LMFDB - Number field 16.16.2111549459996061430585501425390625.1

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

Normalized defining polynomial

comment:Define the number field

sage:x = polygen(QQ); K.<a> = NumberField(x^16 - 74*x^14 - 6*x^13 + 2275*x^12 + 372*x^11 - 37413*x^10 - 9315*x^9 + 351975*x^8 + 120534*x^7 - 1864956*x^6 - 852984*x^5 + 5001568*x^4 + 3165600*x^3 - 4743104*x^2 - 4932096*x - 1200896)

gp:K = bnfinit(y^16 - 74*y^14 - 6*y^13 + 2275*y^12 + 372*y^11 - 37413*y^10 - 9315*y^9 + 351975*y^8 + 120534*y^7 - 1864956*y^6 - 852984*y^5 + 5001568*y^4 + 3165600*y^3 - 4743104*y^2 - 4932096*y - 1200896, 1)

magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 74*x^14 - 6*x^13 + 2275*x^12 + 372*x^11 - 37413*x^10 - 9315*x^9 + 351975*x^8 + 120534*x^7 - 1864956*x^6 - 852984*x^5 + 5001568*x^4 + 3165600*x^3 - 4743104*x^2 - 4932096*x - 1200896);

oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 74*x^14 - 6*x^13 + 2275*x^12 + 372*x^11 - 37413*x^10 - 9315*x^9 + 351975*x^8 + 120534*x^7 - 1864956*x^6 - 852984*x^5 + 5001568*x^4 + 3165600*x^3 - 4743104*x^2 - 4932096*x - 1200896)

$ x^{16} - 74 x^{14} - 6 x^{13} + 2275 x^{12} + 372 x^{11} - 37413 x^{10} - 9315 x^{9} + 351975 x^{8} + \cdots - 1200896 $ x^16 - 74*x^14 - 6*x^13 + 2275*x^12 + 372*x^11 - 37413*x^10 - 9315*x^9 + 351975*x^8 + 120534*x^7 - 1864956*x^6 - 852984*x^5 + 5001568*x^4 + 3165600*x^3 - 4743104*x^2 - 4932096*x - 1200896

comment:Defining polynomial

sage:K.defining_polynomial()

gp:K.pol

magma:DefiningPolynomial(K);

oscar:defining_polynomial(K)

