LMFDB - Number field 25.1.11843998797823466973474682749162211402125799921.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^25 - x^24 - 19*x^23 - 76*x^22 + 156*x^21 + 1005*x^20 + 1755*x^19 - 1564*x^18 - 17480*x^17 + 14003*x^16 - 40555*x^15 + 158147*x^14 - 336486*x^13 - 456783*x^12 + 2807964*x^11 - 9825280*x^10 + 32382901*x^9 - 51222985*x^8 + 76250162*x^7 - 82760080*x^6 + 46691010*x^5 + 4604688*x^4 - 3014982*x^3 - 12748972*x^2 + 5156073*x + 1384173)

gp: K = bnfinit(y^25 - y^24 - 19*y^23 - 76*y^22 + 156*y^21 + 1005*y^20 + 1755*y^19 - 1564*y^18 - 17480*y^17 + 14003*y^16 - 40555*y^15 + 158147*y^14 - 336486*y^13 - 456783*y^12 + 2807964*y^11 - 9825280*y^10 + 32382901*y^9 - 51222985*y^8 + 76250162*y^7 - 82760080*y^6 + 46691010*y^5 + 4604688*y^4 - 3014982*y^3 - 12748972*y^2 + 5156073*y + 1384173, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^25 - x^24 - 19*x^23 - 76*x^22 + 156*x^21 + 1005*x^20 + 1755*x^19 - 1564*x^18 - 17480*x^17 + 14003*x^16 - 40555*x^15 + 158147*x^14 - 336486*x^13 - 456783*x^12 + 2807964*x^11 - 9825280*x^10 + 32382901*x^9 - 51222985*x^8 + 76250162*x^7 - 82760080*x^6 + 46691010*x^5 + 4604688*x^4 - 3014982*x^3 - 12748972*x^2 + 5156073*x + 1384173);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^25 - x^24 - 19*x^23 - 76*x^22 + 156*x^21 + 1005*x^20 + 1755*x^19 - 1564*x^18 - 17480*x^17 + 14003*x^16 - 40555*x^15 + 158147*x^14 - 336486*x^13 - 456783*x^12 + 2807964*x^11 - 9825280*x^10 + 32382901*x^9 - 51222985*x^8 + 76250162*x^7 - 82760080*x^6 + 46691010*x^5 + 4604688*x^4 - 3014982*x^3 - 12748972*x^2 + 5156073*x + 1384173)

$ x^{25} - x^{24} - 19 x^{23} - 76 x^{22} + 156 x^{21} + 1005 x^{20} + 1755 x^{19} - 1564 x^{18} + \cdots + 1384173 $ x^25 - x^24 - 19*x^23 - 76*x^22 + 156*x^21 + 1005*x^20 + 1755*x^19 - 1564*x^18 - 17480*x^17 + 14003*x^16 - 40555*x^15 + 158147*x^14 - 336486*x^13 - 456783*x^12 + 2807964*x^11 - 9825280*x^10 + 32382901*x^9 - 51222985*x^8 + 76250162*x^7 - 82760080*x^6 + 46691010*x^5 + 4604688*x^4 - 3014982*x^3 - 12748972*x^2 + 5156073*x + 1384173

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$25$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[1, 12]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$11843998797823466973474682749162211402125799921$ 11843998797823466973474682749162211402125799921 $\medspace = 11^{20}\cdot 127^{12}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$69.65$	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:	$11^{4/5}127^{1/2}\approx 76.7389775728696$
Ramified primes:	$11$, $127$ 11, 127	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q$
