LMFDB - Number field 44.0.65347506006797164323533696798768413693852562329201668296707932160000000000000000000000.2

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^44 - 47*x^42 + 1036*x^40 - 14238*x^38 + 136828*x^36 - 977616*x^34 + 5391753*x^32 - 23533371*x^30 + 82732335*x^28 - 237396295*x^26 + 562190611*x^24 - 1110544651*x^22 + 1852435411*x^20 - 2651394691*x^18 + 3329083366*x^16 - 3775046236*x^14 + 3998027671*x^12 - 4080345361*x^10 + 4101904756*x^8 - 4105687106*x^6 + 4106094436*x^4 - 4106117712*x^2 + 4106118241)

gp: K = bnfinit(y^44 - 47*y^42 + 1036*y^40 - 14238*y^38 + 136828*y^36 - 977616*y^34 + 5391753*y^32 - 23533371*y^30 + 82732335*y^28 - 237396295*y^26 + 562190611*y^24 - 1110544651*y^22 + 1852435411*y^20 - 2651394691*y^18 + 3329083366*y^16 - 3775046236*y^14 + 3998027671*y^12 - 4080345361*y^10 + 4101904756*y^8 - 4105687106*y^6 + 4106094436*y^4 - 4106117712*y^2 + 4106118241, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^44 - 47*x^42 + 1036*x^40 - 14238*x^38 + 136828*x^36 - 977616*x^34 + 5391753*x^32 - 23533371*x^30 + 82732335*x^28 - 237396295*x^26 + 562190611*x^24 - 1110544651*x^22 + 1852435411*x^20 - 2651394691*x^18 + 3329083366*x^16 - 3775046236*x^14 + 3998027671*x^12 - 4080345361*x^10 + 4101904756*x^8 - 4105687106*x^6 + 4106094436*x^4 - 4106117712*x^2 + 4106118241);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^44 - 47*x^42 + 1036*x^40 - 14238*x^38 + 136828*x^36 - 977616*x^34 + 5391753*x^32 - 23533371*x^30 + 82732335*x^28 - 237396295*x^26 + 562190611*x^24 - 1110544651*x^22 + 1852435411*x^20 - 2651394691*x^18 + 3329083366*x^16 - 3775046236*x^14 + 3998027671*x^12 - 4080345361*x^10 + 4101904756*x^8 - 4105687106*x^6 + 4106094436*x^4 - 4106117712*x^2 + 4106118241)

$ x^{44} - 47 x^{42} + 1036 x^{40} - 14238 x^{38} + 136828 x^{36} - 977616 x^{34} + 5391753 x^{32} + \cdots + 4106118241 $ x^44 - 47*x^42 + 1036*x^40 - 14238*x^38 + 136828*x^36 - 977616*x^34 + 5391753*x^32 - 23533371*x^30 + 82732335*x^28 - 237396295*x^26 + 562190611*x^24 - 1110544651*x^22 + 1852435411*x^20 - 2651394691*x^18 + 3329083366*x^16 - 3775046236*x^14 + 3998027671*x^12 - 4080345361*x^10 + 4101904756*x^8 - 4105687106*x^6 + 4106094436*x^4 - 4106117712*x^2 + 4106118241