Invariants

Degree:	$16$	comment:Degree over Q sage:K.degree() gp:poldegree(K.pol) magma:Degree(K); oscar:degree(K)
Signature:	$(16, 0)$	comment:Signature sage:K.signature() gp:K.sign magma:Signature(K); oscar:signature(K)
Discriminant:	$2111549459996061430585501425390625$ 2111549459996061430585501425390625 $\medspace = 5^{8}\cdot 7^{2}\cdot 31^{2}\cdot 41^{4}\cdot 14197^{4}$	comment:Discriminant sage:K.disc() gp:K.disc magma:OK := Integers(K); Discriminant(OK); oscar:OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$121.00$	comment:Root discriminant sage:(K.disc().abs())^(1./K.degree()) gp:abs(K.disc)^(1/poldegree(K.pol)) magma:Abs(Discriminant(OK))^(1/Degree(K)); oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
Galois root discriminant:	$5^{1/2}7^{1/2}31^{1/2}41^{1/2}14197^{1/2}\approx 25130.72909805842$
Ramified primes:	$5$, $7$, $31$, $41$, $14197$ 5, 7, 31, 41, 14197	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:	$\Q$
$\Aut(K/\Q)$:	$C_2$	comment:Automorphisms sage:K.automorphisms() magma:Automorphisms(K); oscar:automorphism_group(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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}-\frac{1}{3}a^{4}+\frac{1}{3}a^{2}-\frac{1}{3}$, $\frac{1}{3}a^{7}-\frac{1}{3}a^{5}+\frac{1}{3}a^{3}-\frac{1}{3}a$, $\frac{1}{3}a^{8}-\frac{1}{3}$, $\frac{1}{6}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{6}a$, $\frac{1}{36}a^{10}+\frac{1}{18}a^{8}-\frac{1}{6}a^{7}-\frac{1}{36}a^{6}-\frac{1}{3}a^{5}+\frac{7}{36}a^{4}-\frac{5}{12}a^{3}+\frac{7}{36}a^{2}+\frac{1}{6}a-\frac{1}{9}$, $\frac{1}{72}a^{11}+\frac{1}{36}a^{9}-\frac{1}{12}a^{8}+\frac{11}{72}a^{7}-\frac{1}{6}a^{6}+\frac{31}{72}a^{5}+\frac{7}{24}a^{4}+\frac{19}{72}a^{3}-\frac{5}{12}a^{2}-\frac{2}{9}a$, $\frac{1}{144}a^{12}-\frac{1}{72}a^{10}-\frac{1}{24}a^{9}+\frac{1}{48}a^{8}-\frac{1}{12}a^{7}-\frac{13}{144}a^{6}-\frac{17}{48}a^{5}-\frac{11}{48}a^{4}-\frac{11}{24}a^{3}-\frac{5}{36}a^{2}-\frac{1}{2}a+\frac{4}{9}$, $\frac{1}{288}a^{13}-\frac{1}{144}a^{11}+\frac{1}{144}a^{10}+\frac{1}{96}a^{9}-\frac{11}{72}a^{8}+\frac{35}{288}a^{7}-\frac{11}{288}a^{6}-\frac{9}{32}a^{5}+\frac{43}{144}a^{4}-\frac{11}{72}a^{3}+\frac{1}{9}a^{2}-\frac{4}{9}a-\frac{1}{9}$, $\frac{1}{150912}a^{14}-\frac{13}{25152}a^{13}+\frac{95}{75456}a^{12}-\frac{57}{8384}a^{11}-\frac{649}{150912}a^{10}+\frac{1831}{25152}a^{9}-\frac{1069}{16768}a^{8}+\frac{7385}{50304}a^{7}+\frac{4729}{150912}a^{6}+\frac{777}{4192}a^{5}+\frac{707}{3144}a^{4}-\frac{191}{6288}a^{3}+\frac{443}{3144}a^{2}+\frac{163}{1572}a-\frac{569}{2358}$, $\frac{1}{14931535104}a^{15}-\frac{431}{207382432}a^{14}-\frac{415181}{2488589184}a^{13}-\frac{11479199}{7465767552}a^{12}-\frac{28021861}{14931535104}a^{11}-\frac{92365}{20398272}a^{10}-\frac{715873633}{14931535104}a^{9}-\frac{582940051}{14931535104}a^{8}+\frac{634431619}{14931535104}a^{7}+\frac{15651307}{2488589184}a^{6}-\frac{273229115}{933220944}a^{5}-\frac{11432485}{1866441888}a^{4}+\frac{13261115}{38884206}a^{3}+\frac{9843711}{25922804}a^{2}+\frac{33358543}{116652618}a+\frac{41914043}{116652618}$ 1, a, a^2, a^3, a^4, a^5, 1/3*a^6 - 1/3*a^4 + 1/3*a^2 - 1/3, 1/3*a^7 - 1/3*a^5 + 1/3*a^3 - 1/3*a, 1/3*a^8 - 1/3, 1/6*a^9 - 1/2*a^5 - 1/2*a^3 - 1/2*a^2 - 1/6*a, 1/36*a^10 + 1/18*a^8 - 1/6*a^7 - 1/36*a^6 - 1/3*a^5 + 7/36*a^4 - 5/12*a^3 + 7/36*a^2 + 1/6*a - 1/9, 1/72*a^11 + 1/36*a^9 - 1/12*a^8 + 11/72*a^7 - 1/6*a^6 + 31/72*a^5 + 7/24*a^4 + 19/72*a^3 - 5/12*a^2 - 2/9*a, 1/144*a^12 - 1/72*a^10 - 1/24*a^9 + 1/48*a^8 - 1/12*a^7 - 13/144*a^6 - 17/48*a^5 - 11/48*a^4 - 11/24*a^3 - 5/36*a^2 - 1/2*a + 4/9, 1/288*a^13 - 1/144*a^11 + 1/144*a^10 + 1/96*a^9 - 11/72*a^8 + 35/288*a^7 - 11/288*a^6 - 9/32*a^5 + 43/144*a^4 - 11/72*a^3 + 1/9*a^2 - 4/9*a - 1/9, 1/150912*a^14 - 13/25152*a^13 + 95/75456*a^12 - 57/8384*a^11 - 649/150912*a^10 + 1831/25152*a^9 - 1069/16768*a^8 + 7385/50304*a^7 + 4729/150912*a^6 + 777/4192*a^5 + 707/3144*a^4 - 191/6288*a^3 + 443/3144*a^2 + 163/1572*a - 569/2358, 1/14931535104*a^15 - 431/207382432*a^14 - 415181/2488589184*a^13 - 11479199/7465767552*a^12 - 28021861/14931535104*a^11 - 92365/20398272*a^10 - 715873633/14931535104*a^9 - 582940051/14931535104*a^8 + 634431619/14931535104*a^7 + 15651307/2488589184*a^6 - 273229115/933220944*a^5 - 11432485/1866441888*a^4 + 13261115/38884206*a^3 + 9843711/25922804*a^2 + 33358543/116652618*a + 41914043/116652618

comment:Integral basis

sage:K.integral_basis()

gp:K.zk

magma:IntegralBasis(K);

oscar:basis(OK)