$\card{ \Aut(K/\Q) }$:	$1$	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{4}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{5}$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{6}$, $\frac{1}{3}a^{15}-\frac{1}{3}a^{7}$, $\frac{1}{21}a^{16}+\frac{2}{21}a^{14}+\frac{1}{21}a^{13}+\frac{2}{21}a^{12}-\frac{2}{21}a^{11}+\frac{2}{21}a^{8}+\frac{1}{3}a^{6}-\frac{4}{21}a^{5}+\frac{10}{21}a^{4}-\frac{10}{21}a^{3}+\frac{2}{7}a^{2}-\frac{3}{7}a$, $\frac{1}{63}a^{17}+\frac{1}{63}a^{16}-\frac{5}{63}a^{15}+\frac{10}{63}a^{14}+\frac{10}{63}a^{13}+\frac{1}{9}a^{12}-\frac{2}{63}a^{11}+\frac{1}{9}a^{10}+\frac{2}{63}a^{9}+\frac{2}{63}a^{8}-\frac{4}{9}a^{7}-\frac{25}{63}a^{6}+\frac{20}{63}a^{5}+\frac{2}{9}a^{4}-\frac{4}{63}a^{3}-\frac{31}{63}a^{2}-\frac{1}{7}a$, $\frac{1}{315}a^{18}-\frac{2}{315}a^{17}-\frac{1}{63}a^{16}-\frac{17}{315}a^{15}+\frac{4}{45}a^{14}+\frac{22}{315}a^{13}+\frac{4}{315}a^{12}+\frac{7}{45}a^{11}+\frac{2}{315}a^{10}-\frac{5}{63}a^{9}-\frac{4}{45}a^{8}+\frac{101}{315}a^{7}+\frac{74}{315}a^{6}-\frac{37}{315}a^{5}-\frac{37}{315}a^{4}-\frac{4}{45}a^{3}-\frac{1}{7}a^{2}+\frac{7}{15}a+\frac{2}{5}$, $\frac{1}{315}a^{19}+\frac{1}{315}a^{17}-\frac{2}{315}a^{16}+\frac{7}{45}a^{15}-\frac{2}{315}a^{14}-\frac{47}{315}a^{13}+\frac{52}{315}a^{12}+\frac{10}{63}a^{11}+\frac{7}{45}a^{10}+\frac{47}{315}a^{9}+\frac{19}{63}a^{8}-\frac{109}{315}a^{7}-\frac{139}{315}a^{6}-\frac{76}{315}a^{5}-\frac{22}{315}a^{4}+\frac{8}{105}a^{3}+\frac{152}{315}a^{2}+\frac{2}{7}a-\frac{1}{5}$, $\frac{1}{945}a^{20}-\frac{1}{945}a^{19}-\frac{1}{945}a^{18}+\frac{2}{315}a^{17}+\frac{1}{45}a^{16}-\frac{2}{45}a^{15}+\frac{23}{315}a^{14}-\frac{10}{63}a^{13}-\frac{13}{189}a^{12}+\frac{86}{945}a^{11}-\frac{76}{945}a^{10}+\frac{1}{315}a^{9}-\frac{76}{315}a^{8}-\frac{124}{315}a^{7}-\frac{1}{9}a^{6}-\frac{4}{315}a^{5}-\frac{52}{189}a^{4}-\frac{436}{945}a^{3}-\frac{292}{945}a^{2}+\frac{37}{105}a+\frac{2}{15}$, $\frac{1}{2835}a^{21}+\frac{1}{2835}a^{19}+\frac{2}{2835}a^{18}+\frac{1}{135}a^{17}+\frac{2}{315}a^{16}-\frac{22}{189}a^{15}+\frac{4}{135}a^{14}+\frac{418}{2835}a^{13}+\frac{10}{189}a^{12}-\frac{362}{2835}a^{11}-\frac{37}{2835}a^{10}+\frac{92}{945}a^{9}+\frac{86}{315}a^{8}-\frac{334}{945}a^{7}+\frac{188}{945}a^{6}+\frac{76}{2835}a^{5}-\frac{452}{945}a^{4}+\frac{298}{2835}a^{3}-\frac{523}{2835}a^{2}+\frac{8}{35}a-\frac{22}{45}$, $\frac{1}{2835}a^{22}+\frac{1}{2835}a^{20}+\frac{2}{2835}a^{19}+\frac{1}{945}a^{18}+\frac{1}{315}a^{17}-\frac{1}{189}a^{16}-\frac{22}{189}a^{15}+\frac{4}{2835