LMFDB - Number field 16.16.20542690914744340526341740071734336.1

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

Normalized defining polynomial

comment:Define the number field

sage:x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 61*x^14 + 110*x^13 + 1499*x^12 - 2405*x^11 - 18857*x^10 + 26688*x^9 + 126882*x^8 - 158448*x^7 - 431612*x^6 + 476792*x^5 + 629344*x^4 - 563968*x^3 - 315264*x^2 + 136192*x + 59648)

gp:K = bnfinit(y^16 - 2*y^15 - 61*y^14 + 110*y^13 + 1499*y^12 - 2405*y^11 - 18857*y^10 + 26688*y^9 + 126882*y^8 - 158448*y^7 - 431612*y^6 + 476792*y^5 + 629344*y^4 - 563968*y^3 - 315264*y^2 + 136192*y + 59648, 1)

magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 - 61*x^14 + 110*x^13 + 1499*x^12 - 2405*x^11 - 18857*x^10 + 26688*x^9 + 126882*x^8 - 158448*x^7 - 431612*x^6 + 476792*x^5 + 629344*x^4 - 563968*x^3 - 315264*x^2 + 136192*x + 59648);

oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 2*x^15 - 61*x^14 + 110*x^13 + 1499*x^12 - 2405*x^11 - 18857*x^10 + 26688*x^9 + 126882*x^8 - 158448*x^7 - 431612*x^6 + 476792*x^5 + 629344*x^4 - 563968*x^3 - 315264*x^2 + 136192*x + 59648)

$ x^{16} - 2 x^{15} - 61 x^{14} + 110 x^{13} + 1499 x^{12} - 2405 x^{11} - 18857 x^{10} + 26688 x^{9} + \cdots + 59648 $ x^16 - 2*x^15 - 61*x^14 + 110*x^13 + 1499*x^12 - 2405*x^11 - 18857*x^10 + 26688*x^9 + 126882*x^8 - 158448*x^7 - 431612*x^6 + 476792*x^5 + 629344*x^4 - 563968*x^3 - 315264*x^2 + 136192*x + 59648

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:	$20542690914744340526341740071734336$ 20542690914744340526341740071734336 $\medspace = 2^{6}\cdot 7^{2}\cdot 13^{8}\cdot 17^{4}\cdot 17609^{4}$	comment:Discriminant sage:K.disc() gp:K.disc magma:OK := Integers(K); Discriminant(OK); oscar:OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$139.49$	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:	$2^{3/2}7^{1/2}13^{1/2}17^{1/2}17609^{1/2}\approx 14762.417959128512$
Ramified primes:	$2$, $7$, $13$, $17$, $17609$ 2, 7, 13, 17, 17609	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{4}a^{6}+\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{11}-\frac{1}{8}a^{9}-\frac{1}{8}a^{7}-\frac{3}{8}a^{6}-\frac{3}{8}a^{5}+\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{16}a^{12}-\frac{1}{16}a^{10}-\frac{1}{4}a^{9}-\frac{1}{16}a^{8}-\frac{7}{16}a^{7}+\frac{5}{16}a^{6}+\frac{3}{8}a^{5}+\frac{1}{8}a^{4}+\frac{1}{4}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{32}a^{13}-\frac{1}{32}a^{11}-\frac{1}{8}a^{10}+\frac{7}{32}a^{9}-\frac{7}{32}a^{8}-\frac{3}{32}a^{7}-\frac{5}{16}a^{6}-\frac{3}{16}a^{5}-\frac{1}{8}a^{4}-\frac{3}{8}a^{3}-\frac{1}{2}a$, $\frac{1}{2684289728}a^{14}+\frac{7286171}{1342144864}a^{13}-\frac{1188473}{50646976}a^{12}-\frac{721645}{1342144864}a^{11}-\frac{317661669}{2684289728}a^{10}-\frac{45411117}{2684289728}a^{9}+\frac{69489743}{2684289728}a^{8}+\frac{35103247}{335536216}a^{7}+\frac{145144449}{1342144864}a^{6}+\frac{7193408}{41942027}a^{5}-\frac{200131355}{671072432}a^{4}-\frac{51891025}{335536216}a^{3}-\frac{68357}{791359}a^{2}+\frac{2798212}{41942027}a+\frac{3914863}{41942027