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$44$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[0, 22]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$653\!\cdots\!000$ 65347506006797164323533696798768413693852562329201668296707932160000000000000000000000 $\medspace = 2^{44}\cdot 5^{22}\cdot 23^{42}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$89.20$	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:	$2\cdot 5^{1/2}23^{21/22}\approx 89.19615099241642$
Ramified primes:	$2$, $5$, $23$ 2, 5, 23	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q$
$\card{ \Gal(K/\Q) }$:	$44$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is Galois and abelian over $\Q$.
Conductor:	$460=2^{2}\cdot 5\cdot 23$
Dirichlet character group:	$\lbrace$$\chi_{460}(1,·)$, $\chi_{460}(131,·)$, $\chi_{460}(261,·)$, $\chi_{460}(129,·)$, $\chi_{460}(141,·)$, $\chi_{460}(271,·)$, $\chi_{460}(19,·)$, $\chi_{460}(149,·)$, $\chi_{460}(151,·)$, $\chi_{460}(389,·)$, $\chi_{460}(419,·)$, $\chi_{460}(31,·)$, $\chi_{460}(41,·)$, $\chi_{460}(71,·)$, $\chi_{460}(301,·)$, $\chi_{460}(309,·)$, $\chi_{460}(311,·)$, $\chi_{460}(441,·)$, $\chi_{460}(159,·)$, $\chi_{460}(189,·)$, $\chi_{460}(319,·)$, $\chi_{460}(331,·)$, $\chi_{460}(211,·)$, $\chi_{460}(199,·)$, $\chi_{460}(329,·)$, $\chi_{460}(459,·)$, $\chi_{460}(79,·)$, $\chi_{460}(81,·)$, $\chi_{460}(339,·)$, $\chi_{460}(121,·)$, $\chi_{460}(429,·)$, $\chi_{460}(351,·)$, $\chi_{460}(99,·)$, $\chi_{460}(101,·)$, $\chi_{460}(231,·)$, $\chi_{460}(361,·)$, $\chi_{460}(359,·)$, $\chi_{460}(109,·)$, $\chi_{460}(229,·)$, $\chi_{460}(371,·)$, $\chi_{460}(89,·)$, $\chi_{460}(249,·)$, $\chi_{460}(379,·)$, $\chi_{460}(381,·)$$\rbrace$
This is a CM field.
Reflex fields:	unavailable$^{2097152}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{64079}a^{23}-\frac{23}{64079}a^{21}+\frac{230}{64079}a^{19}-\frac{1311}{64079}a^{17}+\frac{4692}{64079}a^{15}-\frac{10948}{64079}a^{13}+\frac{16744}{64079}a^{11}-\frac{16445}{64079}a^{9}+\frac{9867}{64079}a^{7}-\frac{3289}{64079}a^{5}+\frac{506}{64079}a^{3}-\frac{23}{64079}a$, $\frac{1}{1836311903}a^{24}-\frac{701408757}{1836311903}a^{22}+\frac{740497154}{1836311903}a^{20}+\frac{310525523}{1836311903}a^{18}-\frac{797202147}{1836311903}a^{16}+\frac{821033279}{1836311903}a^{14}-\frac{396952661}{1836311903}a^{12}+\frac{316277499}{1836311903}a^{10}-\frac{8768956}{1836311903}a^{8}-\frac{913769829}{1836311903}a^{6}+\frac{328469460}{1836311903}a^{4}-\frac{400109299}{1836311903}a^{2}-\frac{6765}{28657}$, $\frac{1}{1836311903}a^{25}-\frac{25}{1836311903}a^{23}-\frac{701408458}{1836311903}a^{21}+\frac{39086419}{1836311903}a^{19}-\frac{351786396}{1836311903}a^{17}-\frac{676438256}{1836311903}a^{15}+\frac{36627749}{1836311903}a^{13}-\frac{346845481}{1836311903}a^{11}-\frac{800303953}{1836311903}a^{9}+\frac{662938311}{1836311903}a^{7}-\frac{197099920}{1836311903}a^{5}+\frac{104511814}{1836311903}a^{3}-\frac{39088144}{1836311903}a$, $\frac{1}{1836311903