}a^{14}-\frac{142}{945}a^{13}-\frac{209}{2835}a^{12}-\frac{424}{2835}a^{11}-\frac{5}{189}a^{10}+\frac{1}{15}a^{9}-\frac{16}{945}a^{8}+\frac{317}{945}a^{7}-\frac{1076}{2835}a^{6}+\frac{11}{189}a^{5}+\frac{199}{2835}a^{4}-\frac{649}{2835}a^{3}-\frac{19}{45}a^{2}+\frac{62}{315}a+\frac{1}{5}$, $\frac{1}{99225}a^{23}-\frac{1}{6615}a^{22}+\frac{8}{99225}a^{21}+\frac{2}{99225}a^{20}+\frac{64}{99225}a^{19}-\frac{82}{99225}a^{18}-\frac{148}{33075}a^{17}+\frac{556}{33075}a^{16}-\frac{269}{3969}a^{15}+\frac{418}{33075}a^{14}-\frac{8686}{99225}a^{13}-\frac{14026}{99225}a^{12}+\frac{1396}{99225}a^{11}-\frac{181}{2025}a^{10}+\frac{1144}{33075}a^{9}-\frac{5692}{33075}a^{8}-\frac{27836}{99225}a^{7}+\frac{9679}{33075}a^{6}-\frac{22723}{99225}a^{5}-\frac{20599}{99225}a^{4}-\frac{6844}{19845}a^{3}-\frac{5741}{19845}a^{2}-\frac{1441}{3675}a+\frac{677}{1575}$, $\frac{1}{31\!\cdots\!25}a^{24}-\frac{11\!\cdots\!02}{31\!\cdots\!25}a^{23}+\frac{10\!\cdots\!01}{10\!\cdots\!75}a^{22}-\frac{14\!\cdots\!49}{15\!\cdots\!25}a^{21}-\frac{10\!\cdots\!12}{21\!\cdots\!75}a^{20}+\frac{83\!\cdots\!59}{60\!\cdots\!65}a^{19}-\frac{43\!\cdots\!94}{90\!\cdots\!75}a^{18}+\frac{67\!\cdots\!87}{10\!\cdots\!75}a^{17}-\frac{36\!\cdots\!96}{31\!\cdots\!25}a^{16}-\frac{26\!\cdots\!91}{31\!\cdots\!25}a^{15}+\frac{58\!\cdots\!62}{10\!\cdots\!75}a^{14}+\frac{44\!\cdots\!96}{15\!\cdots\!25}a^{13}-\frac{23\!\cdots\!49}{10\!\cdots\!75}a^{12}+\frac{21\!\cdots\!73}{10\!\cdots\!75}a^{11}+\frac{14\!\cdots\!93}{63\!\cdots\!25}a^{10}-\frac{58\!\cdots\!88}{60\!\cdots\!65}a^{9}-\frac{14\!\cdots\!09}{31\!\cdots\!25}a^{8}+\frac{32\!\cdots\!01}{10\!\cdots\!25}a^{7}-\frac{37\!\cdots\!43}{35\!\cdots\!25}a^{6}-\frac{37\!\cdots\!01}{10\!\cdots\!75}a^{5}-\frac{58\!\cdots\!58}{35\!\cdots\!25}a^{4}-\frac{40\!\cdots\!67}{23\!\cdots\!75}a^{3}-\frac{73\!\cdots\!87}{31\!\cdots\!25}a^{2}+\frac{10\!\cdots\!52}{70\!\cdots\!25}a+\frac{22\!\cdots\!61}{50\!\cdots\!75}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, 1/3*a^9 - 1/3*a, 1/3*a^10 - 1/3*a^2, 1/3*a^11 - 1/3*a^3, 1/3*a^12 - 1/3*a^4, 1/3*a^13 - 1/3*a^5, 1/3*a^14 - 1/3*a^6, 1/3*a^15 - 1/3*a^7, 1/21*a^16 + 2/21*a^14 + 1/21*a^13 + 2/21*a^12 - 2/21*a^11 + 2/21*a^8 + 1/3*a^6 - 4/21*a^5 + 10/21*a^4 - 10/21*a^3 + 2/7*a^2 - 3/7*a, 1/63*a^17 + 1/63*a^16 - 5/63*a^15 + 10/63*a^14 + 10/63*a^13 + 1/9*a^12 - 2/63*a^11 + 1/9*a^10 + 2/63*a^9 + 2/63*a^8 - 4/9*a^7 - 25/63*a^6 + 20/63*a^5 + 2/9*a^4 - 4/63*a^3 - 31/63*a^2 - 1/7*a, 1/315*a^18 - 2/315*a^17 - 1/63*a^16 - 17/315*a^15 + 4/45*a^14 + 22/315*a^13 + 4/315*a^12 + 7/45*a^11 + 2/315*a^10 - 5/63*a^9 - 4/45*a^8 + 101/315*a^7 + 74/315*a^6 - 37/315*a^5 - 37/315*a^4 - 4/45*a^3 - 1/7*a^2 + 7/15*a + 2/5, 1/315*a^19 + 1/315*a^17 - 2/315*a^16 + 7/45*a^15 - 2/315*a^14 - 47/315*a^13 + 52/315*a^12 + 10/63*a^11 + 7/45*a^10 + 47/315*a^9 + 19/63*a^8 - 109/315*a^7 - 139/315*a^6 - 76/315*a^5 - 22/315*a^4 + 8/105*a^3 + 152/315*a^2 + 2/7*a - 1/5, 1/945*a^20 - 1/945*a^19 - 1/945*a^18 + 2/315*a^17 + 1/45*a^16 - 2/45*a^15 + 23/315*a^14 - 10/63*a^13 - 13/189*a^12 + 86/945*a^11 - 76/945*a^10 + 1/315*a^9 - 76/315*a^8 - 124/315*a^7 - 1/9*a^6 - 4/315*a^5 - 52/189*a^4 - 436/945*a^3 - 292/945*a^2 + 37/105*a + 2/15, 1/2835*a^21 + 1/2835*a^19 + 2/2835*a^18 + 1/135*a^17 + 2/315*a^16 - 22/189*a^15 + 4/135*a^14 + 418/2835*a^13 + 10/189*a^12 - 362/2835*a^11 - 37/2835*a^10 + 92/945*a^9 + 86/315*a^8 - 334/945*a^7 + 188/945*a^6 + 76/2835*a^5 - 452/945*a^4 + 298/2835*a^3 - 523/2835*a^2 + 8/35*a - 22/45, 1/2835*a^22 + 1/2835*a^20 + 2/2835*a^19 + 1/945*a^18 + 1/315*a^17 - 1/189*a^16 - 22/189*a^15 + 4/2835*a^14 - 142/945*a^13 - 209/2835*a^12 - 424/2835*a^11 - 5/189*a^10 + 1/15*a^9 - 16/945*a^8 + 317/945*a^7 - 1076/2835*a^6 + 11/189*a^5 + 199/2835*a^4 - 649/2835*a^3 - 19/45*a^2 + 62/315*a + 1/5, 1/99225*a^23 - 1/6615*a^22 + 8/99225*a^21 + 2/99225*a^20 + 64/99225*a^19 - 82/99225*a^18 - 148/33075*a^17 + 556/33075*a^16 - 269/3969*a^15 + 418/33075*a^14 - 8686/99225*a^13 - 14026/99225*a^12 + 1396/99225*a^11 - 181/2025*a^10 + 1144/33075*a^9 - 5692/33075*a^8 - 27836/99225*a^7 + 9679/33075*a^6 - 22723/99225*a^5 - 20599/99225*a^4 - 6844/19845*a^3 - 5741/19845*a^2 - 1441/3675*a + 677/1575, 1/31699329657966559650457223065658001010144770301662226597181476826005980188070349125*a^24 - 114038051769016907003597104696069426180403623637213159984620212684488281517902/31699329657966559650457223065658001010144770301662226597181476826005980188070349125*a^23 + 1096801369275434024937853981645152665819844761354960461308917461537399967818201/10566443219322186550152407688552667003381590100554075532393825608668660062690116375*a^22 - 141884572453253625828486929481793855841066321217670658232720927254114006326049/1509491888474598078593201098364666714768798585793439361770546515524094294670016625*a^21 - 1001720635871612617634988994055977219291149221612311431087697225382684705543212/2113288643864437310030481537710533400676318020110815106478765121733732012538023275*a^20 + 83466462083233775378695182875508104006468584067069351489017025600942237040559/60379675538983923143728043934586668590751943431737574470821860620963771786800665*a^19 - 