Monogenic:		No
Index:		Not computed
Inessential primes:		$2$, $3$

Class group and class number

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:		$C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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:	$15$	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:	$\frac{43}{143838}a^{15}-\frac{1739}{1534272}a^{14}-\frac{7087}{383568}a^{13}+\frac{159755}{2301408}a^{12}+\frac{1049011}{2301408}a^{11}-\frac{127631}{75456}a^{10}-\frac{1092091}{191784}a^{9}+\frac{31762243}{1534272}a^{8}+\frac{173708767}{4602816}a^{7}-\frac{603610807}{4602816}a^{6}-\frac{32908727}{255712}a^{5}+\frac{3113275}{7991}a^{4}+\frac{129399305}{575352}a^{3}-\frac{108042941}{287676}a^{2}-\frac{12679093}{47946}a-\frac{2277605}{71919}$, $\frac{3367}{3824676}a^{15}-\frac{312721}{122389632}a^{14}-\frac{3450359}{61194816}a^{13}+\frac{9601229}{61194816}a^{12}+\frac{9968845}{6799424}a^{11}-\frac{52145527}{13598848}a^{10}-\frac{1215199927}{61194816}a^{9}+\frac{1920839903}{40796544}a^{8}+\frac{2036621177}{13598848}a^{7}-\frac{12192758575}{40796544}a^{6}-\frac{1201319551}{1912338}a^{5}+\frac{1121587541}{1274892}a^{4}+\frac{21601951231}{15298704}a^{3}-\frac{5650622671}{7649352}a^{2}-\frac{1922763995}{1274892}a-\frac{905427719}{1912338}$, $\frac{19321}{81593088}a^{15}+\frac{28547}{61194816}a^{14}+\frac{1843349}{122389632}a^{13}-\frac{1180637}{40796544}a^{12}-\frac{94981057}{244779264}a^{11}+\frac{10927909}{15298704}a^{10}+\frac{1269659051}{244779264}a^{9}-\frac{2164982291}{244779264}a^{8}-\frac{9382248893}{244779264}a^{7}+\frac{6899214451}{122389632}a^{6}+\frac{4750558757}{30597408}a^{5}-\frac{5027934581}{30597408}a^{4}-\frac{1255257745}{3824676}a^{3}+\frac{79919389}{637446}a^{2}+\frac{102824896}{318723}a+\frac{203073419}{1912338}$, $\frac{157525}{244779264}a^{15}+\frac{28403}{13598848}a^{14}+\frac{5057369}{122389632}a^{13}-\frac{15660547}{122389632}a^{12}-\frac{263897363}{244779264}a^{11}+\frac{381886471}{122389632}a^{10}+\frac{3591140657}{244779264}a^{9}-\frac{9360057127}{244779264}a^{8}-\frac{27276932293}{244779264}a^{7}+\frac{14839530637}{61194816}a^{6}+\frac{14470554059}{30597408}a^{5}-\frac{21890166403}{30597408}a^{4}-\frac{5526973417}{5099568}a^{3}+\frac{4691590003}{7649352}a^{2}+\frac{1511464411}{1274892}a+\frac{116740327}{318723}$, $\frac{7949911}{14931535104}a^{15}+\frac{31393}{14247648}a^{14}+\frac{247090409}{7465767552}a^{13}-\frac{334393709}{2488589184}a^{12}-\frac{4105356623}{4977178368}a^{11}+\frac{199420853}{61194816}a^{10}+\frac{52247001485}{4977178368}a^{9}-\frac{592472383771}{14931535104}a^{8}-\frac{358767844567}{4977178368}a^{7}+\frac{1866076547273}{7465767552}a^{6}+\frac{41031397099}{155536824}a^{5}-\frac{1376235512713}{1866441888}a^{4}-\frac{120957922981}{233305236}a^{3}+\frac{13490728610}{19442103}a^{2}+\frac{71998099919}{116652618}a+\frac{13366968157}{116652618}$, $\frac{16510019}{14931535104}a^{15}+\frac{15241493}{3732883776}a^{14}+\frac{523930477}{7465767552}a^{13}-\frac{1869601519}{7465767552}a^{12}-\frac{8967336307}{4977178368}a^{11}+\frac{187042561}{30597408}a^{10}+\frac{119273006305}{4977178368}a^{9}-\frac{1119797725187}{14931535104}a^{8}-\frac{292326406133}{1659059456}a^{7}+\frac{3557205916415}{7465767552}a^{6}+\frac{148584294357}{207382432}a^{5}