}$, $\frac{1}{316746187904}a^{15}-\frac{1}{39593273488}a^{14}-\frac{72982221}{5976343168}a^{13}-\frac{2303355115}{79186546976}a^{12}+\frac{4425785543}{316746187904}a^{11}-\frac{8508600895}{316746187904}a^{10}+\frac{65315918101}{316746187904}a^{9}-\frac{10437787105}{158373093952}a^{8}-\frac{4434501811}{158373093952}a^{7}+\frac{28271423975}{79186546976}a^{6}-\frac{5741498867}{79186546976}a^{5}+\frac{14994629}{19796636744}a^{4}+\frac{104480591}{373521448}a^{3}+\frac{3798901135}{9898318372}a^{2}+\frac{241296342}{2474579593}a-\frac{5196769}{46690181}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, 1/2*a^8 - 1/2*a^6 - 1/2*a^4 - 1/2*a^3 - 1/2*a^2, 1/2*a^9 - 1/2*a^7 - 1/2*a^5 - 1/2*a^4 - 1/2*a^3, 1/4*a^10 - 1/4*a^8 - 1/4*a^6 + 1/4*a^5 + 1/4*a^4 - 1/2*a^3 - 1/2*a^2, 1/8*a^11 - 1/8*a^9 - 1/8*a^7 - 3/8*a^6 - 3/8*a^5 + 1/4*a^4 - 1/4*a^3 - 1/2*a, 1/16*a^12 - 1/16*a^10 - 1/4*a^9 - 1/16*a^8 - 7/16*a^7 + 5/16*a^6 + 3/8*a^5 + 1/8*a^4 + 1/4*a^3 - 1/4*a^2, 1/32*a^13 - 1/32*a^11 - 1/8*a^10 + 7/32*a^9 - 7/32*a^8 - 3/32*a^7 - 5/16*a^6 - 3/16*a^5 - 1/8*a^4 - 3/8*a^3 - 1/2*a, 1/2684289728*a^14 + 7286171/1342144864*a^13 - 1188473/50646976*a^12 - 721645/1342144864*a^11 - 317661669/2684289728*a^10 - 45411117/2684289728*a^9 + 69489743/2684289728*a^8 + 35103247/335536216*a^7 + 145144449/1342144864*a^6 + 7193408/41942027*a^5 - 200131355/671072432*a^4 - 51891025/335536216*a^3 - 68357/791359*a^2 + 2798212/41942027*a + 3914863/41942027, 1/316746187904*a^15 - 1/39593273488*a^14 - 72982221/5976343168*a^13 - 2303355115/79186546976*a^12 + 4425785543/316746187904*a^11 - 8508600895/316746187904*a^10 + 65315918101/316746187904*a^9 - 10437787105/158373093952*a^8 - 4434501811/158373093952*a^7 + 28271423975/79186546976*a^6 - 5741498867/79186546976*a^5 + 14994629/19796636744*a^4 + 104480591/373521448*a^3 + 3798901135/9898318372*a^2 + 241296342/2474579593*a - 5196769/46690181

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:		$C_{2}\times C_{2}\times C_{2}\times C_{2}$, which has order $16$ (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:	$\frac{134393}{1769531776}a^{15}-\frac{148821}{884765888}a^{14}-\frac{6385217}{1769531776}a^{13}+\frac{7235091}{884765888}a^{12}+\frac{109855487}{1769531776}a^{11}-\frac{267725573}{1769531776}a^{10}-\frac{782478653}{1769531776}a^{9}+\frac{581511659}{442382944}a^{8}+\frac{720847651}{884765888}a^{7}-\frac{1150859041}{221191472}a^{6}+\frac{1431923927}{442382944}a^{5}+\frac{1580576513}{221191472}a^{4}-\frac{921086959}{110595736}a^{3}-\frac{57581765}{13824467}a^{2}+\frac{32719478}{13824467}a+\frac{58957234}{13824467}$, $\frac{23577}{83884054}a^{15}+\frac{461547}{335536216}a^{14}+\frac{20100385}{1342144864}a^{13}-\frac{23721707}{335536216}a^{12}-\frac{421479817}{1342144864}a^{11}+\frac{117608637}{83884054}a^{10}+\frac{4380336283}{1342144864}a^{9}-\frac{18085714555}{1342144864}a^{8}-\frac{23037057731}{1342144864}a^{7}+\frac{43438673739}{671072432}a^{6}+\frac{27277547785}{671072432}a^{5}-\frac{49018428181}{335536216}a^{4}-\frac{10700436519}{335536216}a^{3}+\frac{11187788595}{83884054}a^{2}-\frac{387015099}{83884054}a-\frac