^{14}+\frac{9075765423}{512}a^{13}+\frac{156669479891}{1024}a^{12}+\frac{151317618561}{512}a^{11}-\frac{157766077521}{256}a^{10}-\frac{273341737259}{128}a^{9}-\frac{3517914615921}{512}a^{8}-\frac{4257777924963}{256}a^{7}-\frac{1037823723303}{64}a^{6}+\frac{178855818903}{32}a^{5}+\frac{448896750669}{16}a^{4}+\frac{228419769381}{8}a^{3}+\frac{29258759817}{2}a^{2}+3912120909a+435934711$, $\frac{823371021}{524288}a^{20}-\frac{289777977}{262144}a^{19}-\frac{1184136721}{65536}a^{18}+\frac{10217637}{65536}a^{17}-\frac{66705886977}{131072}a^{16}-\frac{20985991937}{65536}a^{15}+\frac{155604522609}{32768}a^{14}+\frac{25213377573}{4096}a^{13}+\frac{899710628391}{16384}a^{12}+\frac{850876626807}{8192}a^{11}-\frac{922079630079}{4096}a^{10}-\frac{1547930501039}{2048}a^{9}-\frac{20030923482867}{8192}a^{8}-\frac{24116486191743}{4096}a^{7}-\frac{2877481284901}{512}a^{6}+\frac{1095832197999}{512}a^{5}+\frac{5076416980635}{512}a^{4}+\frac{2540342485393}{256}a^{3}+\frac{321252685173}{64}a^{2}+\frac{5304606975}{4}a+\frac{2335761801}{16}$, $\frac{16963540251}{262144}a^{20}-\frac{5060707383}{131072}a^{19}-\frac{24698898819}{32768}a^{18}-\frac{2215522521}{32768}a^{17}-\frac{1371222211303}{65536}a^{16}-\frac{507013824367}{32768}a^{15}+\frac{3206083407567}{16384}a^{14}+\frac{17575316635}{64}a^{13}+\frac{18651352575905}{8192}a^{12}+\frac{18487921356073}{4096}a^{11}-\frac{18357738232537}{2048}a^{10}-\frac{33110380426333}{1024}a^{9}-\frac{423307301381093}{4096}a^{8}-\frac{515655290679073}{2048}a^{7}-\frac{64469165656813}{256}a^{6}+\frac{19473476990179}{256}a^{5}+\frac{108955011232405}{256}a^{4}+\frac{56574694642803}{128}a^{3}+\frac{7357049204143}{32}a^{2}+\frac{249378764777}{4}a+\frac{56346913515}{8}$, $\frac{18971846289}{262144}a^{20}-\frac{3273160187}{65536}a^{19}-\frac{3416027573}{4096}a^{18}-\frac{112571135}{32768}a^{17}-\frac{1536566954265}{65536}a^{16}-\frac{123569877677}{8192}a^{15}+\frac{3585464292581}{16384}a^{14}+\frac{2348498596951}{8192}a^{13}+\frac{20747185307405}{8192}a^{12}+\frac{9871600841291}{2048}a^{11}-\frac{21154921392365}{2048}a^{10}-\frac{17921097918919}{512}a^{9}-\frac{463065720046303}{4096}a^{8}-\frac{139595101630305}{512}a^{7}-\frac{67044710470611}{256}a^{6}+\frac{24808031105529}{256}a^{5}+\frac{117597070858557}{256}a^{4}+\frac{7392485208197}{16}a^{3}+\frac{7507583284627}{32}a^{2}+\frac{995364225269}{16}a+\frac{13747236329}{2}$, $\frac{4418774625}{32768}a^{20}-\frac{3057684253}{32768}a^{19}-\frac{50909481161}{32768}a^{18}-\frac{61019193}{16384}a^{17}-\frac{178949778591}{4096}a^{16}-\frac{229569948235}{8192}a^{15}+\frac{3340376884465}{8192}a^{14}+\frac{2184863887633}{4096}a^{13}+\frac{2415882231777}{512}a^{12}+\frac{143565312739}{16}a^{11}-\frac{19720400987619}{1024}a^{10}-\frac{33370175109113}{512}a^{9}-\frac{107805936335215}{512}a^{8}-\frac{259937348521937}{512}a^{7}-\frac{249474928624477}{512}a^{6}+\frac{46334848523721}{256}a^{5}+\frac{109479764297923}{128}a^{4}+\frac{55021516220493}{64}a^{3}+436328318756a^{2}+115642886126a+\frac{51085207469}{4}$, $\frac{15846613}{262144}a^{20}+\frac{2743361}{32768}a^{19}-\frac{28200527}{32768}a^{18}-\frac{44143889}{32768}a^{17}-\frac{1229446029}{65536}a^{16}-\frac{880567715}{16384}a^