LMFDB - Number field 20.14.69761895802737851304509440000000000.2

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

Normalized defining polynomial

comment:Define the number field

sage:x = polygen(QQ); K.<a> = NumberField(x^20 - 15*x^18 - 103*x^16 + 954*x^14 + 2017*x^12 - 3645*x^10 - 2285*x^8 + 5318*x^6 - 2429*x^4 + 313*x^2 - 1)

gp:K = bnfinit(y^20 - 15*y^18 - 103*y^16 + 954*y^14 + 2017*y^12 - 3645*y^10 - 2285*y^8 + 5318*y^6 - 2429*y^4 + 313*y^2 - 1, 1)

magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 15*x^18 - 103*x^16 + 954*x^14 + 2017*x^12 - 3645*x^10 - 2285*x^8 + 5318*x^6 - 2429*x^4 + 313*x^2 - 1);

oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 15*x^18 - 103*x^16 + 954*x^14 + 2017*x^12 - 3645*x^10 - 2285*x^8 + 5318*x^6 - 2429*x^4 + 313*x^2 - 1)

$ x^{20} - 15 x^{18} - 103 x^{16} + 954 x^{14} + 2017 x^{12} - 3645 x^{10} - 2285 x^{8} + 5318 x^{6} + \cdots - 1 $ x^20 - 15*x^18 - 103*x^16 + 954*x^14 + 2017*x^12 - 3645*x^10 - 2285*x^8 + 5318*x^6 - 2429*x^4 + 313*x^2 - 1

comment:Defining polynomial

sage:K.defining_polynomial()

gp:K.pol

magma:DefiningPolynomial(K);

oscar:defining_polynomial(K)