{1074169654}{41942027}$, $\frac{628677}{5368579456}a^{15}-\frac{21429}{83884054}a^{14}-\frac{38997925}{5368579456}a^{13}+\frac{20365475}{1342144864}a^{12}+\frac{969296131}{5368579456}a^{11}-\frac{1945226067}{5368579456}a^{10}-\frac{12217603471}{5368579456}a^{9}+\frac{11881253839}{2684289728}a^{8}+\frac{40402771393}{2684289728}a^{7}-\frac{38746537553}{1342144864}a^{6}-\frac{64114448267}{1342144864}a^{5}+\frac{31283322705}{335536216}a^{4}+\frac{17655637305}{335536216}a^{3}-\frac{18700533629}{167768108}a^{2}-\frac{275481987}{83884054}a+\frac{874360717}{41942027}$, $\frac{107819541}{158373093952}a^{15}+\frac{309425487}{158373093952}a^{14}+\frac{6167663885}{158373093952}a^{13}-\frac{16622149315}{158373093952}a^{12}-\frac{140684568119}{158373093952}a^{11}+\frac{88049824905}{39593273488}a^{10}+\frac{809985068351}{79186546976}a^{9}-\frac{3742880075547}{158373093952}a^{8}-\frac{1217022593925}{19796636744}a^{7}+\frac{10458714395401}{79186546976}a^{6}+\frac{871419876573}{4949159186}a^{5}-\frac{14536996170509}{39593273488}a^{4}-\frac{860204672969}{4949159186}a^{3}+\frac{3988125943463}{9898318372}a^{2}-\frac{28220654425}{4949159186}a-\frac{196172488329}{2474579593}$, $\frac{23577}{83884054}a^{15}+\frac{461547}{335536216}a^{14}+\frac{20100385}{1342144864}a^{13}-\frac{23721707}{335536216}a^{12}-\frac{421479817}{1342144864}a^{11}+\frac{117608637}{83884054}a^{10}+\frac{4380336283}{1342144864}a^{9}-\frac{18085714555}{1342144864}a^{8}-\frac{23037057731}{1342144864}a^{7}+\frac{43438673739}{671072432}a^{6}+\frac{27277547785}{671072432}a^{5}-\frac{49018428181}{335536216}a^{4}-\frac{10700436519}{335536216}a^{3}+\frac{11187788595}{83884054}a^{2}-\frac{387015099}{83884054}a-\frac{990285600}{41942027}$, $\frac{1176534597}{316746187904}a^{15}-\frac{1722130257}{158373093952}a^{14}-\frac{67552545401}{316746187904}a^{13}+\frac{92964887331}{158373093952}a^{12}+\frac{1548528822927}{316746187904}a^{11}-\frac{3964357383081}{316746187904}a^{10}-\frac{17949783116253}{316746187904}a^{9}+\frac{5308369838027}{39593273488}a^{8}+\frac{54422644994989}{158373093952}a^{7}-\frac{29951910368861}{39593273488}a^{6}-\frac{78888164944635}{79186546976}a^{5}+\frac{84037834942057}{39593273488}a^{4}+\frac{9923018189413}{9898318372}a^{3}-\frac{23075943006281}{9898318372}a^{2}+\frac{13383148869}{2474579593}a+\frac{1120553508629}{2474579593}$, $\frac{609565345}{316746187904}a^{15}+\frac{343059219}{79186546976}a^{14}+\frac{37321723469}{316746187904}a^{13}-\frac{2458978605}{9898318372}a^{12}-\frac{916440021379}{316746187904}a^{11}+\frac{1808170972435}{316746187904}a^{10}+\frac{11415974525507}{316746187904}a^{9}-\frac{10614628877779}{158373093952}a^{8}-\frac{37259556177783}{158373093952}a^{7}+\frac{33349663718191}{79186546976}a^{6}+\frac{58038084819573}{79186546976}a^{5}-\frac{13072258715515}{9898318372}a^{4}-\frac{3835564928911}{4949159186}a^{3}+\frac{7673025634465}{4949159186}a^{2}-\frac{2525039904}{2474579593}a-\frac{720050795724}{2474579593}$, $\frac{107466717}{79186546976}a^{15}+\frac{113802535}{158373093952}a^{14}-\frac{372824275}{4949159186}a^{13}-\frac{5328729187}{158373093