}a^{26}+\frac{126491647}{1836311903}a^{22}+\frac{188396239}{1836311903}a^{20}+\frac{66104067}{1836311903}a^{18}-\frac{407060998}{1836311903}a^{16}+\frac{363028791}{1836311903}a^{14}+\frac{747209412}{1836311903}a^{12}-\frac{238614090}{1836311903}a^{10}+\frac{443714411}{1836311903}a^{8}+\frac{830709094}{1836311903}a^{6}-\frac{865311201}{1836311903}a^{4}-\frac{860261104}{1836311903}a^{2}+\frac{2817}{28657}$, $\frac{1}{1836311903}a^{27}-\frac{351}{1836311903}a^{23}-\frac{574911613}{1836311903}a^{21}+\frac{353934975}{1836311903}a^{19}+\frac{155877110}{1836311903}a^{17}-\frac{8681156}{1836311903}a^{15}-\frac{833883249}{1836311903}a^{13}+\frac{883307460}{1836311903}a^{11}+\frac{63235422}{1836311903}a^{9}-\frac{410053035}{1836311903}a^{7}+\frac{160380143}{1836311903}a^{5}-\frac{594295487}{1836311903}a^{3}-\frac{582797309}{1836311903}a$, $\frac{1}{1836311903}a^{28}-\frac{703590318}{1836311903}a^{22}-\frac{487854197}{1836311903}a^{20}+\frac{807933406}{1836311903}a^{18}-\frac{707225497}{1836311903}a^{16}+\frac{884140812}{1836311903}a^{14}-\frac{723683826}{1836311903}a^{12}+\frac{897923391}{1836311903}a^{10}+\frac{184667215}{1836311903}a^{8}+\frac{781753189}{1836311903}a^{6}+\frac{847146987}{1836311903}a^{4}+\frac{374855273}{1836311903}a^{2}+\frac{4016}{28657}$, $\frac{1}{1836311903}a^{29}-\frac{3654}{1836311903}a^{23}-\frac{143540342}{1836311903}a^{21}-\frac{798893241}{1836311903}a^{19}+\frac{555545208}{1836311903}a^{17}+\frac{423966706}{1836311903}a^{15}-\frac{262048213}{1836311903}a^{13}-\frac{24144241}{1836311903}a^{11}+\frac{303278538}{1836311903}a^{9}-\frac{23938366}{1836311903}a^{7}+\frac{503606871}{1836311903}a^{5}+\frac{145198075}{1836311903}a^{3}+\frac{601655119}{1836311903}a$, $\frac{1}{1836311903}a^{30}+\frac{400278168}{1836311903}a^{22}+\frac{90274356}{1836311903}a^{20}+\frac{375050196}{1836311903}a^{18}-\frac{162000274}{1836311903}a^{16}-\frac{740096249}{1836311903}a^{14}+\frac{197235835}{1836311903}a^{12}-\frac{895239006}{1836311903}a^{10}-\frac{848401239}{1836311903}a^{8}+\frac{3691359}{1836311903}a^{6}-\frac{575379647}{1836311903}a^{4}+\frac{306551361}{1836311903}a^{2}+\frac{11681}{28657}$, $\frac{1}{1836311903}a^{31}-\frac{2808}{1836311903}a^{23}+\frac{115177289}{1836311903}a^{21}+\frac{126020866}{1836311903}a^{19}-\frac{578844996}{1836311903}a^{17}-\frac{311358872}{1836311903}a^{15}-\frac{803151378}{1836311903}a^{13}-\frac{661455200}{1836311903}a^{11}+\frac{430388729}{1836311903}a^{9}+\frac{338204520}{1836311903}a^{7}-\frac{686884034}{1836311903}a^{5}-\frac{241313165}{1836311903}a^{3}+\frac{773409732}{1836311903}a$, $\frac{1}{1836311903}a^{32}-\frac{914252351}{1836311903}a^{22}+\frac{736955102}{1836311903}a^{20}-\frac{871330337}{1836311903}a^{18}-\frac{390777891}{1836311903}a^{16}+\frac{86857789}{1836311903}a^{14}-\frac{663202167}{1836311903}a^{12}-\frac{237355131}{1836311903}a^{10}-\frac{412969189}{1836311903}a^{8}+\frac{611476528}{1836311903}a^{6}+\frac{272355209}{1836311903}a^{4}-\frac{746929127}{1836