{15}+\frac{2982646339}{16384}a^{14}+\frac{5072723367}{8192}a^{13}+\frac{19479334045}{8192}a^{12}+\frac{8451708975}{1024}a^{11}-\frac{5905159915}{2048}a^{10}-\frac{25951943689}{512}a^{9}-\frac{581038408827}{4096}a^{8}-\frac{403863343489}{1024}a^{7}-\frac{301269128739}{512}a^{6}-\frac{9262786823}{64}a^{5}+\frac{176908135571}{256}a^{4}+\frac{31755989379}{32}a^{3}+\frac{19913300767}{32}a^{2}+\frac{3124930031}{16}a+25027128$, $\frac{2893485569}{524288}a^{20}-\frac{1012897319}{262144}a^{19}-\frac{4162740101}{65536}a^{18}+\frac{20679341}{65536}a^{17}-\frac{234405570397}{131072}a^{16}-\frac{74190570699}{65536}a^{15}+\frac{546765807001}{32768}a^{14}+\frac{177733539003}{8192}a^{13}+\frac{3162700352999}{16384}a^{12}+\frac{2995875254025}{8192}a^{11}-\frac{3235793423195}{4096}a^{10}-\frac{5446775538861}{2048}a^{9}-\frac{70464364045023}{8192}a^{8}-\frac{84867865159269}{4096}a^{7}-\frac{5073125110383}{256}a^{6}+\frac{3826494233877}{512}a^{5}+\frac{17864841544511}{512}a^{4}+\frac{8956288114807}{256}a^{3}+\frac{1134384368357}{64}a^{2}+\frac{75047198741}{16}a+\frac{8276258279}{16}$, $\frac{607239512601}{524288}a^{20}-\frac{208534792075}{262144}a^{19}-\frac{875052345753}{65536}a^{18}-\frac{6228358763}{65536}a^{17}-\frac{49177844850373}{131072}a^{16}-\frac{15902553684983}{65536}a^{15}+\frac{114764278508161}{32768}a^{14}+\frac{37678281052725}{8192}a^{13}+\frac{664178327006247}{16384}a^{12}+\frac{632977227929237}{8192}a^{11}-\frac{676446569769163}{4096}a^{10}-\frac{11\cdots 81}{2048}a^{9}-\frac{14\cdots 07}{8192}a^{8}-\frac{17\cdots 81}{4096}a^{7}-\frac{10\cdots 53}{256}a^{6}+\frac{790900570448269}{512}a^{5}+\frac{37\cdots 43}{512}a^{4}+\frac{18\cdots 31}{256}a^{3}+\frac{241062115714661}{64}a^{2}+\frac{15992032273227}{16}a+\frac{1768241238915}{16}$, $\frac{874785381035}{262144}a^{20}-\frac{301999917419}{131072}a^{19}-\frac{630021701697}{16384}a^{18}-\frac{4789375171}{32768}a^{17}-\frac{70851116093967}{65536}a^{16}-\frac{22778624866523}{32768}a^{15}+\frac{165324298449781}{16384}a^{14}+\frac{54129977870793}{4096}a^{13}+\frac{956627383657113}{8192}a^{12}+\frac{910192997219233}{4096}a^{11}-\frac{975550101814963}{2048}a^{10}-\frac{16\cdots 15}{1024}a^{9}-\frac{21\cdots 37}{4096}a^{8}-\frac{25\cdots 53}{2048}a^{7}-\frac{61\cdots 13}{512}a^{6}+\frac{143048256898457}{32}a^{5}+\frac{54\cdots 23}{256}a^{4}+\frac{27\cdots 97}{128}a^{3}+\frac{346052769931307}{32}a^{2}+2867261903049a+\frac{2534218755141}{8}$, $\frac{389462385291}{262144}a^{20}-\frac{133446128749}{131072}a^{19}-\frac{280663888759}{16384}a^{18}-\frac{4798450759}{32768}a^{17}-\frac{31539955268615}{65536}a^{16}-\frac{10224033950249}{32768}a^{15}+\frac{73605807118177}{16384}a^{14}+\frac{12096989624661}{2048}a^{13}+\frac{426019452967533}{8192}a^{12}+\frac{406286487371023}{4096}a^{11}-\frac{433636576707387}{2048}a^{10}-\frac{737050212658937}{1024}a^{9}-\frac{95\cdots 49}{4096}a^{8}-\frac{11\cdots 51}{2048}a^{7}-\frac{27\cdots 43}{512}a^{6}+\frac{253109752235863}{128}a^{5}+\frac{24\cdots 99}{256}a^{4}+\frac{12\cdots 