952}a^{12}+\frac{16267006903}{9898318372}a^{11}+\frac{78896245015}{158373093952}a^{10}-\frac{2829444938201}{158373093952}a^{9}-\frac{223563758923}{158373093952}a^{8}+\frac{4007815003703}{39593273488}a^{7}-\frac{1849769664461}{79186546976}a^{6}-\frac{5576005493855}{19796636744}a^{5}+\frac{6400703255655}{39593273488}a^{4}+\frac{5772539690257}{19796636744}a^{3}-\frac{619965032560}{2474579593}a^{2}-\frac{70823453116}{2474579593}a+\frac{115056819213}{2474579593}$, $\frac{352528659}{158373093952}a^{15}-\frac{906451169}{158373093952}a^{14}-\frac{21255479843}{158373093952}a^{13}+\frac{50451168933}{158373093952}a^{12}+\frac{514061319969}{158373093952}a^{11}-\frac{279294878699}{39593273488}a^{10}-\frac{3154400974749}{79186546976}a^{9}+\frac{12545631521205}{158373093952}a^{8}+\frac{634105620778}{2474579593}a^{7}-\frac{37513922862063}{79186546976}a^{6}-\frac{15597649360795}{19796636744}a^{5}+\frac{56185366683159}{39593273488}a^{4}+\frac{8276432966513}{9898318372}a^{3}-\frac{16115208616625}{9898318372}a^{2}-\frac{39892278879}{2474579593}a+\frac{796151341504}{2474579593}$, $\frac{12807507}{19796636744}a^{15}+\frac{381943341}{158373093952}a^{14}+\frac{2650868659}{79186546976}a^{13}-\frac{19647493081}{158373093952}a^{12}-\frac{53270435985}{79186546976}a^{11}+\frac{393075581495}{158373093952}a^{10}+\frac{1044568557063}{158373093952}a^{9}-\frac{3863957968365}{158373093952}a^{8}-\frac{79751167937}{2474579593}a^{7}+\frac{9694821018137}{79186546976}a^{6}+\frac{1388105168509}{19796636744}a^{5}-\frac{65413863129}{221191472}a^{4}-\frac{934298967311}{19796636744}a^{3}+\frac{1452620195767}{4949159186}a^{2}-\frac{85707848673}{4949159186}a-\frac{143451342224}{2474579593}$, $\frac{48396195}{39593273488}a^{15}-\frac{19240607}{9898318372}a^{14}-\frac{740505215}{9898318372}a^{13}+\frac{4298685921}{39593273488}a^{12}+\frac{4541443054}{2474579593}a^{11}-\frac{48030060653}{19796636744}a^{10}-\frac{452158228969}{19796636744}a^{9}+\frac{137320819937}{4949159186}a^{8}+\frac{369546087880}{2474579593}a^{7}-\frac{6773897538951}{39593273488}a^{6}-\frac{9348740950661}{19796636744}a^{5}+\frac{10550306362807}{19796636744}a^{4}+\frac{5429905644463}{9898318372}a^{3}-\frac{6089790296375}{9898318372}a^{2}-\frac{463884896629}{4949159186}a+\frac{226746485493}{2474579593}$, $\frac{9438647}{158373093952}a^{15}-\frac{82052271}{158373093952}a^{14}+\frac{1025260457}{158373093952}a^{13}+\frac{3877754483}{158373093952}a^{12}-\frac{36535766647}{158373093952}a^{11}-\frac{8637689341}{19796636744}a^{10}+\frac{151580421159}{39593273488}a^{9}+\frac{578920441789}{158373093952}a^{8}-\frac{2570436428223}{79186546976}a^{7}-\frac{1182578107267}{79186546976}a^{6}+\frac{5469575817653}{39593273488}a^{5}+\frac{1302242463495}{39593273488}a^{4}-\frac{5198635686471}{19796636744}a^{3}-\frac{672583470067}{9898318372}a^{2}+\frac{8573964233}{46690181}a+\frac{178262825152}{2474579593}$, $\frac{1080339799}{316746187904}a^{15}-\frac{598694361}{39593273488}a^{14}-\frac{52842278599}{316746187904}a^{13}+\frac{60066001403}{79186546976}a^{12}+\frac{971060164673}{316746187904}a^{11}-\frac{4610785401577}{316746