Invariants

Degree:	$20$	comment:Degree over Q sage:K.degree() gp:poldegree(K.pol) magma:Degree(K); oscar:degree(K)
Signature:	$[14, 3]$	comment:Signature sage:K.signature() gp:K.sign magma:Signature(K); oscar:signature(K)
Discriminant:	$-69761895802737851304509440000000000$ -69761895802737851304509440000000000 $\medspace = -\,2^{44}\cdot 5^{10}\cdot 67^{8}$	comment:Discriminant sage:K.disc() gp:K.disc magma:OK := Integers(K); Discriminant(OK); oscar:OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$55.23$	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:	$2$, $5$, $67$ 2, 5, 67	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{-1}) $
$\Aut(K/\Q)$:	$C_2$	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{10}a^{10}-\frac{1}{5}a^{8}-\frac{1}{10}a^{6}-\frac{1}{5}a^{4}-\frac{1}{2}a^{2}+\frac{2}{5}$, $\frac{1}{10}a^{11}-\frac{1}{5}a^{9}-\frac{1}{10}a^{7}-\frac{1}{5}a^{5}-\frac{1}{2}a^{3}+\frac{2}{5}a$, $\frac{1}{40}a^{12}-\frac{1}{40}a^{10}+\frac{1}{20}a^{8}-\frac{3}{40}a^{6}-\frac{1}{2}a^{5}-\frac{3}{10}a^{4}-\frac{1}{2}a^{3}-\frac{11}{40}a^{2}-\frac{1}{2}a-\frac{11}{40}$, $\frac{1}{40}a^{13}-\frac{1}{40}a^{11}+\frac{1}{20}a^{9}-\frac{3}{40}a^{7}-\frac{1}{2}a^{6}-\frac{3}{10}a^{5}-\frac{1}{2}a^{4}-\frac{11}{40}a^{3}-\frac{1}{2}a^{2}-\frac{11}{40}a$, $\frac{1}{40}a^{14}+\frac{1}{40}a^{10}-\frac{1}{40}a^{8}-\frac{1}{2}a^{7}-\frac{3}{8}a^{6}+\frac{17}{40}a^{4}+\frac{9}{20}a^{2}-\frac{1}{2}a-\frac{11}{40}$, $\frac{1}{40}a^{15}+\frac{1}{40}a^{11}-\frac{1}{40}a^{9}-\frac{3}{8}a^{7}+\frac{17}{40}a^{5}-\frac{1}{2}a^{4}+\frac{9}{20}a^{3}-\frac{1}{2}a^{2}-\frac{11}{40}a-\frac{1}{2}$, $\frac{1}{200}a^{16}+\frac{1}{100}a^{14}+\frac{1}{100}a^{10}-\frac{39}{200}a^{8}+\frac{7}{20}a^{6}+\frac{11}{50}a^{4}+\frac{19}{50}a^{2}-\frac{31}{200}$, $\frac{1}{200}a^{17}+\frac{1}{100}a^{15}+\frac{1}{100}a^{11}-\frac{39}{200}a^{9}+\frac{7}{20}a^{7}+\frac{11}{50}a^{5}+\frac{19}{50}a^{3}-\frac{31}{200}a$, $\frac{1}{8265765400}a^{18}+\frac{16987463}{8265765400}a^{16}+\frac{82974027}{8265765400}a^{14}+\frac{30686411}{4132882700}a^{12}-\frac{51509359}{1033220675}a^{10}-\frac{276299547}{4132882700}a^{8}-\frac{1}{2}a^{7}-\frac{1825197681}{8265765400}a^{6}-\frac{272725083}{1653153080}a^{4}-\frac{115215837}{330630616}a^{2}-\frac{1}{2}a-\frac{685376413}{4132882700}$, $\frac{1}{8265765400}a^{19}+\frac{16987463}{8265765400}a^{17}+\frac{82974027}{8265765400}a^{15}+\frac{30686411}{4132882700}a^{13}-\frac{51509359}{1033220675}a^{11}-\frac{276299547}{4132882700}a^{9}-\frac{1825197681}{8265765400}a^{7}-\frac{272725083}{1653153080}a^{5}-\frac{1}{2}a^{4}-\frac{115215837}{330630616}a^{3}-\frac{1}{2}a^{2}-\frac{685376413}{4132882700}a-\frac{1}{2}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, 1/2*a^8 - 1/2*a^4 - 1/2, 1/2*a^9 - 1/2*a^5 - 1/2*a, 1/10*a^10 - 1/5*a^8 - 1/10*a^6 - 1/5*a^4 - 1/2*a^2 + 2/5, 1/10*a^11 - 1/5*a^9 - 1/10*a^7 - 1/5*a^5 - 1/2*a^3 + 2/5*a, 1/40*a^12 - 1/40*a^10 + 1/20*a^8 - 3/40*a^6 - 1/2*a^5 - 3/10*a^4 - 1/2*a^3 - 11/40*a^2 - 1/2*a - 11/40, 1/40*a^13 - 1/40*a^11 + 1/20*a^9 - 3/40*a^7 - 1/2*a^6 - 3/10*a^5 - 1/2*a^4 - 11/40*a^3 - 1/2*a^2 - 11/40*a, 1/40*a^14 + 1/40*a^10 - 1/40*a^8 - 1/2*a^7 - 3/8*a^6 + 17/40*a^4 + 9/20*a^2 - 1/2*a - 11/40, 1/40*a^15 + 1/40*a^11 - 1/40*a^9 - 3/8*a^7 + 17/40*a^5 - 1/2*a^4 + 9/20*a^3 - 1/2*a^2 - 11/40*a - 1/2, 1/200*a^16 + 1/100*a^14 + 1/100*a^10 - 39/200*a^8 + 7/20*a^6 + 11/50*a^4 + 19/50*a^2 - 31/200, 1/200*a^17 + 1/100*a^15 + 1/100*a^11 - 39/200*a^9 + 7/20*a^7 + 11/50*a^5 + 19/50*a^3 - 31/200*a, 1/8265765400*a^18 + 16987463/8265765400*a^16 + 82974027/8265765400*a^14 + 30686411/4132882700*a^12 - 51509359/1033220675*a^10 - 276299547/4132882700*a^8 - 1/2*a^7 - 1825197681/8265765400*a^6 - 272725083/1653153080*a^4 - 115215837/330630616*a^2 - 1/2*a - 685376413/4132882700, 1/8265765400*a^19 + 16987463/8265765400*a^17 + 82974027/8265765400*a^15 + 30686411/4132882700*a^13 - 51509359/1033220675*a^11 - 276299547/4132882700*a^9 - 1825197681/8265765400*a^7 - 272725083/1653153080*a^5 - 1/2*a^4 - 115215837/330630616*a^3 - 1/2*a^2 - 685376413/4132882700*a - 1/2