-\frac{2642263513583}{1866441888}a^{4}-\frac{92510310272}{58326309}a^{3}+\frac{300920060359}{233305236}a^{2}+\frac{210489065737}{116652618}a+\frac{56953508857}{116652618}$, $\frac{20823221}{14931535104}a^{15}+\frac{282716}{58326309}a^{14}+\frac{654978871}{7465767552}a^{13}-\frac{2216079917}{7465767552}a^{12}-\frac{33290485559}{14931535104}a^{11}+\frac{442777565}{61194816}a^{10}+\frac{437536750733}{14931535104}a^{9}-\frac{1323739301401}{14931535104}a^{8}-\frac{3170470663135}{14931535104}a^{7}+\frac{4200443391227}{7465767552}a^{6}+\frac{1581524761283}{1866441888}a^{5}-\frac{3117198399565}{1866441888}a^{4}-\frac{427896837643}{233305236}a^{3}+\frac{88899760981}{58326309}a^{2}+\frac{240727092023}{116652618}a+\frac{63673776107}{116652618}$, $\frac{1041461}{14931535104}a^{15}-\frac{107395}{622147296}a^{14}-\frac{6010633}{2488589184}a^{13}+\frac{82363925}{7465767552}a^{12}-\frac{151491497}{14931535104}a^{11}-\frac{17382079}{61194816}a^{10}+\frac{22824868027}{14931535104}a^{9}+\frac{56134286369}{14931535104}a^{8}-\frac{397693559809}{14931535104}a^{7}-\frac{203203335403}{7465767552}a^{6}+\frac{93843354877}{466610472}a^{5}+\frac{204369334847}{1866441888}a^{4}-\frac{13291720397}{19442103}a^{3}-\frac{62176288777}{233305236}a^{2}+\frac{94083693911}{116652618}a+\frac{16621363909}{38884206}$, $\frac{1678409}{1866441888}a^{15}-\frac{181363}{933220944}a^{14}+\frac{58181225}{933220944}a^{13}+\frac{6608219}{933220944}a^{12}-\frac{3357217327}{1866441888}a^{11}-\frac{166877}{5099568}a^{10}+\frac{5782854541}{207382432}a^{9}-\frac{3125272699}{1866441888}a^{8}-\frac{465681278333}{1866441888}a^{7}+\frac{1341942099}{51845608}a^{6}+\frac{198741053077}{155536824}a^{5}-\frac{49385596001}{466610472}a^{4}-\frac{265832762023}{77768412}a^{3}-\frac{45050278909}{233305236}a^{2}+\frac{417645134935}{116652618}a+\frac{9841324868}{6480701}$, $\frac{1970041}{2488589184}a^{15}+\frac{211169}{77768412}a^{14}+\frac{57658843}{1244294592}a^{13}-\frac{606327407}{3732883776}a^{12}-\frac{7942966201}{7465767552}a^{11}+\frac{117850063}{30597408}a^{10}+\frac{89462232307}{7465767552}a^{9}-\frac{340711779887}{7465767552}a^{8}-\frac{499211931425}{7465767552}a^{7}+\frac{1035105457513}{3732883776}a^{6}+\frac{74815079195}{466610472}a^{5}-\frac{360646986523}{466610472}a^{4}-\frac{30431762591}{233305236}a^{3}+\frac{75732892639}{116652618}a^{2}+\frac{31768076629}{116652618}a+\frac{6714776}{6480701}$, $\frac{133840099}{7465767552}a^{15}+\frac{51336653}{829529728}a^{14}+\frac{1056573623}{933220944}a^{13}-\frac{3538614767}{933220944}a^{12}-\frac{215526893651}{7465767552}a^{11}+\frac{3771407441}{40796544}a^{10}+\frac{2838199629377}{7465767552}a^{9}-\frac{528285876487}{466610472}a^{8}-\frac{10272410687159}{3732883776}a^{7}+\frac{17854524616985}{2488589184}a^{6}+\frac{10176732061745}{933220944}a^{5}-\frac{19807427066917}{933220944}a^{4}-\frac{2408095417533}{103691216}a^{3}+\frac{996264567601}{51845608}a^{2}+\frac{666166101119}{25922804}a+\frac{783639423481}{116652618}$, $\frac{9145043}{4977178368}a^{15}-\frac{6947753}{829529728}a^{14}+\frac{89048597}{829529728}a^{13}+\frac{3856277057}{7465767552}a^{12}-\frac{12017144773}{4977178368}a^{11}-\frac{517978643}{40796544}a^{10}+\frac{126107976055}{4977178368}a^{9}+\frac{2352900542173}{14931535104}a^{8}-\frac{537