187904}a^{10}-\frac{8131373823053}{316746187904}a^{9}+\frac{21076800444233}{158373093952}a^{8}+\frac{267670042439}{2988171584}a^{7}-\frac{873116245799}{1494085792}a^{6}-\frac{79778265849}{1494085792}a^{5}+\frac{21713923967625}{19796636744}a^{4}-\frac{2721278904667}{19796636744}a^{3}-\frac{7694613270219}{9898318372}a^{2}+\frac{241945747215}{2474579593}a+\frac{343491993620}{2474579593}$, $\frac{163486899}{79186546976}a^{15}-\frac{15341761}{2474579593}a^{14}-\frac{1956051309}{19796636744}a^{13}+\frac{12014291401}{39593273488}a^{12}+\frac{68715042177}{39593273488}a^{11}-\frac{447150270927}{79186546976}a^{10}-\frac{260060382549}{19796636744}a^{9}+\frac{3911918481131}{79186546976}a^{8}+\frac{2643219910339}{79186546976}a^{7}-\frac{493478811376}{2474579593}a^{6}+\frac{1596128215609}{39593273488}a^{5}+\frac{2916954059095}{9898318372}a^{4}-\frac{3061619138021}{19796636744}a^{3}-\frac{1143215915141}{9898318372}a^{2}+\frac{98401567725}{2474579593}a+\frac{44540747247}{2474579593}$, $\frac{295192477}{316746187904}a^{15}-\frac{14764059}{158373093952}a^{14}-\frac{16977429725}{316746187904}a^{13}+\frac{871865377}{158373093952}a^{12}+\frac{386774551299}{316746187904}a^{11}-\frac{54187083221}{316746187904}a^{10}-\frac{4420642674685}{316746187904}a^{9}+\frac{61748920049}{19796636744}a^{8}+\frac{13171273645255}{158373093952}a^{7}-\frac{295895305277}{9898318372}a^{6}-\frac{18954212124477}{79186546976}a^{5}+\frac{5080531507247}{39593273488}a^{4}+\frac{4925294305671}{19796636744}a^{3}-\frac{432043525362}{2474579593}a^{2}-\frac{1026385276}{46690181}a+\frac{57594434662}{2474579593}$ 134393/1769531776a^15 - 148821/884765888a^14 - 6385217/1769531776a^13 + 7235091/884765888a^12 + 109855487/1769531776a^11 - 267725573/1769531776a^10 - 782478653/1769531776a^9 + 581511659/442382944a^8 + 720847651/884765888a^7 - 1150859041/221191472a^6 + 1431923927/442382944a^5 + 1580576513/221191472a^4 - 921086959/110595736a^3 - 57581765/13824467a^2 + 32719478/13824467a + 58957234/13824467, -23577/83884054a^15 + 461547/335536216a^14 + 20100385/1342144864a^13 - 23721707/335536216a^12 - 421479817/1342144864a^11 + 117608637/83884054a^10 + 4380336283/1342144864a^9 - 18085714555/1342144864a^8 - 23037057731/1342144864a^7 + 43438673739/671072432a^6 + 27277547785/671072432a^5 - 49018428181/335536216a^4 - 10700436519/335536216a^3 + 11187788595/83884054a^2 - 387015099/83884054a - 1074169654/41942027, 628677/5368579456a^15 - 21429/83884054a^14 - 38997925/5368579456a^13 + 20365475/1342144864a^12 + 969296131/5368579456a^11 - 1945226067/5368579456a^10 - 12217603471/5368579456a^9 + 11881253839/2684289728a^8 + 40402771393/2684289728a^7 - 38746537553/1342144864a^6 - 64114448267/1342144864a^5 + 31283322705/335536216a^4 + 17655637305/335536216a^3 - 18700533629/167768108a^2 - 275481987/83884054a + 874360717/41942027, -107819541/158373093952a^15 + 309425487/158373093952a^14 + 6167663885/158373093952a^13 - 16622149315/158373093952a^12 - 140684568119/158373093952a^11 + 88049824905/39593273488a^10 + 809985068351/79186546976a^9 - 3742880075547/158373093952a^8 - 1217022593925/19796636744a^7 + 