comment:Integral basis

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

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:		$C_{2}$, which has order $2$ (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:	$\frac{37753734}{1033220675}a^{18}-\frac{536620331}{1033220675}a^{16}-\frac{4315087188}{1033220675}a^{14}+\frac{32688786728}{1033220675}a^{12}+\frac{102358476866}{1033220675}a^{10}-\frac{61080305704}{1033220675}a^{8}-\frac{145924616204}{1033220675}a^{6}+\frac{93771352748}{1033220675}a^{4}-\frac{2670646858}{1033220675}a^{2}-\frac{1664034941}{1033220675}$, $\frac{37753734}{1033220675}a^{19}-\frac{536620331}{1033220675}a^{17}-\frac{4315087188}{1033220675}a^{15}+\frac{32688786728}{1033220675}a^{13}+\frac{102358476866}{1033220675}a^{11}-\frac{61080305704}{1033220675}a^{9}-\frac{145924616204}{1033220675}a^{7}+\frac{93771352748}{1033220675}a^{5}-\frac{2670646858}{1033220675}a^{3}-\frac{1664034941}{1033220675}a$, $\frac{2240635493}{8265765400}a^{19}-\frac{838237709}{206644135}a^{17}-\frac{28999094254}{1033220675}a^{15}+\frac{532351922459}{2066441350}a^{13}+\frac{4596678134391}{8265765400}a^{11}-\frac{2002403169589}{2066441350}a^{9}-\frac{2720047882739}{4132882700}a^{7}+\frac{1459826293933}{1033220675}a^{5}-\frac{5024983991549}{8265765400}a^{3}+\frac{144640567279}{2066441350}a$, $\frac{1561322129}{8265765400}a^{18}-\frac{562036596}{206644135}a^{16}-\frac{87137132823}{4132882700}a^{14}+\frac{346003369817}{2066441350}a^{12}+\frac{3974702048813}{8265765400}a^{10}-\frac{1628648390889}{4132882700}a^{8}-\frac{1342455107741}{2066441350}a^{6}+\frac{2510454157111}{4132882700}a^{4}-\frac{980826136157}{8265765400}a^{2}+\frac{3832265489}{4132882700}$, $\frac{1254826113}{4132882700}a^{19}-\frac{217148039}{1653153080}a^{18}-\frac{9286405539}{2066441350}a^{17}+\frac{159563909}{82657654}a^{16}-\frac{265696855711}{8265765400}a^{15}+\frac{5831539913}{413288270}a^{14}+\frac{1169357230311}{4132882700}a^{13}-\frac{24993075027}{206644135}a^{12}+\frac{1101222025837}{1653153080}a^{11}-\frac{497659553169}{1653153080}a^{10}-\frac{7890821951293}{8265765400}a^{9}+\frac{79049060947}{206644135}a^{8}-\frac{6812820872981}{8265765400}a^{7}+\frac{331819861139}{826576540}a^{6}+\frac{11670275024709}{8265765400}a^{5}-\frac{234617380751}{413288270}a^{4}-\frac{2213693319897}{4132882700}a^{3}+\frac{276110582283}{1653153080}a^{2}+\frac{423136013043}{8265765400}a-\frac{787232672}{206644135}$, $\frac{284368527}{1033220675}a^{19}-\frac{16516776627}{4132882700}a^{17}-\frac{62466009393}{2066441350}a^{15}+\frac{1023170046071}{4132882700}a^{13}+\frac{2768143764287}{4132882700}a^{11}-\frac{2691793102283}{4132882700}a^{9}-\frac{3496519468963}{4132882700}a^{7}+\frac{2062602461213}{2066441350}a^{5}-\frac{1322658581221}{4132882700}a^{3}+\frac{68027350669}{2066441350}a$, $\frac{1254826113}{4132882700}a^{19}+\frac{217148039}{1653153080}a^{18}-\frac{9286405539}{2066441350}a^{17}-\frac{159563909}{82657654}a^{16}-\frac{265696855711}{8265765400}a^{15}-\frac{5831539913}{413288270}a^{14}+\frac{1169357230311}{4132882700}a^{13}+\frac{24993075027}{206644135}a^{12}+\frac{1101222025837}{1653153080}a^{11}+\frac{497659553169}{1653153080}a^{10}-\frac{7890821951293}{8265765400}a^{9}-\frac{79049060947}{206644135}a^{8}-\frac{6812820872981}{8265765400}a^{7}-\frac{331819861139}{826576540}a^{6}+\frac{11670275024709}{8265765400}a^{5}+\frac{234617380751}{413288270}a^{4}-\frac{2213693319897}{4132882700}a^{3}-\frac{276110582283}{1653153080}a^{2}+\frac{423136013043}{8265765400}a+\frac{787232672}{206644135}$, $\frac{77002853}{4132882700}a^{19}-\frac{99806913}{413288270}a^{17}