LMFDB - Number field 16.8.891610044825600000000.2

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 2*x^14 + 8*x^13 + x^12 - 56*x^11 + 56*x^10 + 184*x^9 - 464*x^8 + 192*x^7 + 576*x^6 - 840*x^5 + 321*x^4 + 324*x^3 - 450*x^2 + 144*x + 9)

gp: K = bnfinit(y^16 - 4*y^15 + 2*y^14 + 8*y^13 + y^12 - 56*y^11 + 56*y^10 + 184*y^9 - 464*y^8 + 192*y^7 + 576*y^6 - 840*y^5 + 321*y^4 + 324*y^3 - 450*y^2 + 144*y + 9, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 + 2*x^14 + 8*x^13 + x^12 - 56*x^11 + 56*x^10 + 184*x^9 - 464*x^8 + 192*x^7 + 576*x^6 - 840*x^5 + 321*x^4 + 324*x^3 - 450*x^2 + 144*x + 9);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 4*x^15 + 2*x^14 + 8*x^13 + x^12 - 56*x^11 + 56*x^10 + 184*x^9 - 464*x^8 + 192*x^7 + 576*x^6 - 840*x^5 + 321*x^4 + 324*x^3 - 450*x^2 + 144*x + 9)

$ x^{16} - 4 x^{15} + 2 x^{14} + 8 x^{13} + x^{12} - 56 x^{11} + 56 x^{10} + 184 x^{9} - 464 x^{8} + \cdots + 9 $ x^16 - 4*x^15 + 2*x^14 + 8*x^13 + x^12 - 56*x^11 + 56*x^10 + 184*x^9 - 464*x^8 + 192*x^7 + 576*x^6 - 840*x^5 + 321*x^4 + 324*x^3 - 450*x^2 + 144*x + 9

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$16$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[8, 4]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$891610044825600000000$ 891610044825600000000 $\medspace = 2^{32}\cdot 3^{12}\cdot 5^{8}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$20.39$	sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) magma: Abs(Discriminant(OK))^(1/Degree(K)); oscar: (1.0 * dK)^(1/degree(K))
Galois root discriminant:	$2^{2}3^{3/4}5^{1/2}\approx 20.38853093816547$
Ramified primes:	$2$, $3$, $5$ 2, 3, 5	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q$