10458714395401/79186546976a^6 + 871419876573/4949159186a^5 - 14536996170509/39593273488a^4 - 860204672969/4949159186a^3 + 3988125943463/9898318372a^2 - 28220654425/4949159186a - 196172488329/2474579593, -23577/83884054a^15 + 461547/335536216a^14 + 20100385/1342144864a^13 - 23721707/335536216a^12 - 421479817/1342144864a^11 + 117608637/83884054a^10 + 4380336283/1342144864a^9 - 18085714555/1342144864a^8 - 23037057731/1342144864a^7 + 43438673739/671072432a^6 + 27277547785/671072432a^5 - 49018428181/335536216a^4 - 10700436519/335536216a^3 + 11187788595/83884054a^2 - 387015099/83884054a - 990285600/41942027, 1176534597/316746187904a^15 - 1722130257/158373093952a^14 - 67552545401/316746187904a^13 + 92964887331/158373093952a^12 + 1548528822927/316746187904a^11 - 3964357383081/316746187904a^10 - 17949783116253/316746187904a^9 + 5308369838027/39593273488a^8 + 54422644994989/158373093952a^7 - 29951910368861/39593273488a^6 - 78888164944635/79186546976a^5 + 84037834942057/39593273488a^4 + 9923018189413/9898318372a^3 - 23075943006281/9898318372a^2 + 13383148869/2474579593a + 1120553508629/2474579593, -609565345/316746187904a^15 + 343059219/79186546976a^14 + 37321723469/316746187904a^13 - 2458978605/9898318372a^12 - 916440021379/316746187904a^11 + 1808170972435/316746187904a^10 + 11415974525507/316746187904a^9 - 10614628877779/158373093952a^8 - 37259556177783/158373093952a^7 + 33349663718191/79186546976a^6 + 58038084819573/79186546976a^5 - 13072258715515/9898318372a^4 - 3835564928911/4949159186a^3 + 7673025634465/4949159186a^2 - 2525039904/2474579593a - 720050795724/2474579593, 107466717/79186546976a^15 + 113802535/158373093952a^14 - 372824275/4949159186a^13 - 5328729187/158373093952a^12 + 16267006903/9898318372a^11 + 78896245015/158373093952a^10 - 2829444938201/158373093952a^9 - 223563758923/158373093952a^8 + 4007815003703/39593273488a^7 - 1849769664461/79186546976a^6 - 5576005493855/19796636744a^5 + 6400703255655/39593273488a^4 + 5772539690257/19796636744a^3 - 619965032560/2474579593a^2 - 70823453116/2474579593a + 115056819213/2474579593, 352528659/158373093952a^15 - 906451169/158373093952a^14 - 21255479843/158373093952a^13 + 50451168933/158373093952a^12 + 514061319969/158373093952a^11 - 279294878699/39593273488a^10 - 3154400974749/79186546976a^9 + 12545631521205/158373093952a^8 + 634105620778/2474579593a^7 - 37513922862063/79186546976a^6 - 15597649360795/19796636744a^5 + 56185366683159/39593273488a^4 + 8276432966513/9898318372a^3 - 16115208616625/9898318372a^2 - 39892278879/2474579593a + 796151341504/2474579593, -12807507/19796636744a^15 + 381943341/158373093952a^14 + 2650868659/79186546976a^13 - 19647493081/158373093952a^12 - 53270435985/79186546976a^11 + 393075581495/158373093952a^10 + 1044568557063/158373093952a^9 - 3863957968365/158373093952a^8 - 79751167937/2474579593a^7 + 9694821018137/79186546976a^6 + 1388105168509/19796636744a^5 - 65413863129/221191472a^4 - 934298967311/19796636744a^3 + 1452620195767/4949159186a^2 - 85707848673/4949159186a - 143451342224/2474579593, 48396195/39593273488a^15 - 19240607/9898318372a^14 - 