311903}a^{2}+\frac{3471}{28657}$, $\frac{1}{1836311903}a^{33}-\frac{8080}{1836311903}a^{23}-\frac{91232198}{1836311903}a^{21}+\frac{65295051}{1836311903}a^{19}+\frac{146655487}{1836311903}a^{17}+\frac{96371913}{1836311903}a^{15}-\frac{73297822}{1836311903}a^{13}+\frac{372695085}{1836311903}a^{11}+\frac{561855980}{1836311903}a^{9}-\frac{340680954}{1836311903}a^{7}-\frac{634466899}{1836311903}a^{5}-\frac{889927557}{1836311903}a^{3}-\frac{605769091}{1836311903}a$, $\frac{1}{1836311903}a^{34}-\frac{615456100}{1836311903}a^{22}+\frac{578119397}{1836311903}a^{20}+\frac{790821829}{1836311903}a^{18}+\frac{485179877}{1836311903}a^{16}-\frac{719309041}{1836311903}a^{14}-\frac{804223157}{1836311903}a^{12}-\frac{62121076}{1836311903}a^{10}+\frac{422318783}{1836311903}a^{8}-\frac{84523256}{1836311903}a^{6}-\frac{2355328}{13210877}a^{4}+\frac{256356172}{1836311903}a^{2}-\frac{12301}{28657}$, $\frac{1}{1836311903}a^{35}+\frac{10289}{1836311903}a^{23}-\frac{723424229}{1836311903}a^{21}-\frac{884237135}{1836311903}a^{19}-\frac{250330685}{1836311903}a^{17}+\frac{366676631}{1836311903}a^{15}+\frac{334434081}{1836311903}a^{13}-\frac{75303296}{1836311903}a^{11}+\frac{828761014}{1836311903}a^{9}+\frac{38873786}{1836311903}a^{7}+\frac{855684996}{1836311903}a^{5}-\frac{490674504}{1836311903}a^{3}-\frac{253467502}{1836311903}a$, $\frac{1}{1836311903}a^{36}-\frac{634502246}{1836311903}a^{22}+\frac{834942809}{1836311903}a^{20}-\frac{64725612}{1836311903}a^{18}-\frac{25703587}{1836311903}a^{16}-\frac{242219750}{1836311903}a^{14}+\frac{212953461}{1836311903}a^{12}+\frac{594265919}{1836311903}a^{10}+\frac{283378823}{1836311903}a^{8}+\frac{716512217}{1836311903}a^{6}+\frac{537264979}{1836311903}a^{4}-\frac{540176617}{1836311903}a^{2}-\frac{2768}{28657}$, $\frac{1}{1836311903}a^{37}-\frac{7609}{1836311903}a^{23}-\frac{904250521}{1836311903}a^{21}+\frac{800400561}{1836311903}a^{19}+\frac{1119365}{1836311903}a^{17}+\frac{145022291}{1836311903}a^{15}+\frac{533596634}{1836311903}a^{13}-\frac{328202911}{1836311903}a^{11}-\frac{56693796}{1836311903}a^{9}-\frac{548493734}{1836311903}a^{7}-\frac{265274306}{1836311903}a^{5}-\frac{840473320}{1836311903}a^{3}-\frac{80252099}{1836311903}a$, $\frac{1}{1836311903}a^{38}+\frac{235219487}{1836311903}a^{22}-\frac{397984960}{1836311903}a^{20}-\frac{543595289}{1836311903}a^{18}-\frac{427898623}{1836311903}a^{16}+\frac{642722539}{1836311903}a^{14}-\frac{7920025}{1836311903}a^{12}-\frac{906108738}{1836311903}a^{10}+\frac{672060473}{1836311903}a^{8}-\frac{863038409}{1836311903}a^{6}-\frac{736852163}{1836311903}a^{4}+\frac{93226984}{1836311903}a^{2}-\frac{6913}{28657}$, $\frac{1}{1836311903}a^{39}+\frac{2831}{1836311903}a^{23}-\frac{496937581}{1836311903}a^{21}+\frac{445930921}{1836311903}a^{19}-\frac{559262311}{1836311903}a^{17}+\frac{629626290}{1836311903}a^{15}+\frac{634741857}{1836311903}a^{13}-\frac{484764867}{1836311903}a^{11}-\frac{299211228}{1836311903}a^{9}-\frac{647537769}{1836311903}a^{7}-\frac{196581