436531739746908728232344476102737761106355700682234769562705579997026729664394/905695133084758847155920659018800028861279151476063617062327909314456576802009975*a^18 + 67112406102248529182802296705687736765124607661480789504065823703312131810290687/10566443219322186550152407688552667003381590100554075532393825608668660062690116375*a^17 - 362795515572153725574676067766853179965908117962682891483640293797203276914218896/31699329657966559650457223065658001010144770301662226597181476826005980188070349125*a^16 - 2619217938397126706193822228648252997943498618336118908095496169671796714608810191/31699329657966559650457223065658001010144770301662226597181476826005980188070349125*a^15 + 581887380683782930852000233810839543018233743852164934162712403793719312519961962/10566443219322186550152407688552667003381590100554075532393825608668660062690116375*a^14 + 44613118863561986303415054838666671581392842597365143604119681380571020269664196/1509491888474598078593201098364666714768798585793439361770546515524094294670016625*a^13 - 238037889710868554591280160730436127835104842991113278440147122771973032192269149/10566443219322186550152407688552667003381590100554075532393825608668660062690116375*a^12 + 21764497158157458569695077322634669184809404588545529586056376115388147223316473/10566443219322186550152407688552667003381590100554075532393825608668660062690116375*a^11 + 146352066842279714126086990342174826993102677331049785493761485291835255323788593/6339865931593311930091444613131600202028954060332445319436295365201196037614069825*a^10 - 5844104571805726318853200079792419255423778132368431442014798201806760517745888/60379675538983923143728043934586668590751943431737574470821860620963771786800665*a^9 - 14711729059731392032487599114095057779883212018158164258358014986675721414884138909/31699329657966559650457223065658001010144770301662226597181476826005980188070349125*a^8 + 326644600159037772040900512175827293108705226812503154675542189413120087915626401/1093080333033329643119214588470965552073957596609042296454533683655378627174839625*a^7 - 375487488537515340547817665443228574746541563259005233506143245708016249280228343/3522147739774062183384135896184222334460530033518025177464608536222886687563372125*a^6 - 3701124123645145901987597482819852290580638889906861527781620163768978242505687001/10566443219322186550152407688552667003381590100554075532393825608668660062690116375*a^5 - 587030055661696122366780576869103587419806023415608547415166336629786447443330658/3522147739774062183384135896184222334460530033518025177464608536222886687563372125*a^4 - 40865886478882511379837858078629980172794838409189725291064844025466122033525867/234809849318270812225609059745614822297368668901201678497640569081525779170891475*a^3 - 738948698312638487157725146919395992501776507129115715375531390173860488806729287/31699329657966559650457223065658001010144770301662226597181476826005980188070349125*a^2 + 105329329135378258872114272047569892114300010652028853408912800812374596488776252/704429547954812436676827179236844466892106006703605035492921707244577337512674425*a + 224799166070650145047364619736808923067092649479685029477350587142834931101508561/503163962824866026197733699454888904922932861931146453923515505174698098223338875

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		No
Index:		Not computed
Inessential primes:		$3$, $5$

Class group and class number

$C_{5}$, which has order $5$ (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:	$12$	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:	$\frac{14\!\cdots\!09}{58\!\cdots\!75}a^{24}-\frac{12\!\cdots\!43}{58\!\cdots\!75}a^{23}-\frac{28\!\cdots\!48}{58\!\cdots\!75}a^{22}-\frac{13\!\cdots\!54}{64\!\cdots\!75}a^{21}+\frac{86\!\cdots\!19}{23\!\cdots\!75}a^{20}+\frac{60\!\cdots\!12}{23\!\cdots\!55}a^{19}+\frac{57\!\cdots\!03}{11\!\cdots\!75}a^{18}-\frac{79\!\cdots\!81}{27\!\cdots\!75}a^{17}-\frac{25\!\cdots\!64}{58\!\cdots\!75}a^{16}+\frac{23\!\cdots\!83}{83\!\cdots\!25}a^{15}-\frac{61\!\cdots\!76}{58\!\cdots\!75}a^{14}+\frac{76\!\cdots\!73}{19\!\cdots\!25}a^{13}-\frac{47\!\cdots\!23}{58\!\cdots\!75}a^{12}-\frac{73\!\cdots\!79}{58\!\cdots\!75}a^{11}+\frac{80\!\cdots\!92}{11\!\cdots\!75}a^{10}-\frac{18\!\cdots\!17}{77\!\cdots\!85}a^{9}+\frac{46\!\cdots\!69}{58\!\cdots\!75}a^{8}-\frac{24\!\cdots\!66}{20\!\cdots\!75}a^{7}+\frac{10\!\cdots\!92}{58\!\cdots\!75}a^{6}-\frac{37\!\cdots\!34}{19\!\cdots\!25}a^{5}+\frac{61\!\cdots\!52}{58\!\cdots\!75}a^{4}+\frac{19\!\cdots\!09}{11\!\cdots\!75}a^{3}-\frac{15\!\cdots\!33}{58\!\cdots\!75}a^{2}-\frac{99\!\cdots\!74}{43\!\cdots\!25}a-\frac{95\!\cdots\!51}{92\!\cdots\!25}$, $\frac{25\!\cdots\!41}{19\!\cdots\!25}a^{24}-\frac{77\!\cdots\!86}{58\!\cdots\!75}a^{23}-\frac{14\!\cdots\!06}{58\!\cdots\!75}a^{22}-\frac{57\!\cdots\!87}{58\!\cdots\!75}a^{21}+\frac{78\!\cdots\!32}{38\!\cdots\!25}a^{20}+\frac{50\!\cdots\!81}{38\!\cdots\!25}a^{19}+\frac{26\!\cdots\!57}{11\!\cdots\!75}a^{18}-\frac{37\!\cdots\!79}{19\!\cdots\!25}a^{17}-\frac{14\!\cdots\!67}{64\!\cdots\!75}a^{16}+\frac{10\!\cdots\!32}{58\!\cdots\!75}a^{15}-\frac{32\!\cdots\!57}{58\!