517890547}{4977178368}a^{7}-\frac{425571854859}{414764864}a^{6}-\frac{17413086377}{207382432}a^{5}+\frac{6047071485793}{1866441888}a^{4}+\frac{609231996025}{311073648}a^{3}-\frac{554277430939}{155536824}a^{2}-\frac{315754335835}{77768412}a-\frac{64242733159}{58326309}$, $\frac{20022003}{1659059456}a^{15}+\frac{380264567}{7465767552}a^{14}+\frac{5296610647}{7465767552}a^{13}-\frac{22786541729}{7465767552}a^{12}-\frac{81804411511}{4977178368}a^{11}+\frac{8895832715}{122389632}a^{10}+\frac{2816077893407}{14931535104}a^{9}-\frac{12982004818669}{14931535104}a^{8}-\frac{5458809837101}{4977178368}a^{7}+\frac{10073767632733}{1866441888}a^{6}+\frac{342096736289}{116652618}a^{5}-\frac{29640125761135}{1866441888}a^{4}-\frac{1015123189825}{311073648}a^{3}+\frac{7509724389619}{466610472}a^{2}+\frac{1143947081623}{233305236}a-\frac{80184102050}{58326309}$, $\frac{28499893}{7465767552}a^{15}+\frac{23043749}{2488589184}a^{14}-\frac{420620993}{1866441888}a^{13}-\frac{1038002077}{1866441888}a^{12}+\frac{39075711233}{7465767552}a^{11}+\frac{1633837969}{122389632}a^{10}-\frac{148442852225}{2488589184}a^{9}-\frac{606965269585}{3732883776}a^{8}+\frac{308113703471}{933220944}a^{7}+\frac{7828991514653}{7465767552}a^{6}-\frac{197553539191}{311073648}a^{5}-\frac{1558653541135}{466610472}a^{4}-\frac{106024741549}{103691216}a^{3}+\frac{1812066508843}{466610472}a^{2}+\frac{893757406283}{233305236}a+\frac{13085611889}{12961402}$, $\frac{49159787}{14931535104}a^{15}+\frac{28430285}{7465767552}a^{14}+\frac{1498531895}{7465767552}a^{13}-\frac{1700635085}{7465767552}a^{12}-\frac{73228118381}{14931535104}a^{11}+\frac{220313575}{40796544}a^{10}+\frac{911570091631}{14931535104}a^{9}-\frac{317294342915}{4977178368}a^{8}-\frac{6060470776651}{14931535104}a^{7}+\frac{473842505845}{1244294592}a^{6}+\frac{2570239098697}{1866441888}a^{5}-\frac{617120864075}{622147296}a^{4}-\frac{1959505688503}{933220944}a^{3}+\frac{247415358191}{466610472}a^{2}+\frac{301015465025}{233305236}a+\frac{20029537205}{58326309}$ 43/143838a^15 - 1739/1534272a^14 - 7087/383568a^13 + 159755/2301408a^12 + 1049011/2301408a^11 - 127631/75456a^10 - 1092091/191784a^9 + 31762243/1534272a^8 + 173708767/4602816a^7 - 603610807/4602816a^6 - 32908727/255712a^5 + 3113275/7991a^4 + 129399305/575352a^3 - 108042941/287676a^2 - 12679093/47946a - 2277605/71919, 3367/3824676a^15 - 312721/122389632a^14 - 3450359/61194816a^13 + 9601229/61194816a^12 + 9968845/6799424a^11 - 52145527/13598848a^10 - 1215199927/61194816a^9 + 1920839903/40796544a^8 + 2036621177/13598848a^7 - 12192758575/40796544a^6 - 1201319551/1912338a^5 + 1121587541/1274892a^4 + 21601951231/15298704a^3 - 5650622671/7649352a^2 - 1922763995/1274892a - 905427719/1912338, -19321/81593088a^15 + 28547/61194816a^14 + 1843349/122389632a^13 - 1180637/40796544a^12 - 94981057/244779264a^11 + 10927909/15298704a^10 + 1269659051/244779264a^9 - 2164982291/244779264a^8 - 9382248893/244779264a^7 + 6899214451/122389632a^6 + 4750558757/30597408a^5 - 5027934581/30597408a^4 - 1255257745/3824676a^3 + 79919389/637446a^2 + 102824896/318723a + 203073419/1912338, -157525/244779264a^15 + 28403/13598848a^14 + 5057369/122389632a^13 - 15660547/122389632a^12 - 263897363/244779264a^11 + 381886471/122389632a^10 + 3591140657/244779264a^9 - 