742}{1836311903}a^{5}+\frac{433872743}{1836311903}a^{3}-\frac{541930748}{1836311903}a$, $\frac{1}{1836311903}a^{40}+\frac{138086343}{1836311903}a^{22}-\frac{669630730}{1836311903}a^{20}-\frac{63616387}{1836311903}a^{18}+\frac{681575660}{1836311903}a^{16}-\frac{775913697}{1836311903}a^{14}-\frac{534666212}{1836311903}a^{12}+\frac{439397767}{1836311903}a^{10}+\frac{305321928}{1836311903}a^{8}-\frac{677667170}{1836311903}a^{6}-\frac{289345599}{1836311903}a^{4}-\frac{836949430}{1836311903}a^{2}+\frac{8839}{28657}$, $\frac{1}{1836311903}a^{41}-\frac{11740}{1836311903}a^{23}+\frac{670313276}{1836311903}a^{21}-\frac{1320766}{3983323}a^{19}-\frac{66715924}{1836311903}a^{17}-\frac{514017374}{1836311903}a^{15}+\frac{78450303}{1836311903}a^{13}+\frac{41781892}{1836311903}a^{11}-\frac{189527148}{1836311903}a^{9}-\frac{748020105}{1836311903}a^{7}+\frac{346209347}{1836311903}a^{5}+\frac{901584789}{1836311903}a^{3}+\frac{70026384}{1836311903}a$, $\frac{1}{1836311903}a^{42}+\frac{154079148}{1836311903}a^{22}-\frac{272833968}{1836311903}a^{20}+\frac{423796641}{1836311903}a^{18}+\frac{14546437}{1836311903}a^{16}+\frac{207966916}{1836311903}a^{14}+\frac{377151566}{1836311903}a^{12}-\frac{114356754}{1836311903}a^{10}-\frac{862096977}{1836311903}a^{8}+\frac{422554213}{1836311903}a^{6}+\frac{878048889}{1836311903}a^{4}+\frac{72703998}{1836311903}a^{2}-\frac{12553}{28657}$, $\frac{1}{1836311903}a^{43}-\frac{9541}{1836311903}a^{23}-\frac{401417927}{1836311903}a^{21}-\frac{126675672}{1836311903}a^{19}+\frac{30508386}{1836311903}a^{17}+\frac{730727910}{1836311903}a^{15}-\frac{230520119}{1836311903}a^{13}-\frac{157141655}{1836311903}a^{11}+\frac{852279391}{1836311903}a^{9}+\frac{495715534}{1836311903}a^{7}+\frac{853661782}{1836311903}a^{5}-\frac{771072710}{1836311903}a^{3}+\frac{903344257}{1836311903}a$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, a^10, a^11, a^12, a^13, a^14, a^15, a^16, a^17, a^18, a^19, a^20, a^21, a^22, 1/64079*a^23 - 23/64079*a^21 + 230/64079*a^19 - 1311/64079*a^17 + 4692/64079*a^15 - 10948/64079*a^13 + 16744/64079*a^11 - 16445/64079*a^9 + 9867/64079*a^7 - 3289/64079*a^5 + 506/64079*a^3 - 23/64079*a, 1/1836311903*a^24 - 701408757/1836311903*a^22 + 740497154/1836311903*a^20 + 310525523/1836311903*a^18 - 797202147/1836311903*a^16 + 821033279/1836311903*a^14 - 396952661/1836311903*a^12 + 316277499/1836311903*a^10 - 8768956/1836311903*a^8 - 913769829/1836311903*a^6 + 328469460/1836311903*a^4 - 400109299/1836311903*a^2 - 6765/28657, 1/1836311903*a^25 - 25/1836311903*a^23 - 701408458/1836311903*a^21 + 39086419/1836311903*a^19 - 351786396/1836311903*a^17 - 676438256/1836311903*a^15 + 36627749/1836311903*a^13 - 346845481/1836311903*a^11 - 800303953/1836311903*a^9 + 662938311/1836311903*a^7 - 197099920/1836311903*a^5 + 104511814/1836311903*a^3 - 39088144/1836311903*a, 1/1836311903*a^26 + 126491647/1836311903*a^22 + 188396239/1836311903*a^20 + 66104067/1836311903*a^18 - 407060998/1836311903*a^16 + 363028791/1836311903*a^14 + 747209412/1836311903*a^12 - 238614090/1836311903*a^10 + 443714411/1836311903*a^8 + 830709094/1836311903*a^6 - 865311201/1836311903*a^4 - 860261104/1836311903*a^2 + 2817/28657, 1/1836311903*a^27 - 351/1836311903*a^23 - 574911613/1836311903*a^21 + 353934975/1836311903*a^19 + 155877110/1836311903*a^17 - 8681156/1836311903*a^15 - 833883249/1836311903*a^13 + 883307460/1836311903*a^11 + 63235422/1836311903*a^9 - 410053035/1836311903*a^7 + 160380143/1836311903*a^5 - 594295487/1836311903*a^3 - 582797309/1836311903*a, 1/1836311903*a^28 - 703590318/1836311903*a^22 - 487854197/1836311903*a^20 + 807933406/1836311903*a^18 - 707225497/1836311903*a^16 + 884140812/1836311903*a^14 - 723683826/1836311903*a^12 + 897923391/1836311903*a^10 + 184667215/1836311903*a^8 + 781753189/1836311903*a^6 + 847146987/1836311903*a^4 + 374855273/1836311903*a^2 + 4016/28657, 1/1836311903*a^29 - 3654/1836311903*a^23 - 143540342/1836311903*a^21 - 798893241/1836311903*a^19 + 555545208/1836311903*a^17 + 423966706/1836311903*a^15 - 262048213/1836311903*a^13 - 24144241/1836311903*a^11 + 303278538/1836311903*a^9 - 23938366/1836311903*a^7 + 503606871/1836311903*a^5 + 145198075/1836311903*a^3 + 601655119/1836311903*a, 1/1836311903*a^30 + 400278168/1836311903*a^22 + 90274356/1836311903*a^20 + 375050196/1836311903*a^18 - 162000274/1836311903*a^16 - 740096249/1836311903*a^14 + 197235835/1836311903*a^12 - 895239006/1836311903*a^10 - 848401239/1836311903*a^8 + 3691359/1836311903*a^6 - 575379647/1836311903*a^4 + 306551361/1836311903*a^2 + 11681/28657, 1/1836311903*a^31 - 2808/1836311903*a^23 + 115177289/1836311903*a^21 + 126020866/1836311903*a^19 - 578844996/1836311903*a^17 - 311358872/1836311903*a^15 - 803151378/1836311903*a^13 - 661455200/1836311903*a^11 + 430388729/1836311903*a^9 + 338204520/1836311903*a^7 - 686884034/1836311903*a^5 - 241313165/1836311903*a^3 + 773409732/1836311903*a, 1/1836311903*a^32 - 914252351/1836311903*a^22 + 736955102/1836311903*a^20 - 871330337/1836311903*a^18 - 390777891/1836311903*a^16 + 86857789/1836311903*a^14 - 663202167/1836311903*a^12 - 237355131/1836311903*a^10 - 412969189/1836311903*a^8 + 611476528/1836311903*a^6 + 272355209/1836311903*a^4 - 746929127/1836311903*a^2 + 3471/28657, 1/1836311903*a^33 - 8080/1836311903*a^23 - 91232198/1836311903*a^21 + 65295051/1836311903*a^19 + 146655487/1836311903*a^17 + 96371913/1836311903*a^15 - 73297822/1836311903*a^13 + 372695085/1836311903*a^11 + 561855980/1836311903*a^9 - 340680954/1836311903*a^7 - 634466899/1836311903*a^5 - 889927557/1836311903*a^3 - 605769091/1836311903*a, 1/1836311903*a^34 - 615456100/1836311903*a^22 + 578119397/1836311903*a^20 + 790821829/1836311903*a^18 + 485179877/1836311903*a^16 - 719309041/1836311903*a^14 - 804223157/1836311903*a^12 - 62121076/1836311903*a^10 + 422318783/1836311903*a^8 - 84523256/1836311903*a^6 - 2355328/13210877*a^4 + 256356172/1836311903*a^2 - 12301/28657, 1/1836311903*a^35 + 10289/1836311903*a^23 - 723424229/1836311903*a^21 - 