$\card{ \Aut(K/\Q) }$:	$8$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{6}a^{12}+\frac{1}{3}a^{11}+\frac{1}{3}a^{10}+\frac{1}{3}a^{9}-\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{2}$, $\frac{1}{6}a^{13}-\frac{1}{3}a^{11}-\frac{1}{3}a^{10}+\frac{1}{3}a^{8}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{2}a$, $\frac{1}{1710}a^{14}+\frac{67}{855}a^{13}-\frac{71}{855}a^{12}+\frac{154}{855}a^{11}-\frac{364}{855}a^{10}-\frac{94}{855}a^{9}+\frac{20}{171}a^{8}+\frac{386}{855}a^{7}+\frac{41}{855}a^{6}-\frac{14}{285}a^{5}+\frac{67}{285}a^{4}+\frac{4}{57}a^{3}+\frac{253}{570}a^{2}-\frac{33}{95}a+\frac{3}{95}$, $\frac{1}{286326990007650}a^{15}-\frac{1716058253}{28632699000765}a^{14}+\frac{1121830640209}{15907055000425}a^{13}+\frac{11941578457}{2074833260925}a^{12}+\frac{946225423084}{5726539800153}a^{11}+\frac{12603452157464}{47721165001275}a^{10}+\frac{1394659786364}{7534920789675}a^{9}-\frac{2427420375476}{7534920789675}a^{8}+\frac{20413743103279}{47721165001275}a^{7}+\frac{1851007490731}{7534920789675}a^{6}-\frac{731734656934}{2074833260925}a^{5}-\frac{18925178455403}{47721165001275}a^{4}+\frac{9921722165061}{31814110000850}a^{3}+\frac{40762299481}{100465610529}a^{2}-\frac{230222701142}{3181411000085}a+\frac{7553770763628}{15907055000425}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, a^10, a^11, 1/6*a^12 + 1/3*a^11 + 1/3*a^10 + 1/3*a^9 - 1/3*a^8 - 1/3*a^7 + 1/3*a^6 - 1/3*a^5 - 1/3*a^4 - 1/2, 1/6*a^13 - 1/3*a^11 - 1/3*a^10 + 1/3*a^8 + 1/3*a^5 - 1/3*a^4 - 1/2*a, 1/1710*a^14 + 67/855*a^13 - 71/855*a^12 + 154/855*a^11 - 364/855*a^10 - 94/855*a^9 + 20/171*a^8 + 386/855*a^7 + 41/855*a^6 - 14/285*a^5 + 67/285*a^4 + 4/57*a^3 + 253/570*a^2 - 33/95*a + 3/95, 1/286326990007650*a^15 - 1716058253/28632699000765*a^14 + 1121830640209/15907055000425*a^13 + 11941578457/2074833260925*a^12 + 946225423084/5726539800153*a^11 + 12603452157464/47721165001275*a^10 + 1394659786364/7534920789675*a^9 - 2427420375476/7534920789675*a^8 + 20413743103279/47721165001275*a^7 + 1851007490731/7534920789675*a^6 - 731734656934/2074833260925*a^5 - 18925178455403/47721165001275*a^4 + 9921722165061/31814110000850*a^3 + 40762299481/100465610529*a^2 - 230222701142/3181411000085*a + 7553770763628/15907055000425