-\frac{2527414053}{1033220675}a^{15}+\frac{54803048031}{4132882700}a^{13}+\frac{142362955213}{2066441350}a^{11}+\frac{46602276186}{1033220675}a^{9}-\frac{259753762193}{4132882700}a^{7}-\frac{43037749174}{1033220675}a^{5}+\frac{5036437119}{1033220675}a^{3}+\frac{31322934301}{4132882700}a$, $\frac{6606000459}{8265765400}a^{19}-\frac{109492341}{1653153080}a^{18}-\frac{99158749121}{8265765400}a^{17}+\frac{1642839569}{1653153080}a^{16}-\frac{169850409357}{2066441350}a^{15}+\frac{563999093}{82657654}a^{14}+\frac{3154715770329}{4132882700}a^{13}-\frac{104627022413}{1653153080}a^{12}+\frac{13260163140031}{8265765400}a^{11}-\frac{110756862117}{826576540}a^{10}-\frac{24232318837019}{8265765400}a^{9}+\frac{407307055549}{1653153080}a^{8}-\frac{3717388011501}{2066441350}a^{7}+\frac{276752112513}{1653153080}a^{6}+\frac{17681008353969}{4132882700}a^{5}-\frac{291422555673}{826576540}a^{4}-\frac{16369064879233}{8265765400}a^{3}+\frac{29735508524}{206644135}a^{2}+\frac{2141334823919}{8265765400}a-\frac{12200793977}{826576540}$, $\frac{4613778461}{8265765400}a^{19}+\frac{114412489}{826576540}a^{18}-\frac{33949489919}{4132882700}a^{17}-\frac{16226108613}{8265765400}a^{16}-\frac{24698726463}{413288270}a^{15}-\frac{131178058201}{8265765400}a^{14}+\frac{4254553100207}{8265765400}a^{13}+\frac{98506676299}{826576540}a^{12}+\frac{1307040541702}{1033220675}a^{11}+\frac{3120774818159}{8265765400}a^{10}-\frac{671360226003}{413288270}a^{9}-\frac{834552349349}{4132882700}a^{8}-\frac{13047258010911}{8265765400}a^{7}-\frac{833207561507}{1653153080}a^{6}+\frac{4999731861479}{2066441350}a^{5}+\frac{2716461027713}{8265765400}a^{4}-\frac{3642396552903}{4132882700}a^{3}-\frac{152766694269}{4132882700}a^{2}+\frac{28598169513}{330630616}a+\frac{13275201079}{4132882700}$, $\frac{4613778461}{8265765400}a^{19}-\frac{114412489}{826576540}a^{18}-\frac{33949489919}{4132882700}a^{17}+\frac{16226108613}{8265765400}a^{16}-\frac{24698726463}{413288270}a^{15}+\frac{131178058201}{8265765400}a^{14}+\frac{4254553100207}{8265765400}a^{13}-\frac{98506676299}{826576540}a^{12}+\frac{1307040541702}{1033220675}a^{11}-\frac{3120774818159}{8265765400}a^{10}-\frac{671360226003}{413288270}a^{9}+\frac{834552349349}{4132882700}a^{8}-\frac{13047258010911}{8265765400}a^{7}+\frac{833207561507}{1653153080}a^{6}+\frac{4999731861479}{2066441350}a^{5}-\frac{2716461027713}{8265765400}a^{4}-\frac{3642396552903}{4132882700}a^{3}+\frac{152766694269}{4132882700}a^{2}+\frac{28598169513}{330630616}a-\frac{13275201079}{4132882700}$, $\frac{247568881}{2066441350}a^{19}+\frac{16479389}{330630616}a^{18}-\frac{3643096849}{2066441350}a^{17}-\frac{118592593}{165315308}a^{16}-\frac{106049466463}{8265765400}a^{15}-\frac{4602639093}{826576540}a^{14}+\frac{456598207329}{4132882700}a^{13}+\frac{36497515367}{826576540}a^{12}+\frac{2246402239091}{8265765400}a^{11}+\frac{210309463279}{1653153080}a^{10}-\frac{2887445063209}{8265765400}a^{9}-\frac{42822224291}{413288270}a^{8}-\frac{2829698393829}{8265765400}a^{7}-\frac{143401093983}{826576540}a^{6}+\frac{4279072818993}{8265765400}a^{5}+\frac{130870566891}{826576540}a^{4}-\frac{194632289131}{1033220675}a^{3}-\frac{49901232807}{1653153080}a^{2}+\frac{176403066579}{8265765400}a+\frac{930136629}{826576540}$, $\frac{724151471}{8265765400}a^{19}-\frac{77941553}{413288270}a^{18}-\frac{5694566399}{4132882700}a^{17}+\frac{903599741}{330630616}a^{16}-\frac{13390196779}{1653153080}a^{15}+\frac{34401750167}{1653153080}a^{14}+\frac{748970191197}{8265765400}a^{13}-\frac{55966972043}{330630616}a^{12}+\frac{987325875061}{8265765400}a^{11}-\frac{385155042219}{826576