884237135/1836311903*a^19 - 250330685/1836311903*a^17 + 366676631/1836311903*a^15 + 334434081/1836311903*a^13 - 75303296/1836311903*a^11 + 828761014/1836311903*a^9 + 38873786/1836311903*a^7 + 855684996/1836311903*a^5 - 490674504/1836311903*a^3 - 253467502/1836311903*a, 1/1836311903*a^36 - 634502246/1836311903*a^22 + 834942809/1836311903*a^20 - 64725612/1836311903*a^18 - 25703587/1836311903*a^16 - 242219750/1836311903*a^14 + 212953461/1836311903*a^12 + 594265919/1836311903*a^10 + 283378823/1836311903*a^8 + 716512217/1836311903*a^6 + 537264979/1836311903*a^4 - 540176617/1836311903*a^2 - 2768/28657, 1/1836311903*a^37 - 7609/1836311903*a^23 - 904250521/1836311903*a^21 + 800400561/1836311903*a^19 + 1119365/1836311903*a^17 + 145022291/1836311903*a^15 + 533596634/1836311903*a^13 - 328202911/1836311903*a^11 - 56693796/1836311903*a^9 - 548493734/1836311903*a^7 - 265274306/1836311903*a^5 - 840473320/1836311903*a^3 - 80252099/1836311903*a, 1/1836311903*a^38 + 235219487/1836311903*a^22 - 397984960/1836311903*a^20 - 543595289/1836311903*a^18 - 427898623/1836311903*a^16 + 642722539/1836311903*a^14 - 7920025/1836311903*a^12 - 906108738/1836311903*a^10 + 672060473/1836311903*a^8 - 863038409/1836311903*a^6 - 736852163/1836311903*a^4 + 93226984/1836311903*a^2 - 6913/28657, 1/1836311903*a^39 + 2831/1836311903*a^23 - 496937581/1836311903*a^21 + 445930921/1836311903*a^19 - 559262311/1836311903*a^17 + 629626290/1836311903*a^15 + 634741857/1836311903*a^13 - 484764867/1836311903*a^11 - 299211228/1836311903*a^9 - 647537769/1836311903*a^7 - 196581742/1836311903*a^5 + 433872743/1836311903*a^3 - 541930748/1836311903*a, 1/1836311903*a^40 + 138086343/1836311903*a^22 - 669630730/1836311903*a^20 - 63616387/1836311903*a^18 + 681575660/1836311903*a^16 - 775913697/1836311903*a^14 - 534666212/1836311903*a^12 + 439397767/1836311903*a^10 + 305321928/1836311903*a^8 - 677667170/1836311903*a^6 - 289345599/1836311903*a^4 - 836949430/1836311903*a^2 + 8839/28657, 1/1836311903*a^41 - 11740/1836311903*a^23 + 670313276/1836311903*a^21 - 1320766/3983323*a^19 - 66715924/1836311903*a^17 - 514017374/1836311903*a^15 + 78450303/1836311903*a^13 + 41781892/1836311903*a^11 - 189527148/1836311903*a^9 - 748020105/1836311903*a^7 + 346209347/1836311903*a^5 + 901584789/1836311903*a^3 + 70026384/1836311903*a, 1/1836311903*a^42 + 154079148/1836311903*a^22 - 272833968/1836311903*a^20 + 423796641/1836311903*a^18 + 14546437/1836311903*a^16 + 207966916/1836311903*a^14 + 377151566/1836311903*a^12 - 114356754/1836311903*a^10 - 862096977/1836311903*a^8 + 422554213/1836311903*a^6 + 878048889/1836311903*a^4 + 72703998/1836311903*a^2 - 12553/28657, 1/1836311903*a^43 - 9541/1836311903*a^23 - 401417927/1836311903*a^21 - 126675672/1836311903*a^19 + 30508386/1836311903*a^17 + 730727910/1836311903*a^15 - 230520119/1836311903*a^13 - 157141655/1836311903*a^11 + 852279391/1836311903*a^9 + 495715534/1836311903*a^7 + 853661782/1836311903*a^5 - 771072710/1836311903*a^3 + 903344257/1836311903*a