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

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

oscar: class_group(K)

Unit group

sage: UK = K.unit_group()

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

oscar: UK, fUK = unit_group(OK)

Rank:	$11$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	$ -1 $ -1 (order $2$)	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:	$\frac{335472907}{21225128985}a^{15}-\frac{171030019}{2830017198}a^{14}+\frac{259930864}{21225128985}a^{13}+\frac{140791094}{922831695}a^{12}+\frac{73982428}{1415008599}a^{11}-\frac{19486398037}{21225128985}a^{10}+\frac{4510912598}{7075042995}a^{9}+\frac{24096032413}{7075042995}a^{8}-\frac{143008547692}{21225128985}a^{7}+\frac{7816523521}{21225128985}a^{6}+\frac{3379691114}{307610565}a^{5}-\frac{76226337622}{7075042995}a^{4}+\frac{3991482581}{7075042995}a^{3}+\frac{18011038159}{2830017198}a^{2}-\frac{2791921310}{471669533}a+\frac{671245817}{2358347665}$, $\frac{9728896771}{12448999565550}a^{15}-\frac{400362947}{55328886958}a^{14}+\frac{140090616236}{6224499782775}a^{13}-\frac{79245453187}{6224499782775}a^{12}-\frac{18771255632}{414966652185}a^{11}-\frac{149332133213}{6224499782775}a^{10}+\frac{239218374799}{691611086975}a^{9}-\frac{190808877586}{691611086975}a^{8}-\frac{7729610608268}{6224499782775}a^{7}+\frac{18045197321249}{6224499782775}a^{6}-\frac{1681786775167}{2074833260925}a^{5}-\frac{8639856391928}{2074833260925}a^{4}+\frac{20458578097343}{4149666521850}a^{3}+\frac{145858091393}{829933304370}a^{2}-\frac{83806061489}{27664443479}a+\frac{1224609145253}{691611086975}$, $\frac{331257515257}{6224499782775}a^{15}-\frac{130657544363}{829933304370}a^{14}-\frac{827279861257}{12448999565550}a^{13}+\frac{4723518981449}{12448999565550}a^{12}+\frac{39628454753}{82993330437}a^{11}-\frac{16045137348487}{6224499782775}a^{10}+\frac{464968472708}{2074833260925}a^{9}+\frac{7224158257501}{691611086975}a^{8}-\frac{84660166007782}{6224499782775}a^{7}-\frac{38598735760829}{6224499782775}a^{6}+\frac{54858076672202}{2074833260925}a^{5}-\frac{29981758107892}{2074833260925}a^{4}-\frac{10004816129269}{2074833260925}a^{3}+\frac{2525270962667}{165986660874}a^{2}-\frac{1577945560391}{276644434790}a-\frac{2711410801841}{1383222173950}$, $\frac{455320275811}{12448999565550}a^{15}-\frac{153701910191}{1244899956555}a^{14}+\frac{10825352902}{2074833260925}a^{13}+\frac{28309944074}{109201750575}a^{12}+\frac{15255711314}{65521050345}a^{11}-\frac{3942526806181}{2074833260925}a^{10}+\frac{5895911845496}{6224499782775}a^{9}+\frac{42528237574291}{6224499782775}a^{8}-\frac{25087539775591}{2074833260925}a^{7}+\frac{1275133755724}{6224499782775}a^{6}+\frac{12703681182656}{691611086975}a^{5}-\frac{11056770591656}{691611086975}a^{4}+\frac{811012199971}{1383222173950}a^{3}+\frac{4521457780781}{414966652185}a^{2}-\frac{874452453484}{138322217395}a-\frac{519047258182}{691611086975}$, $\frac{4439278088327}{95442330002550}a^{15}-\frac{2741945311049}{19088466000510}a^{14}-\frac{2071653247733}{47721165001275}a^{13}+\frac{252127810954}{691611086975}a^{12}+\frac{1043190836734}{3181411000085}a^{11}-\frac{110446817053901}{47721165001275}a^{10}+\frac{23467133865247}{47721165001275}a^{9}+\frac{450092315280112}{47721165001275}a^{8}-\frac{220007340531012}{15907055000425}a^{7}-\frac{195195294002557}{47721165001275}a^{6}+\frac{54646878868336}{2074833260925}a^{5}-\frac{902120068871678}{47721165001275}a^{4}-\frac{3679975059361}{1674426842150}a^{3}+\frac{101984031879789}{6362822000170}a^{2}-\frac{28445857147119}{3181411000085}a-\frac{4533337410672}{15907055000425}$, $\frac{1612368409043}{47721165001275}a^{15}-\frac{2980684766843}{28632699000765}a^{14}-\frac{2992558955732}{143163495003825}a^{13}+\frac{72979243217}{327605251725}a^{12}+\frac{400575342617}{1506984157935}a^{11}-\frac{236597839614979}{143163495003825}a^{10}+\frac{64371471988913}{143163495003825}a^{9}+\frac{892819270811473}{143163495003825}a^{8}-\frac{13\!