540}a^{10}-\frac{788365270061}{1653153080}a^{9}+\frac{369256718603}{826576540}a^{8}-\frac{48761252462}{1033220675}a^{7}+\frac{267447686781}{413288270}a^{6}+\frac{5704510527561}{8265765400}a^{5}-\frac{1096894343203}{1653153080}a^{4}-\frac{904265360979}{2066441350}a^{3}+\frac{229393483479}{1653153080}a^{2}+\frac{14115838813}{206644135}a-\frac{6671504773}{1653153080}$, $\frac{321221019}{8265765400}a^{19}+\frac{88323071}{1033220675}a^{18}-\frac{2165317829}{4132882700}a^{17}-\frac{9906264303}{8265765400}a^{16}-\frac{20119985771}{4132882700}a^{15}-\frac{20632935917}{2066441350}a^{14}+\frac{253764260653}{8265765400}a^{13}+\frac{593807750961}{8265765400}a^{12}+\frac{548473975151}{4132882700}a^{11}+\frac{402511389419}{1653153080}a^{10}-\frac{14058563073}{4132882700}a^{9}-\frac{639052136459}{8265765400}a^{8}-\frac{2202787100879}{8265765400}a^{7}-\frac{2378756995803}{8265765400}a^{6}-\frac{17443352461}{826576540}a^{5}+\frac{714685787221}{4132882700}a^{4}+\frac{22517532399}{165315308}a^{3}-\frac{171072448667}{8265765400}a^{2}-\frac{207702779439}{8265765400}a-\frac{2163165279}{2066441350}$, $\frac{306623509}{826576540}a^{19}+\frac{1559568173}{8265765400}a^{18}-\frac{22337115637}{4132882700}a^{17}-\frac{4518839437}{1653153080}a^{16}-\frac{83720405967}{2066441350}a^{15}-\frac{86115657291}{4132882700}a^{14}+\frac{277873048911}{826576540}a^{13}+\frac{1399758498031}{8265765400}a^{12}+\frac{1840624989353}{2066441350}a^{11}+\frac{1932539621743}{4132882700}a^{10}-\frac{3930407449507}{4132882700}a^{9}-\frac{3714910171691}{8265765400}a^{8}-\frac{1000048334273}{826576540}a^{7}-\frac{5522994289513}{8265765400}a^{6}+\frac{2891597304111}{2066441350}a^{5}+\frac{1356152466631}{2066441350}a^{4}-\frac{381398842913}{1033220675}a^{3}-\frac{487006668557}{4132882700}a^{2}+\frac{30524025073}{1033220675}a+\frac{7935452923}{4132882700}$, $\frac{7032954259}{4132882700}a^{19}+\frac{379498031}{1653153080}a^{18}-\frac{212189627579}{8265765400}a^{17}-\frac{3503923354}{1033220675}a^{16}-\frac{143163240759}{826576540}a^{15}-\frac{100879284687}{4132882700}a^{14}+\frac{13554863549951}{8265765400}a^{13}+\frac{88180361119}{413288270}a^{12}+\frac{27322192402613}{8265765400}a^{11}+\frac{4223095139911}{8265765400}a^{10}-\frac{10888841197221}{1653153080}a^{9}-\frac{2952863336141}{4132882700}a^{8}-\frac{30017511584923}{8265765400}a^{7}-\frac{139966420657}{206644135}a^{6}+\frac{19774204695517}{2066441350}a^{5}+\frac{4258742397281}{4132882700}a^{4}-\frac{37550949227543}{8265765400}a^{3}-\frac{2890055225687}{8265765400}a^{2}+\frac{241595842367}{413288270}a+\frac{140740712961}{4132882700}$ 37753734/1033220675a^18 - 536620331/1033220675a^16 - 4315087188/1033220675a^14 + 32688786728/1033220675a^12 + 102358476866/1033220675a^10 - 61080305704/1033220675a^8 - 145924616204/1033220675a^6 + 93771352748/1033220675a^4 - 2670646858/1033220675a^2 - 1664034941/1033220675, 37753734/1033220675a^19 - 536620331/1033220675a^17 - 4315087188/1033220675a^15 + 32688786728/1033220675a^13 + 102358476866/1033220675a^11 - 61080305704/1033220675a^9 - 145924616204/1033220675a^7 + 93771352748/1033220675a^5 - 2670646858/1033220675a^3 - 1664034941/1033220675a, 2240635493/8265765400a^19 - 838237709/206644135a^17 - 28999094254/1033220675a^15 + 532351922459/2066441350a^13 + 4596678134391/8265765400a^11 - 2002403169589/2066441350a^9 - 2720047882739/4132882700a^7 + 1459826293933/1033220675a^5 - 5024983991549/8265765400a^3 + 144640567279/2066441350a, 1561322129/8265765400a^18 - 562036596/206644135a^16 - 87137132823/4132882700a^14 + 346003369817/2066441350a^12 + 3974702048813/8265765400a^10 - 