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

not computed

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:	$21$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	$ -\frac{1}{64079} a^{23} + \frac{23}{64079} a^{21} - \frac{230}{64079} a^{19} + \frac{1311}{64079} a^{17} - \frac{4692}{64079} a^{15} + \frac{10948}{64079} a^{13} - \frac{16744}{64079} a^{11} + \frac{16445}{64079} a^{9} - \frac{9867}{64079} a^{7} + \frac{3289}{64079} a^{5} - \frac{506}{64079} a^{3} + \frac{23}{64079} a $ -(1)/(64079)a^(23) + (23)/(64079)a^(21) - (230)/(64079)a^(19) + (1311)/(64079)a^(17) - (4692)/(64079)a^(15) + (10948)/(64079)a^(13) - (16744)/(64079)a^(11) + (16445)/(64079)a^(9) - (9867)/(64079)a^(7) + (3289)/(64079)a^(5) - (506)/(64079)a^(3) + (23)/(64079)a (order $4$)	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:	not computed	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:	not computed	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

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

x = polygen(QQ); K.<a> = NumberField(x^44 - 47*x^42 + 1036*x^40 - 14238*x^38 + 136828*x^36 - 977616*x^34 + 5391753*x^32 - 23533371*x^30 + 82732335*x^28 - 237396295*x^26 + 562190611*x^24 - 1110544651*x^22 + 1852435411*x^20 - 2651394691*x^18 + 3329083366*x^16 - 3775046236*x^14 + 3998027671*x^12 - 4080345361*x^10 + 4101904756*x^8 - 4105687106*x^6 + 4106094436*x^4 - 4106117712*x^2 + 4106118241)

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^44 - 47*x^42 + 1036*x^40 - 14238*x^38 + 136828*x^36 - 977616*x^34 + 5391753*x^32 - 23533371*x^30 + 82732335*x^28 - 237396295*x^26 + 562190611*x^24 - 1110544651*x^22 + 1852435411*x^20 - 2651394691*x^18 + 3329083366*x^16 - 3775046236*x^14 + 3998027671*x^12 - 4080345361*x^10 + 4101904756*x^8 - 4105687106*x^6 + 4106094436*x^4 - 4106117712*x^2 + 4106118241, 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^44 - 47*x^42 + 1036*x^40 - 14238*x^38 + 136828*x^36 - 977616*x^34 + 5391753*x^32 - 23533371*x^30 + 82732335*x^28 - 237396295*x^26 + 562190611*x^24 - 1110544651*x^22 + 1852435411*x^20 - 2651394691*x^18 + 3329083366*x^16 - 3775046236*x^14 + 3998027671*x^12 - 4080345361*x^10 + 4101904756*x^8 - 4105687106*x^6 + 4106094436*x^4 - 4106117712*x^2 + 4106118241);

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^44 - 47*x^42 + 1036*x^40 - 14238*x^38 + 136828*x^36 - 977616*x^34 + 5391753*x^32 - 23533371*x^30 + 82732335*x^28 - 237396295*x^26 + 562190611*x^24 - 1110544651*x^22 + 1852435411*x^20 - 2651394691*x^18 + 3329083366*x^16 - 3775046236*x^14 + 3998027671*x^12 - 4080345361*x^10 + 4101904756*x^8 - 4105687106*x^6 + 4106094436*x^4 - 4106117712*x^2 + 4106118241);

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

$C_2\times C_{22}$ (as 44T2):

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)

An abelian group of order 44

The 44 conjugacy class representatives for $C_2\times C_{22}$

Character table for $C_2\times C_{22}$

Intermediate fields

$\Q(\sqrt{-1}) $, $\Q(\sqrt{115}) $, $\Q(\sqrt{-115}) $, $\Q(i, \sqrt{115})$, $\Q(\zeta_{23})^+$, 22.0.7198079267989980836471065337135104.1, 22.22.8083780427918435509708715954790400000000000.1, 22.0.1927323443393334271838358868310546875.1

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]

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	$22^{2}$	R	$22^{2}$	$22^{2}$	$22^{2}$	${\href{/padicField/17.11.0.1}{11} }^{4}$	$22^{2}$	R	${\href{/padicField/29.11.0.1}{11} }^{4}$	$22^{2}$	${\href{/padicField/37.11.0.1}{11} }^{4}$	${\href{/padicField/41.11.0.1}{11} }^{4}$	$22^{2}$	${\href{/padicField/47.2.0.1}{2} }^{22}$	${\href{/padicField/53.11.0.1}{11} }^{4}$	$22^{2}$

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		Deg $44$	$2$	$22$	$44$
$5$ 5	5.22.11.2	$x^{22} + 29296875 x^{2} - 146484375$	$2$	$11$	$11$	22T1	$[\ ]_{2}^{11}$
$5$ 5	5.22.11.2	$x^{22} + 29296875 x^{2} - 146484375$	$2$	$11$	$11$	22T1	$[\ ]_{2}^{11}$
$23$ 23		Deg $44$	$22$	$2$	$42$

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