39}{128}a^{3}+\frac{154854909253863}{32}a^{2}+\frac{10277306353989}{8}a+\frac{1136841025167}{8}$, $\frac{51053112485}{262144}a^{20}-\frac{512447987}{4096}a^{19}-\frac{18486754359}{8192}a^{18}-\frac{3548067935}{32768}a^{17}-\frac{4130723682309}{65536}a^{16}-\frac{715039999489}{16384}a^{15}+\frac{9649057445637}{16384}a^{14}+\frac{6549744670693}{8192}a^{13}+\frac{55983339805621}{8192}a^{12}+\frac{13602787968463}{1024}a^{11}-\frac{56075062021273}{2048}a^{10}-\frac{12260584802563}{128}a^{9}-\frac{12\cdots 99}{4096}a^{8}-\frac{763868479951699}{1024}a^{7}-\frac{93672665742149}{128}a^{6}+\frac{62631258213115}{256}a^{5}+\frac{322306089874137}{256}a^{4}+\frac{82400488535945}{64}a^{3}+\frac{21186618251243}{32}a^{2}+\frac{2842374799419}{16}a+\frac{79446548603}{4}$, $\frac{236513390003}{524288}a^{20}-\frac{75460651183}{262144}a^{19}-\frac{342741195515}{65536}a^{18}-\frac{17794989253}{65536}a^{17}-\frac{19134679362751}{131072}a^{16}-\frac{6666780631607}{65536}a^{15}+\frac{44700976694231}{32768}a^{14}+\frac{7609744562013}{4096}a^{13}+\frac{259418998893897}{16384}a^{12}+\frac{252604895925809}{8192}a^{11}-\frac{259418087741593}{4096}a^{10}-\frac{455075779360697}{2048}a^{9}-\frac{58\cdots 05}{8192}a^{8}-\frac{70\cdots 21}{4096}a^{7}-\frac{870823451690549}{512}a^{6}+\frac{288391796280777}{512}a^{5}+\frac{14\cdots 29}{512}a^{4}+\frac{765854911975863}{256}a^{3}+\frac{98568020519011}{64}a^{2}+\frac{1654707491865}{4}a+\frac{740778942095}{16}$ 18206775/16384a^20 - 11291481/16384a^19 - 211511331/16384a^18 - 3608793/4096a^17 - 736211673/2048a^16 - 1053220185/4096a^15 + 13764756015/4096a^14 + 4747994229/1024a^13 + 19990387037/512a^12 + 4904340183/64a^11 - 79424405991/512a^10 - 70501511461/128a^9 - 451804038165/256a^8 - 1098007945005/256a^7 - 1087403811047/256a^6 + 43300927053/32a^5 + 463658121315/64a^4 + 119422921803/16a^3 + 30876190659/8a^2 + 4163485941/4a + 116956393, 415055421/262144a^20 - 136934631/131072a^19 - 299988603/16384a^18 - 19190925/32768a^17 - 33594531057/65536a^16 - 11329402851/32768a^15 + 78444924903/16384a^14 + 13141258479/2048a^13 + 454678980291/8192a^12 + 438545699181/4096a^11 - 458430060717/2048a^10 - 792567185551/1024a^9 - 10203694323363/4096a^8 - 12345517380621/2048a^7 - 3004901869805/512a^6 + 130388518017/64a^5 + 2602934776233/256a^4 + 1322939996241/128a^3 + 169302753693/32a^2 + 11308533039/8a + 1258983689/8, 238538277/262144a^20 - 88055181/131072a^19 - 170850627/16384a^18 + 13930947/32768a^17 - 19339045569/65536a^16 - 5743150317/32768a^15 + 45080975835/16384a^14 + 14221132107/4096a^13 + 260127361975/8192a^12 + 242185499127/4096a^11 - 270044793117/2048a^10 - 442956460529/1024a^9 - 5754959145243/4096a^8 - 6901793275971/2048a^7 - 1620259858451/512a^6 + 20729033325/16a^5 + 1450996964577/256a^4 + 716722924755/128a^3 + 89705646477/32a^2 + 733211433a + 639008555/8, 1761820119/262144a^20 - 613261809/131072a^19 - 2536052795/32768a^18 + 3774519/32768a^17 - 142711490331/65536a^16 - 45463047741/32768a^15 + 332961785343/16384a^14 + 108542608647/4096a^13 + 1926015726289/8192a^12 + 1827831086415/4096a^11 - 1968293136069/2048a^10 - 3321325789631/1024a^9 - 42940386870921/4096a^8 - 51743760379203/2048a^7 - 3097858934183/128a^6 + 2321921278089/256a^5 + 10894999038801/256a^4 + 5467211076757/128a^3 + 692873430327/32a^2 + 45857445057/8a + 5058574613/8, 142988247/32768a^20 - 46949625/16384a^19 - 103383081/2048a^18 - 7217721/4096a^17 - 11572746231/8192a^16 - 3921442609/4096a^15 + 27024103989/2048a^14 + 9075765423/512a^13 + 156669479891/1024a^12 + 151317618561/512a^11 - 157766077521/256a^10 - 273341737259/128a^9 - 3517914615921/512a^8 - 4257777924963/256a^7 - 1037823723303/64a^6 + 178855818903/32a^5 + 448896750669/16a^4 + 228419769381/8a^3 + 29258759817/2a^2 + 3912120909a + 435934711, 823371021/524288a^20 - 289777977/262144a^19 - 1184136721/65536a^18 + 10217637/65536a^17 - 66705886977/131072a^16 - 20985991937/65536a^15 + 155604522609/32768a^14 + 25213377573/4096a^13 + 899710628391/16384a^12 + 850876626807/8192a^11 - 922079630079/4096a^10 - 1547930501039/2048a^9 - 20030923482867/8192a^8 - 24116486191743/4096a^7 - 2877481284901/512a^6 + 1095832197999/512a^5 + 5076416980635/512a^4 + 2540342485393/256a^3 + 321252685173/64a^2 + 5304606975/4a + 2335761801/16, 16963540251/262144a^20 - 5060707383/131072a^19 - 24698898819/32768a^18 - 2215522521/32768a^17 - 1371222211303/65536a^16 - 507013824367/32768a^15 + 3206083407567/16384a^14 + 17575316635/64a^13 + 18651352575905/8192a^12 + 18487921356073/4096a^11 - 18357738232537/2048a^10 - 33110380426333/1024a^9 - 423307301381093/4096a^8 - 515655290679073/2048a^7 - 64469165656813/256a^6 + 19473476990179/256a^5 + 108955011232405/256a^4 + 56574694642803/128a^3 + 7357049204143/32a^2 + 249378764777/4a + 56346913515/8, 18971846289/262144a^20 - 3273160187/65536a^19 - 3416027573/4096a^18 - 112571135/32768a^17 - 1536566954265/65536a^16 - 123569877677/8192a^15 + 3585464292581/16384a^14 + 2348498596951/8192a^13 + 20747185307405/8192a^12 + 9871600841291/2048a^11 - 21154921392365/2048a^10 - 17921097918919/512a^9 - 463065720046303/4096a^8 - 139595101630305/512a^7 - 67044710470611/256a^6 + 24808031105529/256a^5 + 117597070858557/256a^4 + 7392485208197/16a^3 + 7507583284627/32a^2 + 995364225269/16a + 13747236329/2, 4418774625/32768a^20 - 3057684253/32768a^19 - 50909481161/32768a^18 - 61019193/16384a^17 - 178949778591/4096a^16 - 229569948235/8192a^15 + 3340376884465/8192a^14 + 2184863887633/4096a^13 + 2415882231777/512a^12 + 143565312739/16a^11 - 19720400987619/1024a^10 - 33370175109113/512a^9 - 107805936335215/512a^8 - 259937348521937/512a^7 - 249474928624477/512a^6 + 46334848523721/256a^5 + 109479764297923/128a^4 + 55021516220493/64a^3 + 436328318756a^2 + 115642886126a + 51085207469/4, 15846613/262144a^20 + 2743361/32768a^19 - 28200527/32768a^18 - 44143889/32768a^17 - 1229446029/65536a^16 - 880567715/16384a^15 + 2982646339/16384a^14 + 5072723367/8192a^13 + 19479334045/8192a^12 + 8451708975/1024a^11 - 5905159915/2048a^10 - 25951943689/512a^9 - 581038408827/4096a^8 - 403863343489/1024a^7 - 301269128739/512a^6 - 9262786823/64a^5 + 176908135571/256a^4 + 31755989379/32a^3 + 19913300767/32a^2 + 3124930031/16a + 25027128, 2893485569/524288a^20 - 1012897319/262144a^19 - 4162740101/65536a^18 + 20679341/65536a^17 - 234405570397/131072a^16 - 74190570699/65536a^15 + 546765807001/32768a^14 + 177733539003/8192a^13 + 3162700352999/16384a^12 + 2995875254025/8192a^11 - 3235793423195/4096a^10 - 5446775538861/2048a^9 - 70464364045023/8192a^8 - 84867865159269/4096a^7 - 5073125110383/256a^6 + 3826494233877/512a^5 + 17864841544511/512a^4 + 8956288114807/256a^3 + 1134384368357/64a^2 + 75047198741/16a + 8276258279/16, 607239512601/524288a^20 - 208534792075/262144a^19 - 875052345753/65536a^18 - 6228358763/65536a^17 - 49177844850373/131072a^16 - 15902553684983/65536a^15 + 114764278508161/32768a^14 + 37678281052725/8192a^13 + 664178327006247/16384a^12 + 632977227929237/8192a^11 - 676446569769163/4096a^10 - 1148561161406281/2048a^9 - 14832796512364807/8192a^8 - 17892648580315481/4096a^7 - 1075732721712453/256a^6 + 790900570448269/512a^5 + 3768794502301343/512a^4 + 1897392560623331/256a^3 + 241062115714661/64a^2 + 15992032273227/16a + 1768241238915/16, 874785381035/262144a^20 - 301999917419/131072a^19 - 630021701697/16384a^18 - 4789375171/32768a^17 - 70851116093967/65536a^16 - 22778624866523/32768a^15 + 165324298449781/16384a^14 + 54129977870793/4096a^13 + 956627383657113/8192a^12 + 910192997219233/4096a^11 - 975550101814963/2048a^10 - 1652467787882215/1024a^9 - 21350064068270037/4096a^8 - 25743622502541653/2048a^7 - 6181113722200013/512a^6 + 143048256898457/32a^5 + 5421631232467223/256a^4 + 2726227381661197/128a^3 + 346052769931307/32a^2 + 2867261903049a + 2534218755141/8, 389462385291/262144a^20 - 133446128749/131072a^19 - 280663888759/16384a^18 - 4798450759/32768a^17 - 31539955268615/65536a^16 - 10224033950249/32768a^15 + 73605807118177/16384a^14 + 12096989624661/2048a^13 + 426019452967533/8192a^12 + 406286487371023/4096a^11 - 433636576707387/2048a^10 - 737050212658937/1024a^9 - 9516768961806549/4096a^8 - 11481951584127351/2048a^7 - 2763192761032843/512a^6 + 253109752235863/128a^5 + 2418617874778599/256a^4 + 1218329857350239/128a^3 + 154854909253863/32a^2 + 10277306353989/8a + 1136841025167/8, 51053112485/262144a^20 - 512447987/4096a^19 - 18486754359/8192a^18 - 3548067935/32768a^17 - 4130723682309/65536a^16 - 715039999489/16384a^15 + 9649057445637/16384a^14 + 6549744670693/8192a^13 + 55983339805621/8192a^12 + 13602787968463/1024a^11 - 56075062021273/2048a^10 - 12260584802563/128a^9 - 1260286724163499/4096a^8 - 763868479951699/1024a^7 - 93672665742149/128a^6 + 62631258213115/256a^5 + 322306089874137/256a^4 + 82400488535945/64a^3 + 21186618251243/32a^2 + 2842374799419/16a + 79446548603/4, 236513390003/524288a^20 - 75460651183/262144a^19 - 342741195515/65536a^18 - 17794989253/65536a^17 - 19134679362751/131072a^16 - 6666780631607/65536a^15 + 44700976694231/32768a^14 + 7609744562013/4096a^13 + 259418998893897/16384a^12 + 252604895925809/8192a^11 - 259418087741593/4096a^10 - 455075779360697/2048a^9 - 5844488426736205/8192a^8 - 7088054628598121/4096a^7 - 870823451690549/512a^6 + 288391796280777/512a^5 + 1495580546157629/512a^4 + 765854911975863/256a^3 + 98568020519011/64a^2 + 1654707491865/4a + 740778942095/16 (assuming GRH)	comment:Fundamental units 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:	$ 616524970790000000 $ (assuming GRH)	comment:Regulator sage:K.regulator() gp:K.reg magma:Regulator(K); oscar:regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{13}\cdot(2\pi)^{4}\cdot 616524970790000000 \cdot 1}{2\cdot\sqrt{8830806594437225156672627375869213526379397644288}}\cr\approx \mathstrut & 1.32443273411247 \end{aligned}\] (assuming GRH)