1628648390889/4132882700a^8 - 1342455107741/2066441350a^6 + 2510454157111/4132882700a^4 - 980826136157/8265765400a^2 + 3832265489/4132882700, 1254826113/4132882700a^19 - 217148039/1653153080a^18 - 9286405539/2066441350a^17 + 159563909/82657654a^16 - 265696855711/8265765400a^15 + 5831539913/413288270a^14 + 1169357230311/4132882700a^13 - 24993075027/206644135a^12 + 1101222025837/1653153080a^11 - 497659553169/1653153080a^10 - 7890821951293/8265765400a^9 + 79049060947/206644135a^8 - 6812820872981/8265765400a^7 + 331819861139/826576540a^6 + 11670275024709/8265765400a^5 - 234617380751/413288270a^4 - 2213693319897/4132882700a^3 + 276110582283/1653153080a^2 + 423136013043/8265765400a - 787232672/206644135, 284368527/1033220675a^19 - 16516776627/4132882700a^17 - 62466009393/2066441350a^15 + 1023170046071/4132882700a^13 + 2768143764287/4132882700a^11 - 2691793102283/4132882700a^9 - 3496519468963/4132882700a^7 + 2062602461213/2066441350a^5 - 1322658581221/4132882700a^3 + 68027350669/2066441350a, 1254826113/4132882700a^19 + 217148039/1653153080a^18 - 9286405539/2066441350a^17 - 159563909/82657654a^16 - 265696855711/8265765400a^15 - 5831539913/413288270a^14 + 1169357230311/4132882700a^13 + 24993075027/206644135a^12 + 1101222025837/1653153080a^11 + 497659553169/1653153080a^10 - 7890821951293/8265765400a^9 - 79049060947/206644135a^8 - 6812820872981/8265765400a^7 - 331819861139/826576540a^6 + 11670275024709/8265765400a^5 + 234617380751/413288270a^4 - 2213693319897/4132882700a^3 - 276110582283/1653153080a^2 + 423136013043/8265765400a + 787232672/206644135, 77002853/4132882700a^19 - 99806913/413288270a^17 - 2527414053/1033220675a^15 + 54803048031/4132882700a^13 + 142362955213/2066441350a^11 + 46602276186/1033220675a^9 - 259753762193/4132882700a^7 - 43037749174/1033220675a^5 + 5036437119/1033220675a^3 + 31322934301/4132882700a, 6606000459/8265765400a^19 - 109492341/1653153080a^18 - 99158749121/8265765400a^17 + 1642839569/1653153080a^16 - 169850409357/2066441350a^15 + 563999093/82657654a^14 + 3154715770329/4132882700a^13 - 104627022413/1653153080a^12 + 13260163140031/8265765400a^11 - 110756862117/826576540a^10 - 24232318837019/8265765400a^9 + 407307055549/1653153080a^8 - 3717388011501/2066441350a^7 + 276752112513/1653153080a^6 + 17681008353969/4132882700a^5 - 291422555673/826576540a^4 - 16369064879233/8265765400a^3 + 29735508524/206644135a^2 + 2141334823919/8265765400a - 12200793977/826576540, 4613778461/8265765400a^19 + 114412489/826576540a^18 - 33949489919/4132882700a^17 - 16226108613/8265765400a^16 - 24698726463/413288270a^15 - 131178058201/8265765400a^14 + 4254553100207/8265765400a^13 + 98506676299/826576540a^12 + 1307040541702/1033220675a^11 + 3120774818159/8265765400a^10 - 671360226003/413288270a^9 - 834552349349/4132882700a^8 - 13047258010911/8265765400a^7 - 833207561507/1653153080a^6 + 4999731861479/2066441350a^5 + 2716461027713/8265765400a^4 - 3642396552903/4132882700a^3 - 152766694269/4132882700a^2 + 28598169513/330630616a + 13275201079/4132882700, 4613778461/8265765400a^19 - 114412489/826576540a^18 - 33949489919/4132882700a^17 + 16226108613/8265765400a^16 - 24698726463/413288270a^15 + 131178058201/8265765400a^14 + 4254553100207/8265765400a^13 - 98506676299/826576540a^12 + 1307040541702/1033220675a^11 - 3120774818159/8265765400a^10 - 671360226003/413288270a^9 + 834552349349/4132882700a^8 - 13047258010911/8265765400a^7 + 833207561507/1653153080a^6 + 4999731861479/2066441350a^5 - 2716461027713/8265765400a^4 - 3642396552903/4132882700a^3 + 152766694269/4132882700a^2 + 28598169513/330630616a - 13275201079/4132882700, 