9360057127/244779264a^8 - 27276932293/244779264a^7 + 14839530637/61194816a^6 + 14470554059/30597408a^5 - 21890166403/30597408a^4 - 5526973417/5099568a^3 + 4691590003/7649352a^2 + 1511464411/1274892a + 116740327/318723, -7949911/14931535104a^15 + 31393/14247648a^14 + 247090409/7465767552a^13 - 334393709/2488589184a^12 - 4105356623/4977178368a^11 + 199420853/61194816a^10 + 52247001485/4977178368a^9 - 592472383771/14931535104a^8 - 358767844567/4977178368a^7 + 1866076547273/7465767552a^6 + 41031397099/155536824a^5 - 1376235512713/1866441888a^4 - 120957922981/233305236a^3 + 13490728610/19442103a^2 + 71998099919/116652618a + 13366968157/116652618, -16510019/14931535104a^15 + 15241493/3732883776a^14 + 523930477/7465767552a^13 - 1869601519/7465767552a^12 - 8967336307/4977178368a^11 + 187042561/30597408a^10 + 119273006305/4977178368a^9 - 1119797725187/14931535104a^8 - 292326406133/1659059456a^7 + 3557205916415/7465767552a^6 + 148584294357/207382432a^5 - 2642263513583/1866441888a^4 - 92510310272/58326309a^3 + 300920060359/233305236a^2 + 210489065737/116652618a + 56953508857/116652618, -20823221/14931535104a^15 + 282716/58326309a^14 + 654978871/7465767552a^13 - 2216079917/7465767552a^12 - 33290485559/14931535104a^11 + 442777565/61194816a^10 + 437536750733/14931535104a^9 - 1323739301401/14931535104a^8 - 3170470663135/14931535104a^7 + 4200443391227/7465767552a^6 + 1581524761283/1866441888a^5 - 3117198399565/1866441888a^4 - 427896837643/233305236a^3 + 88899760981/58326309a^2 + 240727092023/116652618a + 63673776107/116652618, 1041461/14931535104a^15 - 107395/622147296a^14 - 6010633/2488589184a^13 + 82363925/7465767552a^12 - 151491497/14931535104a^11 - 17382079/61194816a^10 + 22824868027/14931535104a^9 + 56134286369/14931535104a^8 - 397693559809/14931535104a^7 - 203203335403/7465767552a^6 + 93843354877/466610472a^5 + 204369334847/1866441888a^4 - 13291720397/19442103a^3 - 62176288777/233305236a^2 + 94083693911/116652618a + 16621363909/38884206, -1678409/1866441888a^15 - 181363/933220944a^14 + 58181225/933220944a^13 + 6608219/933220944a^12 - 3357217327/1866441888a^11 - 166877/5099568a^10 + 5782854541/207382432a^9 - 3125272699/1866441888a^8 - 465681278333/1866441888a^7 + 1341942099/51845608a^6 + 198741053077/155536824a^5 - 49385596001/466610472a^4 - 265832762023/77768412a^3 - 45050278909/233305236a^2 + 417645134935/116652618a + 9841324868/6480701, -1970041/2488589184a^15 + 211169/77768412a^14 + 57658843/1244294592a^13 - 606327407/3732883776a^12 - 7942966201/7465767552a^11 + 117850063/30597408a^10 + 89462232307/7465767552a^9 - 340711779887/7465767552a^8 - 499211931425/7465767552a^7 + 1035105457513/3732883776a^6 + 74815079195/466610472a^5 - 360646986523/466610472a^4 - 30431762591/233305236a^3 + 75732892639/116652618a^2 + 31768076629/116652618a + 6714776/6480701, -133840099/7465767552a^15 + 51336653/829529728a^14 + 1056573623/933220944a^13 - 3538614767/933220944a^12 - 215526893651/7465767552a^11 + 3771407441/40796544a^10 + 2838199629377/7465767552a^9 - 528285876487/466610472a^8 - 10272410687159/3732883776a^7 + 17854524616985/2488589184a^6 + 10176732061745/933220944a^5 - 19807427066917/933220944a^4 - 2408095417533/103691216a^3 + 996264567601/51845608a^2 + 666166101119/25922804a + 783639423481/116652618, -9145043/4977178368a^15 - 6947753/829529728a^14 + 89048597/829529728a^13 + 3856277057/7465767552a^12 - 12017144773/4977178368a^11 - 517978643/40796544a^10 + 126107976055/4977178368a^9 + 2352900542173/14931535104a^8 - 537517890547/4977178368a^7 - 425571854859/414764864a^6 - 17413086377/207382432a^5 + 6047071485793/1866441888a^4 + 609231996025/311073648a^3 - 554277430939/155536824a^2 - 315754335835/77768412a - 64242733159/58326309, -20022003/1659059456a^15 + 380264567/7465767552a^14 + 5296610647/7465767552a^13 - 22786541729/7465767552a^12 - 81804411511/4977178368a^11 + 8895832715/122389632a^10 + 2816077893407/14931535104a^9 - 12982004818669/14931535104a^8 - 5458809837101/4977178368a^7 + 10073767632733/1866441888a^6 + 342096736289/116652618a^5 - 29640125761135/1866441888a^4 - 1015123189825/311073648a^3 + 7509724389619/466610472a^2 + 1143947081623/233305236a - 80184102050/58326309, 28499893/7465767552a^15 + 23043749/2488589184a^14 - 420620993/1866441888a^13 - 1038002077/1866441888a^12 + 39075711233/7465767552a^11 + 1633837969/122389632a^10 - 148442852225/2488589184a^9 - 606965269585/3732883776a^8 + 308113703471/933220944a^7 + 7828991514653/7465767552a^6 - 197553539191/311073648a^5 - 1558653541135/466610472a^4 - 106024741549/103691216a^3 + 1812066508843/466610472a^2 + 893757406283/233305236a + 13085611889/12961402, -49159787/14931535104a^15 + 28430285/7465767552a^14 + 1498531895/7465767552a^13 - 1700635085/7465767552a^12 - 73228118381/14931535104a^11 + 220313575/40796544a^10 + 911570091631/14931535104a^9 - 317294342915/4977178368a^8 - 6060470776651/14931535104a^7 + 473842505845/1244294592a^6 + 2570239098697/1866441888a^5 - 617120864075/622147296a^4 - 1959505688503/933220944a^3 + 247415358191/466610472a^2 + 301015465025/233305236*a + 20029537205/58326309 (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:	$ 789769674267.3774 $ (assuming GRH)	comment:Regulator sage:K.regulator() gp:K.reg magma:Regulator(K); oscar:regulator(K)
Unit signature rank:	$ 13 $ (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{16}\cdot(2\pi)^{0}\cdot 789769674267.3774 \cdot 1}{2\cdot\sqrt{2111549459996061430585501425390625}}\cr\approx \mathstrut & 0.563183288336982 \end{aligned}\] (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^16 - 74*x^14 - 6*x^13 + 2275*x^12 + 372*x^11 - 37413*x^10 - 9315*x^9 + 351975*x^8 + 120534*x^7 - 1864956*x^6 - 852984*x^5 + 5001568*x^4 + 3165600*x^3 - 4743104*x^2 - 4932096*x - 1200896) 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^16 - 74*x^14 - 6*x^13 + 2275*x^12 + 372*x^11 - 37413*x^10 - 9315*x^9 + 351975*x^8 + 120534*x^7 - 1864956*x^6 - 852984*x^5 + 5001568*x^4 + 3165600*x^3 - 4743104*x^2 - 4932096*x - 1200896, 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^16 - 74*x^14 - 6*x^13 + 2275*x^12 + 372*x^11 - 37413*x^10 - 9315*x^9 + 351975*x^8 + 120534*x^7 - 1864956*x^6 - 852984*x^5 + 5001568*x^4 + 3165600*x^3 - 4743104*x^2 - 4932096*x - 1200896); 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 = polynomial_ring(QQ); K, a = number_field(x^16 - 74*x^14 - 6*x^13 + 2275*x^12 + 372*x^11 - 37413*x^10 - 9315*x^9 + 351975*x^8 + 120534*x^7 - 1864956*x^6 - 852984*x^5 + 5001568*x^4 + 3165600*x^3 - 4743104*x^2 - 4932096*x - 1200896); 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