\cdots\!44}{143163495003825}a^{7}-\frac{240244566085678}{143163495003825}a^{6}+\frac{10972848068166}{691611086975}a^{5}-\frac{182574630585868}{15907055000425}a^{4}-\frac{31352616223918}{47721165001275}a^{3}+\frac{92605113934363}{9544233000255}a^{2}-\frac{16958480812662}{3181411000085}a-\frac{9675322180471}{15907055000425}$, $\frac{789099165971}{19088466000510}a^{15}-\frac{422958036174}{3181411000085}a^{14}-\frac{124912401543}{6362822000170}a^{13}+\frac{255720599083}{829933304370}a^{12}+\frac{2669662471844}{9544233000255}a^{11}-\frac{19663834607942}{9544233000255}a^{10}+\frac{2205701183294}{3181411000085}a^{9}+\frac{76573740024631}{9544233000255}a^{8}-\frac{122771822540482}{9544233000255}a^{7}-\frac{2768553460621}{1908846600051}a^{6}+\frac{8984073332564}{414966652185}a^{5}-\frac{173499733444483}{9544233000255}a^{4}+\frac{9274899273073}{6362822000170}a^{3}+\frac{40002385556264}{3181411000085}a^{2}-\frac{54466149856263}{6362822000170}a+\frac{88597794279}{334885368430}$, $\frac{624872705384}{15907055000425}a^{15}-\frac{636535184179}{5726539800153}a^{14}-\frac{497526387107}{7534920789675}a^{13}+\frac{1788484375907}{6224499782775}a^{12}+\frac{10248681677363}{28632699000765}a^{11}-\frac{262892502150736}{143163495003825}a^{10}-\frac{10800089013823}{143163495003825}a^{9}+\frac{11\!\cdots\!97}{143163495003825}a^{8}-\frac{71902038607759}{7534920789675}a^{7}-\frac{806450231905747}{143163495003825}a^{6}+\frac{41926440209962}{2074833260925}a^{5}-\frac{526923272313641}{47721165001275}a^{4}-\frac{163716924505027}{47721165001275}a^{3}+\frac{120591640068643}{9544233000255}a^{2}-\frac{3901391566195}{636282200017}a-\frac{14591700635559}{15907055000425}$, $\frac{2552082246131}{286326990007650}a^{15}-\frac{1903395407441}{57265398001530}a^{14}+\frac{161671969554}{15907055000425}a^{13}+\frac{4305579387}{72801167050}a^{12}+\frac{18258892739}{301396831587}a^{11}-\frac{23014659730016}{47721165001275}a^{10}+\frac{46961109392146}{143163495003825}a^{9}+\frac{223648454969186}{143163495003825}a^{8}-\frac{150144444599176}{47721165001275}a^{7}+\frac{119021845485284}{143163495003825}a^{6}+\frac{7084231981121}{2074833260925}a^{5}-\frac{174861451446868}{47721165001275}a^{4}+\frac{59208007309841}{31814110000850}a^{3}+\frac{2621002348147}{3817693200102}a^{2}-\frac{5383857355822}{3181411000085}a+\frac{6201011009211}{31814110000850}$, $\frac{813978801848}{47721165001275}a^{15}-\frac{1716663772598}{28632699000765}a^{14}-\frac{596736546229}{286326990007650}a^{13}+\frac{50851192937}{327605251725}a^{12}+\frac{130115741612}{1506984157935}a^{11}-\frac{130068617645494}{143163495003825}a^{10}+\frac{59680112112293}{143163495003825}a^{9}+\frac{527777698053853}{143163495003825}a^{8}-\frac{902977424875984}{143163495003825}a^{7}-\frac{48769553264533}{143163495003825}a^{6}+\frac{7404886776426}{691611086975}a^{5}-\frac{148872888700548}{15907055000425}a^{4}+\frac{52037901711602}{47721165001275}a^{3}+\frac{46773872799088}{9544233000255}a^{2}-\frac{19566634462389}{6362822000170}a-\frac{17176265068856}{15907055000425}$, $\frac{12674959688257}{143163495003825}a^{15}-\frac{14505396612859}{57265398001530}a^{14}-\frac{19839254983291}{143163495003825}a^{13}+\frac{4032786392644}{6224499782775}a^{12}+\frac{4495766907778}{5726539800153}a^{11}-\frac{597759319924862}{143163495003825}a^{10}-\frac{1496959863526}{143163495003825}a^{9}+\frac{25\!