comment:Analytic class number formula

sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^21 - 12*x^19 - 8*x^18 - 324*x^17 - 432*x^16 + 2880*x^15 + 6048*x^14 + 37728*x^13 + 90752*x^12 - 96768*x^11 - 582144*x^10 - 1895872*x^9 - 4845312*x^8 - 6218496*x^7 - 1158144*x^6 + 7271424*x^5 + 10764288*x^4 + 7647232*x^3 + 3096576*x^2 + 688128*x + 65536) 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))))

gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^21 - 12*x^19 - 8*x^18 - 324*x^17 - 432*x^16 + 2880*x^15 + 6048*x^14 + 37728*x^13 + 90752*x^12 - 96768*x^11 - 582144*x^10 - 1895872*x^9 - 4845312*x^8 - 6218496*x^7 - 1158144*x^6 + 7271424*x^5 + 10764288*x^4 + 7647232*x^3 + 3096576*x^2 + 688128*x + 65536, 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)))]

magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 12*x^19 - 8*x^18 - 324*x^17 - 432*x^16 + 2880*x^15 + 6048*x^14 + 37728*x^13 + 90752*x^12 - 96768*x^11 - 582144*x^10 - 1895872*x^9 - 4845312*x^8 - 6218496*x^7 - 1158144*x^6 + 7271424*x^5 + 10764288*x^4 + 7647232*x^3 + 3096576*x^2 + 688128*x + 65536); 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)));

oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 12*x^19 - 8*x^18 - 324*x^17 - 432*x^16 + 2880*x^15 + 6048*x^14 + 37728*x^13 + 90752*x^12 - 96768*x^11 - 582144*x^10 - 1895872*x^9 - 4845312*x^8 - 6218496*x^7 - 1158144*x^6 + 7271424*x^5 + 10764288*x^4 + 7647232*x^3 + 3096576*x^2 + 688128*x + 65536); 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