247568881/2066441350a^19 + 16479389/330630616a^18 - 3643096849/2066441350a^17 - 118592593/165315308a^16 - 106049466463/8265765400a^15 - 4602639093/826576540a^14 + 456598207329/4132882700a^13 + 36497515367/826576540a^12 + 2246402239091/8265765400a^11 + 210309463279/1653153080a^10 - 2887445063209/8265765400a^9 - 42822224291/413288270a^8 - 2829698393829/8265765400a^7 - 143401093983/826576540a^6 + 4279072818993/8265765400a^5 + 130870566891/826576540a^4 - 194632289131/1033220675a^3 - 49901232807/1653153080a^2 + 176403066579/8265765400a + 930136629/826576540, 724151471/8265765400a^19 - 77941553/413288270a^18 - 5694566399/4132882700a^17 + 903599741/330630616a^16 - 13390196779/1653153080a^15 + 34401750167/1653153080a^14 + 748970191197/8265765400a^13 - 55966972043/330630616a^12 + 987325875061/8265765400a^11 - 385155042219/826576540a^10 - 788365270061/1653153080a^9 + 369256718603/826576540a^8 - 48761252462/1033220675a^7 + 267447686781/413288270a^6 + 5704510527561/8265765400a^5 - 1096894343203/1653153080a^4 - 904265360979/2066441350a^3 + 229393483479/1653153080a^2 + 14115838813/206644135a - 6671504773/1653153080, 321221019/8265765400a^19 + 88323071/1033220675a^18 - 2165317829/4132882700a^17 - 9906264303/8265765400a^16 - 20119985771/4132882700a^15 - 20632935917/2066441350a^14 + 253764260653/8265765400a^13 + 593807750961/8265765400a^12 + 548473975151/4132882700a^11 + 402511389419/1653153080a^10 - 14058563073/4132882700a^9 - 639052136459/8265765400a^8 - 2202787100879/8265765400a^7 - 2378756995803/8265765400a^6 - 17443352461/826576540a^5 + 714685787221/4132882700a^4 + 22517532399/165315308a^3 - 171072448667/8265765400a^2 - 207702779439/8265765400a - 2163165279/2066441350, 306623509/826576540a^19 + 1559568173/8265765400a^18 - 22337115637/4132882700a^17 - 4518839437/1653153080a^16 - 83720405967/2066441350a^15 - 86115657291/4132882700a^14 + 277873048911/826576540a^13 + 1399758498031/8265765400a^12 + 1840624989353/2066441350a^11 + 1932539621743/4132882700a^10 - 3930407449507/4132882700a^9 - 3714910171691/8265765400a^8 - 1000048334273/826576540a^7 - 5522994289513/8265765400a^6 + 2891597304111/2066441350a^5 + 1356152466631/2066441350a^4 - 381398842913/1033220675a^3 - 487006668557/4132882700a^2 + 30524025073/1033220675a + 7935452923/4132882700, 7032954259/4132882700a^19 + 379498031/1653153080a^18 - 212189627579/8265765400a^17 - 3503923354/1033220675a^16 - 143163240759/826576540a^15 - 100879284687/4132882700a^14 + 13554863549951/8265765400a^13 + 88180361119/413288270a^12 + 27322192402613/8265765400a^11 + 4223095139911/8265765400a^10 - 10888841197221/1653153080a^9 - 2952863336141/4132882700a^8 - 30017511584923/8265765400a^7 - 139966420657/206644135a^6 + 19774204695517/2066441350a^5 + 4258742397281/4132882700a^4 - 37550949227543/8265765400a^3 - 2890055225687/8265765400a^2 + 241595842367/413288270a + 140740712961/4132882700 (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:	$ 82546344719.8 $ (assuming GRH)	comment:Regulator sage:K.regulator() gp:K.reg magma:Regulator(K); oscar:regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{14}\cdot(2\pi)^{3}\cdot 82546344719.8 \cdot 1}{2\cdot\sqrt{69761895802737851304509440000000000}}\cr\approx \mathstrut & 0.635065117055 \end{aligned}\] (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^20 - 15*x^18 - 103*x^16 + 954*x^14 + 2017*x^12 - 3645*x^10 - 2285*x^8 + 5318*x^6 - 2429*x^4 + 313*x^2 - 1) DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent() hK = K.class_number(); wK = K.unit_group().torsion_generator().order(); 2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))

gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^20 - 15*x^18 - 103*x^16 + 954*x^14 + 2017*x^12 - 3645*x^10 - 2285*x^8 + 5318*x^6 - 2429*x^4 + 313*x^2 - 1, 1); [polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]

magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 15*x^18 - 103*x^16 + 954*x^14 + 2017*x^12 - 3645*x^10 - 2285*x^8 + 5318*x^6 - 2429*x^4 + 313*x^2 - 1); OK := Integers(K); DK := Discriminant(OK); UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK); r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK); hK := #clK; wK := #TorsionSubgroup(UK); 2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));

oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 15*x^18 - 103*x^16 + 954*x^14 + 2017*x^12 - 3645*x^10 - 2285*x^8 + 5318*x^6 - 2429*x^4 + 313*x^2 - 1); OK = ring_of_integers(K); DK = discriminant(OK); UK, fUK = unit_group(OK); clK, fclK = class_group(OK); r1,r2 = signature(K); RK = regulator(K); RR = parent(RK); hK = order(clK); wK = torsion_units_order(K); 2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))

Galois group

$C_2^{10}.S_5$ (as 20T803):

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 non-solvable group of order 122880

The 126 conjugacy class representatives for $C_2^{10}.S_5$

Character table for $C_2^{10}.S_5$

Intermediate fields

$\Q(\sqrt{5}) $, 5.5.5745920.1, 10.10.4126949580800000.1

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 20 siblings:	data not computed
Degree 40 siblings:	data not computed
Minimal sibling:	20.4.35718090651001779867908833280000000.3

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	${\href{/padicField/3.4.0.1}{4} }^{5}$	R	${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$	${\href{/padicField/11.4.0.1}{4} }^{3}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$	${\href{/padicField/13.8.0.1}{8} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$	${\href{/padicField/17.8.0.1}{8} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$	${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }^{2}$	${\href{/padicField/23.4.0.1}{4} }^{5}$	${\href{/padicField/29.5.0.1}{5} }^{4}$	${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.5.0.1}{5} }^{2}$	${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.4.0.1}{4} }^{2}$	${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$	${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$	${\href{/padicField/47.8.0.1}{8} }^{2}{,}\,{\href{/padicField/47.4.0.1}{4} }$	${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$	${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.5.0.1}{5} }^{2}$

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	2.2.2.4a2.2	$x^{4} + 4 x^{3} + 5 x^{2} + 4 x + 7$	$2$	$2$	$4$	$D_{4}$	$$[2, 2]^{2}$$
$2$ 2	2.2.8.40c2.12	$x^{16} + 12 x^{15} + 64 x^{14} + 224 x^{13} + 574 x^{12} + 1152 x^{11} + 1872 x^{10} + 2528 x^{9} + 2875 x^{8} + 2784 x^{7} + 2296 x^{6} + 1608 x^{5} + 942 x^{4} + 452 x^{3} + 172 x^{2} + 48 x + 19$	$8$	$2$	$40$	16T751	$$[\frac{4}{3}, \frac{4}{3}, \frac{8}{3}, \frac{8}{3}, 3, 3]_{3}^{2}$$
$5$ 5	5.1.2.1a1.1	$x^{2} + 5$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
	5.1.2.1a1.1	$x^{2} + 5$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
	5.2.2.2a1.2	$x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$	$2$	$2$	$2$	$C_2^2$	$$[\ ]_{2}^{2}$$
	5.3.2.3a1.2	$x^{6} + 6 x^{4} + 6 x^{3} + 9 x^{2} + 18 x + 14$	$2$	$3$	$3$	$C_6$	$$[\ ]_{2}^{3}$$
	5.3.2.3a1.2	$x^{6} + 6 x^{4} + 6 x^{3} + 9 x^{2} + 18 x + 14$	$2$	$3$	$3$	$C_6$	$$[\ ]_{2}^{3}$$
$67$ 67	67.4.1.0a1.1	$x^{4} + 8 x^{2} + 54 x + 2$	$1$	$4$	$0$	$C_4$	$$[\ ]^{4}$$
	67.4.1.0a1.1	$x^{4} + 8 x^{2} + 54 x + 2$	$1$	$4$	$0$	$C_4$	$$[\ ]^{4}$$
	67.4.3.8a1.3	$x^{12} + 24 x^{10} + 162 x^{9} + 198 x^{8} + 2592 x^{7} + 9356 x^{6} + 11016 x^{5} + 70380 x^{4} + 162648 x^{3} + 17592 x^{2} + 648 x + 75$	$3$	$4$	$8$	$C_{12}$	$$[\ ]_{3}^{4}$$

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