740505215/9898318372a^13 + 4298685921/39593273488a^12 + 4541443054/2474579593a^11 - 48030060653/19796636744a^10 - 452158228969/19796636744a^9 + 137320819937/4949159186a^8 + 369546087880/2474579593a^7 - 6773897538951/39593273488a^6 - 9348740950661/19796636744a^5 + 10550306362807/19796636744a^4 + 5429905644463/9898318372a^3 - 6089790296375/9898318372a^2 - 463884896629/4949159186a + 226746485493/2474579593, -9438647/158373093952a^15 - 82052271/158373093952a^14 + 1025260457/158373093952a^13 + 3877754483/158373093952a^12 - 36535766647/158373093952a^11 - 8637689341/19796636744a^10 + 151580421159/39593273488a^9 + 578920441789/158373093952a^8 - 2570436428223/79186546976a^7 - 1182578107267/79186546976a^6 + 5469575817653/39593273488a^5 + 1302242463495/39593273488a^4 - 5198635686471/19796636744a^3 - 672583470067/9898318372a^2 + 8573964233/46690181a + 178262825152/2474579593, 1080339799/316746187904a^15 - 598694361/39593273488a^14 - 52842278599/316746187904a^13 + 60066001403/79186546976a^12 + 971060164673/316746187904a^11 - 4610785401577/316746187904a^10 - 8131373823053/316746187904a^9 + 21076800444233/158373093952a^8 + 267670042439/2988171584a^7 - 873116245799/1494085792a^6 - 79778265849/1494085792a^5 + 21713923967625/19796636744a^4 - 2721278904667/19796636744a^3 - 7694613270219/9898318372a^2 + 241945747215/2474579593a + 343491993620/2474579593, 163486899/79186546976a^15 - 15341761/2474579593a^14 - 1956051309/19796636744a^13 + 12014291401/39593273488a^12 + 68715042177/39593273488a^11 - 447150270927/79186546976a^10 - 260060382549/19796636744a^9 + 3911918481131/79186546976a^8 + 2643219910339/79186546976a^7 - 493478811376/2474579593a^6 + 1596128215609/39593273488a^5 + 2916954059095/9898318372a^4 - 3061619138021/19796636744a^3 - 1143215915141/9898318372a^2 + 98401567725/2474579593a + 44540747247/2474579593, 295192477/316746187904a^15 - 14764059/158373093952a^14 - 16977429725/316746187904a^13 + 871865377/158373093952a^12 + 386774551299/316746187904a^11 - 54187083221/316746187904a^10 - 4420642674685/316746187904a^9 + 61748920049/19796636744a^8 + 13171273645255/158373093952a^7 - 295895305277/9898318372a^6 - 18954212124477/79186546976a^5 + 5080531507247/39593273488a^4 + 4925294305671/19796636744a^3 - 432043525362/2474579593a^2 - 1026385276/46690181*a + 57594434662/2474579593 (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:	$ 1046189699591.4612 $ (assuming GRH)	comment:Regulator sage:K.regulator() gp:K.reg magma:Regulator(K); oscar:regulator(K)
Unit signature rank:	$ 12 $ (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{16}\cdot(2\pi)^{0}\cdot 1046189699591.4612 \cdot 1}{2\cdot\sqrt{20542690914744340526341740071734336}}\cr\approx \mathstrut & 0.239183771064890 \end{aligned}\] (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 - 2*x^15 - 61*x^14 + 110*x^13 + 1499*x^12 - 2405*x^11 - 18857*x^10 + 26688*x^9 + 126882*x^8 - 158448*x^7 - 431612*x^6 + 476792*x^5 + 629344*x^4 - 563968*x^3 - 315264*x^2 + 136192*x + 59648) 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 - 2*x^15 - 61*x^14 + 110*x^13 + 1499*x^12 - 2405*x^11 - 18857*x^10 + 