\cdots\!84}{143163495003825}a^{8}-\frac{31\!\cdots\!07}{143163495003825}a^{7}-\frac{189926762714356}{15907055000425}a^{6}+\frac{31779630291458}{691611086975}a^{5}-\frac{12\!\cdots\!92}{47721165001275}a^{4}-\frac{345917872508219}{47721165001275}a^{3}+\frac{36610235865527}{1272564400034}a^{2}-\frac{39317996500288}{3181411000085}a-\frac{38442174478258}{15907055000425}$ 335472907/21225128985a^15 - 171030019/2830017198a^14 + 259930864/21225128985a^13 + 140791094/922831695a^12 + 73982428/1415008599a^11 - 19486398037/21225128985a^10 + 4510912598/7075042995a^9 + 24096032413/7075042995a^8 - 143008547692/21225128985a^7 + 7816523521/21225128985a^6 + 3379691114/307610565a^5 - 76226337622/7075042995a^4 + 3991482581/7075042995a^3 + 18011038159/2830017198a^2 - 2791921310/471669533a + 671245817/2358347665, 9728896771/12448999565550a^15 - 400362947/55328886958a^14 + 140090616236/6224499782775a^13 - 79245453187/6224499782775a^12 - 18771255632/414966652185a^11 - 149332133213/6224499782775a^10 + 239218374799/691611086975a^9 - 190808877586/691611086975a^8 - 7729610608268/6224499782775a^7 + 18045197321249/6224499782775a^6 - 1681786775167/2074833260925a^5 - 8639856391928/2074833260925a^4 + 20458578097343/4149666521850a^3 + 145858091393/829933304370a^2 - 83806061489/27664443479a + 1224609145253/691611086975, 331257515257/6224499782775a^15 - 130657544363/829933304370a^14 - 827279861257/12448999565550a^13 + 4723518981449/12448999565550a^12 + 39628454753/82993330437a^11 - 16045137348487/6224499782775a^10 + 464968472708/2074833260925a^9 + 7224158257501/691611086975a^8 - 84660166007782/6224499782775a^7 - 38598735760829/6224499782775a^6 + 54858076672202/2074833260925a^5 - 29981758107892/2074833260925a^4 - 10004816129269/2074833260925a^3 + 2525270962667/165986660874a^2 - 1577945560391/276644434790a - 2711410801841/1383222173950, 455320275811/12448999565550a^15 - 153701910191/1244899956555a^14 + 10825352902/2074833260925a^13 + 28309944074/109201750575a^12 + 15255711314/65521050345a^11 - 3942526806181/2074833260925a^10 + 5895911845496/6224499782775a^9 + 42528237574291/6224499782775a^8 - 25087539775591/2074833260925a^7 + 1275133755724/6224499782775a^6 + 12703681182656/691611086975a^5 - 11056770591656/691611086975a^4 + 811012199971/1383222173950a^3 + 4521457780781/414966652185a^2 - 874452453484/138322217395a - 519047258182/691611086975, 4439278088327/95442330002550a^15 - 2741945311049/19088466000510a^14 - 2071653247733/47721165001275a^13 + 252127810954/691611086975a^12 + 1043190836734/3181411000085a^11 - 110446817053901/47721165001275a^10 + 23467133865247/47721165001275a^9 + 450092315280112/47721165001275a^8 - 220007340531012/15907055000425a^7 - 195195294002557/47721165001275a^6 + 54646878868336/2074833260925a^5 - 902120068871678/47721165001275a^4 - 3679975059361/1674426842150a^3 + 101984031879789/6362822000170a^2 - 28445857147119/3181411000085a - 4533337410672/15907055000425, 1612368409043/47721165001275a^15 - 2980684766843/28632699000765a^14 - 2992558955732/143163495003825a^13 + 72979243217/327605251725a^12 + 400575342617/1506984157935a^11 - 236597839614979/143163495003825a^10 + 