$C_3^7:C_2\wr C_7.C_6$ (as 21T142):

comment:Galois group

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 11757312

The 168 conjugacy class representatives for $C_3^7:C_2\wr C_7.C_6$

Character table for $C_3^7:C_2\wr C_7.C_6$

Intermediate fields

7.7.138584369664.1

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

comment:Intermediate fields

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 42 siblings:	data not computed
Minimal sibling:	This field is its own minimal sibling

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	R	${\href{/padicField/5.14.0.1}{14} }{,}\,{\href{/padicField/5.7.0.1}{7} }$	R	${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$	R	${\href{/padicField/17.9.0.1}{9} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$	${\href{/padicField/19.9.0.1}{9} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{3}{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$	${\href{/padicField/23.9.0.1}{9} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$	${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$	${\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }$	${\href{/padicField/37.9.0.1}{9} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$	$18{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$	${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }$	${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$	${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$	R

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

comment:Frobenius cycle types

sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])

magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];

oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$2$ 2	2.1.3.2a1.1	$x^{3} + 2$	$3$	$1$	$2$	$S_3$	$$[\ ]_{3}^{2}$$
	2.3.2.9a1.6	$x^{6} + 2 x^{4} + 6 x^{3} + x^{2} + 6 x + 15$	$2$	$3$	$9$	$C_6$	$$[3]^{3}$$
	2.3.4.24b17.2	$x^{12} + 6 x^{10} + 4 x^{9} + 12 x^{8} + 20 x^{7} + 18 x^{6} + 32 x^{5} + 33 x^{4} + 28 x^{3} + 26 x^{2} + 20 x + 17$	$4$	$3$	$24$	$C_2^2 \times A_4$	$$[2, 2, 3]^{6}$$
$3$ 3	3.1.21.40a1.648	$x^{21} + 3 x^{20} + 18 x^{8} + 18 x^{7} + 18 x^{5} + 18 x^{4} + 9 x^{2} + 18 x + 3$	$21$	$1$	$40$	21T99	not computed
$7$ 7	$\Q_{7}$	$x + 4$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	7.1.2.1a1.2	$x^{2} + 21$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
	7.6.1.0a1.1	$x^{6} + x^{4} + 5 x^{3} + 4 x^{2} + 6 x + 3$	$1$	$6$	$0$	$C_6$	$$[\ ]^{6}$$
	7.12.1.0a1.1	$x^{12} + 2 x^{8} + 5 x^{7} + 3 x^{6} + 2 x^{5} + 4 x^{4} + 5 x^{2} + 3$	$1$	$12$	$0$	$C_{12}$	$$[\ ]^{12}$$
$13$ 13	13.3.1.0a1.1	$x^{3} + 2 x + 11$	$1$	$3$	$0$	$C_3$	$$[\ ]^{3}$$
	13.1.6.5a1.6	$x^{6} + 104$	$6$	$1$	$5$	$C_6$	$$[\ ]_{6}$$
	13.1.6.5a1.6	$x^{6} + 104$	$6$	$1$	$5$	$C_6$	$$[\ ]_{6}$$
	13.1.6.5a1.6	$x^{6} + 104$	$6$	$1$	$5$	$C_6$	$$[\ ]_{6}$$
$59$ 59	$\Q_{59}$	$x + 57$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	59.1.2.1a1.2	$x^{2} + 118$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
	59.3.1.0a1.1	$x^{3} + 5 x + 57$	$1$	$3$	$0$	$C_3$	$$[\ ]^{3}$$
	59.3.1.0a1.1	$x^{3} + 5 x + 57$	$1$	$3$	$0$	$C_3$	$$[\ ]^{3}$$
	59.3.1.0a1.1	$x^{3} + 5 x + 57$	$1$	$3$	$0$	$C_3$	$$[\ ]^{3}$$
	59.9.1.0a1.1	$x^{9} + x^{3} + 32 x^{2} + 47 x + 57$	$1$	$9$	$0$	$C_9$	$$[\ ]^{9}$$

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