26688*x^9 + 126882*x^8 - 158448*x^7 - 431612*x^6 + 476792*x^5 + 629344*x^4 - 563968*x^3 - 315264*x^2 + 136192*x + 59648, 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 - 2*x^15 - 61*x^14 + 110*x^13 + 1499*x^12 - 2405*x^11 - 18857*x^10 + 26688*x^9 + 126882*x^8 - 158448*x^7 - 431612*x^6 + 476792*x^5 + 629344*x^4 - 563968*x^3 - 315264*x^2 + 136192*x + 59648); 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 - 2*x^15 - 61*x^14 + 110*x^13 + 1499*x^12 - 2405*x^11 - 18857*x^10 + 26688*x^9 + 126882*x^8 - 158448*x^7 - 431612*x^6 + 476792*x^5 + 629344*x^4 - 563968*x^3 - 315264*x^2 + 136192*x + 59648); 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{13}) $, 4.4.17609.1, 8.8.8856105798241.1

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	R	${\href{/padicField/3.8.0.1}{8} }^{2}$	${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.4.0.1}{4} }$	R	${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.4.0.1}{4} }$	R	R	${\href{/padicField/19.8.0.1}{8} }^{2}$	${\href{/padicField/23.8.0.1}{8} }^{2}$	${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$	${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.4.0.1}{4} }$	${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.4.0.1}{4} }$	${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.4.0.1}{4} }$	${\href{/padicField/43.4.0.1}{4} }^{4}$	${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$	${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$	${\href{/padicField/59.2.0.1}{2} }^{8}$

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	2.2.2.6a1.1	$x^{4} + 2 x^{3} + 3 x^{2} + 2 x + 3$	$2$	$2$	$6$	$C_2^2$	$$[3]^{2}$$
	2.6.1.0a1.1	$x^{6} + x^{4} + x^{3} + x + 1$	$1$	$6$	$0$	$C_6$	$$[\ ]^{6}$$
	2.6.1.0a1.1	$x^{6} + x^{4} + x^{3} + x + 1$	$1$	$6$	$0$	$C_6$	$$[\ ]^{6}$$
$7$ 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	$x^{12} + 2 x^{8} + 5 x^{7} + 3 x^{6} + 2 x^{5} + 4 x^{4} + 5 x^{2} + 3$	$1$	$12$	$0$	$C_{12}$	$$[\ ]^{12}$$
$13$ 13	13.4.2.4a1.2	$x^{8} + 6 x^{6} + 24 x^{5} + 13 x^{4} + 72 x^{3} + 156 x^{2} + 48 x + 17$	$2$	$4$	$4$	$C_4\times C_2$	$$[\ ]_{2}^{4}$$
$13$ 13	13.4.2.4a1.2	$x^{8} + 6 x^{6} + 24 x^{5} + 13 x^{4} + 72 x^{3} + 156 x^{2} + 48 x + 17$	$2$	$4$	$4$	$C_4\times C_2$	$$[\ ]_{2}^{4}$$
$17$ 17	$\Q_{17}$	$x + 14$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	$\Q_{17}$	$x + 14$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	17.2.1.0a1.1	$x^{2} + 16 x + 3$	$1$	$2$	$0$	$C_2$	$$[\ ]^{2}$$
	17.1.2.1a1.2	$x^{2} + 51$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
	17.2.1.0a1.1	$x^{2} + 16 x + 3$	$1$	$2$	$0$	$C_2$	$$[\ ]^{2}$$
	17.1.2.1a1.1	$x^{2} + 17$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
	17.2.1.0a1.1	$x^{2} + 16 x + 3$	$1$	$2$	$0$	$C_2$	$$[\ ]^{2}$$
	17.2.2.2a1.2	$x^{4} + 32 x^{3} + 262 x^{2} + 96 x + 26$	$2$	$2$	$2$	$C_2^2$	$$[\ ]_{2}^{2}$$
$17609$ 17609		Deg $4$	$1$	$4$	$0$	$C_4$	$$[\ ]^{4}$$
		Deg $4$	$1$	$4$	$0$	$C_4$	$$[\ ]^{4}$$
		Deg $8$	$2$	$4$	$4$

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$ over number fields
Genus 2 curves
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more