64371471988913/143163495003825a^9 + 892819270811473/143163495003825a^8 - 1349311862183644/143163495003825a^7 - 240244566085678/143163495003825a^6 + 10972848068166/691611086975a^5 - 182574630585868/15907055000425a^4 - 31352616223918/47721165001275a^3 + 92605113934363/9544233000255a^2 - 16958480812662/3181411000085a - 9675322180471/15907055000425, 789099165971/19088466000510a^15 - 422958036174/3181411000085a^14 - 124912401543/6362822000170a^13 + 255720599083/829933304370a^12 + 2669662471844/9544233000255a^11 - 19663834607942/9544233000255a^10 + 2205701183294/3181411000085a^9 + 76573740024631/9544233000255a^8 - 122771822540482/9544233000255a^7 - 2768553460621/1908846600051a^6 + 8984073332564/414966652185a^5 - 173499733444483/9544233000255a^4 + 9274899273073/6362822000170a^3 + 40002385556264/3181411000085a^2 - 54466149856263/6362822000170a + 88597794279/334885368430, 624872705384/15907055000425a^15 - 636535184179/5726539800153a^14 - 497526387107/7534920789675a^13 + 1788484375907/6224499782775a^12 + 10248681677363/28632699000765a^11 - 262892502150736/143163495003825a^10 - 10800089013823/143163495003825a^9 + 1126739869360597/143163495003825a^8 - 71902038607759/7534920789675a^7 - 806450231905747/143163495003825a^6 + 41926440209962/2074833260925a^5 - 526923272313641/47721165001275a^4 - 163716924505027/47721165001275a^3 + 120591640068643/9544233000255a^2 - 3901391566195/636282200017a - 14591700635559/15907055000425, 2552082246131/286326990007650a^15 - 1903395407441/57265398001530a^14 + 161671969554/15907055000425a^13 + 4305579387/72801167050a^12 + 18258892739/301396831587a^11 - 23014659730016/47721165001275a^10 + 46961109392146/143163495003825a^9 + 223648454969186/143163495003825a^8 - 150144444599176/47721165001275a^7 + 119021845485284/143163495003825a^6 + 7084231981121/2074833260925a^5 - 174861451446868/47721165001275a^4 + 59208007309841/31814110000850a^3 + 2621002348147/3817693200102a^2 - 5383857355822/3181411000085a + 6201011009211/31814110000850, 813978801848/47721165001275a^15 - 1716663772598/28632699000765a^14 - 596736546229/286326990007650a^13 + 50851192937/327605251725a^12 + 130115741612/1506984157935a^11 - 130068617645494/143163495003825a^10 + 59680112112293/143163495003825a^9 + 527777698053853/143163495003825a^8 - 902977424875984/143163495003825a^7 - 48769553264533/143163495003825a^6 + 7404886776426/691611086975a^5 - 148872888700548/15907055000425a^4 + 52037901711602/47721165001275a^3 + 46773872799088/9544233000255a^2 - 19566634462389/6362822000170a - 17176265068856/15907055000425, 12674959688257/143163495003825a^15 - 14505396612859/57265398001530a^14 - 19839254983291/143163495003825a^13 + 4032786392644/6224499782775a^12 + 4495766907778/5726539800153a^11 - 597759319924862/143163495003825a^10 - 1496959863526/143163495003825a^9 + 2529200188642084/143163495003825a^8 - 3156935573932907/143163495003825a^7 - 189926762714356/15907055000425a^6 + 31779630291458/691611086975a^5 - 1248467024013692/47721165001275a^4 - 345917872508219/47721165001275a^3 + 36610235865527/1272564400034a^2 - 39317996500288/3181411000085*a - 38442174478258/15907055000425 (assuming GRH)	sage: UK.fundamental_units() gp: K.fu magma: [K\|fUK(g): g in Generators(UK)]; oscar: [K(fUK(a)) for a in gens(UK)]
Regulator:	$ 49825.1110893 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 49825.1110893 \cdot 1}{2\cdot\sqrt{891610044825600000000}}\cr\approx \mathstrut & 0.332881946631 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula

x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 2*x^14 + 8*x^13 + x^12 - 56*x^11 + 56*x^10 + 184*x^9 - 464*x^8 + 192*x^7 + 576*x^6 - 840*x^5 + 321*x^4 + 324*x^3 - 450*x^2 + 144*x + 9)

DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()

hK = K.class_number(); wK = K.unit_group().torsion_generator().order();

2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))

# self-contained Pari/GP code snippet to compute the analytic class number formula

K = bnfinit(x^16 - 4*x^15 + 2*x^14 + 8*x^13 + x^12 - 56*x^11 + 56*x^10 + 184*x^9 - 464*x^8 + 192*x^7 + 576*x^6 - 840*x^5 + 321*x^4 + 324*x^3 - 450*x^2 + 144*x + 9, 1);

[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]

/* self-contained Magma code snippet to compute the analytic class number formula */

Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^16 - 4*x^15 + 2*x^14 + 8*x^13 + x^12 - 56*x^11 + 56*x^10 + 184*x^9 - 464*x^8 + 192*x^7 + 576*x^6 - 840*x^5 + 321*x^4 + 324*x^3 - 450*x^2 + 144*x + 9);

OK := Integers(K); DK := Discriminant(OK);

UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);

r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);

hK := #clK; wK := #TorsionSubgroup(UK);

2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));

# self-contained Oscar code snippet to compute the analytic class number formula

Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 4*x^15 + 2*x^14 + 8*x^13 + x^12 - 56*x^11 + 56*x^10 + 184*x^9 - 464*x^8 + 192*x^7 + 576*x^6 - 840*x^5 + 321*x^4 + 324*x^3 - 450*x^2 + 144*x + 9);

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

$C_2^2\times D_4$ (as 16T25):

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 32

The 20 conjugacy class representatives for $C_2^2 \times D_4$

Character table for $C_2^2 \times D_4$

Intermediate fields

$\Q(\sqrt{15}) $, $\Q(\sqrt{6}) $, $\Q(\sqrt{3}) $, $\Q(\sqrt{30}) $, $\Q(\sqrt{10}) $, $\Q(\sqrt{5}) $, $\Q(\sqrt{2}) $, 4.2.34560.1, 4.2.8640.2, 4.2.34560.2, 4.2.8640.1, $\Q(\sqrt{2}, \sqrt{5})$, $\Q(\sqrt{6}, \sqrt{10})$, $\Q(\sqrt{3}, \sqrt{10})$, $\Q(\sqrt{3}, \sqrt{5})$, $\Q(\sqrt{5}, \sqrt{6})$, $\Q(\sqrt{2}, \sqrt{3})$, $\Q(\sqrt{2}, \sqrt{15})$, 8.8.3317760000.1, 8.4.29859840000.2, 8.4.29859840000.5, 8.4.29859840000.3, 8.4.1866240000.2, 8.4.1194393600.2, 8.4.1194393600.1

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

sage: K.subfields()[1:-1]

gp: L = nfsubfields(K); L[2..length(b)]

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

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

Sibling fields

Galois closure:	deg 32
Degree 16 siblings:	16.0.1426576071720960000.1, 16.0.3482851737600000000.5, 16.0.55725627801600000000.1, 16.0.891610044825600000000.13, 16.0.891610044825600000000.7, 16.0.891610044825600000000.14, 16.0.891610044825600000000.8
Minimal sibling:	16.0.1426576071720960000.1

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	R	R	${\href{/padicField/7.4.0.1}{4} }^{4}$	${\href{/padicField/11.4.0.1}{4} }^{4}$	${\href{/padicField/13.4.0.1}{4} }^{4}$	${\href{/padicField/17.2.0.1}{2} }^{8}$	${\href{/padicField/19.2.0.1}{2} }^{8}$	${\href{/padicField/23.2.0.1}{2} }^{8}$	${\href{/padicField/29.2.0.1}{2} }^{8}$	${\href{/padicField/31.2.0.1}{2} }^{8}$	${\href{/padicField/37.4.0.1}{4} }^{4}$	${\href{/padicField/41.4.0.1}{4} }^{4}$	${\href{/padicField/43.2.0.1}{2} }^{8}$	${\href{/padicField/47.2.0.1}{2} }^{8}$	${\href{/padicField/53.2.0.1}{2} }^{8}$	${\href{/padicField/59.4.0.1}{4} }^{4}$

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

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:

p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:

p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])

// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:

p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:

p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$2$ 2	2.8.16.6	$x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$	$4$	$2$	$16$	$C_2^3$	$[2, 3]^{2}$
$2$ 2	2.8.16.6	$x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$	$4$	$2$	$16$	$C_2^3$	$[2, 3]^{2}$
$3$ 3	3.8.6.2	$x^{8} + 6 x^{5} + 6 x^{4} + 18 x^{2} + 18 x + 9$	$4$	$2$	$6$	$D_4$	$[\ ]_{4}^{2}$
$3$ 3	3.8.6.2	$x^{8} + 6 x^{5} + 6 x^{4} + 18 x^{2} + 18 x + 9$	$4$	$2$	$6$	$D_4$	$[\ ]_{4}^{2}$
$5$ 5	5.4.2.1	$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	5.4.2.1	$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	5.4.2.1	$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	5.4.2.1	$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$

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