$C_2^5:(C_2\times S_4)$ (as 16T1298):

comment:Galois group

sage:K.galois_group()

gp:polgalois(K.pol)

magma:G = GaloisGroup(K);

oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)

A solvable group of order 1536

The 62 conjugacy class representatives for $C_2^5:(C_2\times S_4)$

Character table for $C_2^5:(C_2\times S_4)$

Intermediate fields

$\Q(\sqrt{5}) $, 4.4.14197.1, 8.8.125971755625.1

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(L)]

magma:L := Subfields(K); L[2..#L];

oscar:subfields(K)[2:end-1]

Sibling fields

Degree 16 siblings:	data not computed
Degree 32 siblings:	data not computed
Minimal sibling:	This field is its own minimal sibling

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	${\href{/padicField/2.4.0.1}{4} }^{4}$	${\href{/padicField/3.2.0.1}{2} }^{8}$	R	R	${\href{/padicField/11.8.0.1}{8} }^{2}$	${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.4.0.1}{4} }$	${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.4.0.1}{4} }$	${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$	${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$	${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$	R	${\href{/padicField/37.2.0.1}{2} }^{8}$	R	${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$	${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.4.0.1}{4} }$	${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.4.0.1}{4} }$	${\href{/padicField/59.8.0.1}{8} }^{2}$

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
$5$ 5	5.8.2.8a1.2	$x^{16} + 2 x^{12} + 6 x^{10} + 8 x^{9} + 5 x^{8} + 6 x^{6} + 8 x^{5} + 13 x^{4} + 24 x^{3} + 28 x^{2} + 16 x + 9$	$2$	$8$	$8$	$C_8\times C_2$	$$[\ ]_{2}^{8}$$
$7$ 7	7.2.2.2a1.1	$x^{4} + 12 x^{3} + 42 x^{2} + 43 x + 9$	$2$	$2$	$2$	$C_4$	$$[\ ]_{2}^{2}$$
$7$ 7	7.12.1.0a1.1	$x^{12} + 2 x^{8} + 5 x^{7} + 3 x^{6} + 2 x^{5} + 4 x^{4} + 5 x^{2} + 3$	$1$	$12$	$0$	$C_{12}$	$$[\ ]^{12}$$
$31$ 31	31.2.1.0a1.1	$x^{2} + 29 x + 3$	$1$	$2$	$0$	$C_2$	$$[\ ]^{2}$$
	31.2.1.0a1.1	$x^{2} + 29 x + 3$	$1$	$2$	$0$	$C_2$	$$[\ ]^{2}$$
	31.2.1.0a1.1	$x^{2} + 29 x + 3$	$1$	$2$	$0$	$C_2$	$$[\ ]^{2}$$
	31.2.1.0a1.1	$x^{2} + 29 x + 3$	$1$	$2$	$0$	$C_2$	$$[\ ]^{2}$$
	31.1.2.1a1.1	$x^{2} + 31$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
	31.2.1.0a1.1	$x^{2} + 29 x + 3$	$1$	$2$	$0$	$C_2$	$$[\ ]^{2}$$
	31.1.2.1a1.1	$x^{2} + 31$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
	31.2.1.0a1.1	$x^{2} + 29 x + 3$	$1$	$2$	$0$	$C_2$	$$[\ ]^{2}$$
$41$ 41	41.8.1.0a1.1	$x^{8} + 5 x^{4} + 32 x^{3} + 20 x^{2} + 6 x + 6$	$1$	$8$	$0$	$C_8$	$$[\ ]^{8}$$
$41$ 41	41.4.2.4a1.1	$x^{8} + 46 x^{5} + 12 x^{4} + 529 x^{2} + 317 x + 36$	$2$	$4$	$4$	$C_8$	$$[\ ]_{2}^{4}$$
$14197$ 14197		Deg $4$	$1$	$4$	$0$	$C_4$	$$[\ ]^{4}$$
		Deg $4$	$1$	$4$	$0$	$C_4$	$$[\ ]^{4}$$
		Deg $8$	$2$	$4$	$4$

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