\cdots\!75}a^{14}+\frac{12\!\cdots\!33}{58\!\cdots\!75}a^{13}-\frac{12\!\cdots\!21}{27\!\cdots\!75}a^{12}-\frac{10\!\cdots\!26}{19\!\cdots\!25}a^{11}+\frac{42\!\cdots\!61}{11\!\cdots\!75}a^{10}-\frac{50\!\cdots\!72}{38\!\cdots\!25}a^{9}+\frac{83\!\cdots\!86}{19\!\cdots\!25}a^{8}-\frac{19\!\cdots\!31}{28\!\cdots\!25}a^{7}+\frac{61\!\cdots\!44}{58\!\cdots\!75}a^{6}-\frac{13\!\cdots\!76}{11\!\cdots\!75}a^{5}+\frac{19\!\cdots\!74}{27\!\cdots\!75}a^{4}-\frac{22\!\cdots\!98}{12\!\cdots\!75}a^{3}-\frac{10\!\cdots\!01}{58\!\cdots\!75}a^{2}-\frac{24\!\cdots\!82}{17\!\cdots\!13}a+\frac{11\!\cdots\!73}{92\!\cdots\!25}$, $\frac{89\!\cdots\!01}{35\!\cdots\!25}a^{24}-\frac{76\!\cdots\!49}{45\!\cdots\!75}a^{23}-\frac{22\!\cdots\!39}{45\!\cdots\!75}a^{22}-\frac{75\!\cdots\!11}{31\!\cdots\!25}a^{21}+\frac{39\!\cdots\!97}{21\!\cdots\!75}a^{20}+\frac{11\!\cdots\!14}{42\!\cdots\!55}a^{19}+\frac{44\!\cdots\!88}{63\!\cdots\!25}a^{18}+\frac{15\!\cdots\!33}{10\!\cdots\!75}a^{17}-\frac{48\!\cdots\!38}{10\!\cdots\!75}a^{16}-\frac{23\!\cdots\!19}{31\!\cdots\!25}a^{15}-\frac{28\!\cdots\!26}{31\!\cdots\!25}a^{14}+\frac{10\!\cdots\!94}{31\!\cdots\!25}a^{13}-\frac{59\!\cdots\!91}{10\!\cdots\!75}a^{12}-\frac{19\!\cdots\!68}{10\!\cdots\!75}a^{11}+\frac{35\!\cdots\!72}{63\!\cdots\!25}a^{10}-\frac{15\!\cdots\!06}{84\!\cdots\!31}a^{9}+\frac{95\!\cdots\!89}{15\!\cdots\!25}a^{8}-\frac{76\!\cdots\!16}{10\!\cdots\!25}a^{7}+\frac{53\!\cdots\!06}{45\!\cdots\!75}a^{6}-\frac{30\!\cdots\!52}{31\!\cdots\!25}a^{5}+\frac{27\!\cdots\!34}{10\!\cdots\!75}a^{4}+\frac{29\!\cdots\!11}{70\!\cdots\!25}a^{3}+\frac{51\!\cdots\!42}{31\!\cdots\!25}a^{2}-\frac{11\!\cdots\!92}{70\!\cdots\!25}a-\frac{16\!\cdots\!76}{50\!\cdots\!75}$, $\frac{62\!\cdots\!97}{31\!\cdots\!25}a^{24}-\frac{74\!\cdots\!69}{31\!\cdots\!25}a^{23}-\frac{12\!\cdots\!34}{31\!\cdots\!25}a^{22}-\frac{16\!\cdots\!19}{11\!\cdots\!75}a^{21}+\frac{74\!\cdots\!21}{21\!\cdots\!75}a^{20}+\frac{24\!\cdots\!29}{12\!\cdots\!65}a^{19}+\frac{19\!\cdots\!96}{70\!\cdots\!25}a^{18}-\frac{48\!\cdots\!61}{10\!\cdots\!75}a^{17}-\frac{15\!\cdots\!66}{45\!\cdots\!75}a^{16}+\frac{12\!\cdots\!98}{31\!\cdots\!25}a^{15}-\frac{31\!\cdots\!44}{45\!\cdots\!75}a^{14}+\frac{11\!\cdots\!67}{39\!\cdots\!25}a^{13}-\frac{27\!\cdots\!64}{39\!\cdots\!25}a^{12}-\frac{28\!\cdots\!57}{31\!\cdots\!25}a^{11}+\frac{13\!\cdots\!47}{21\!\cdots\!75}a^{10}-\frac{85\!\cdots\!61}{42\!\cdots\!55}a^{9}+\frac{20\!\cdots\!52}{31\!\cdots\!25}a^{8}-\frac{11\!\cdots\!53}{10\!\cdots\!25}a^{7}+\frac{44\!\cdots\!86}{31\!\cdots\!25}a^{6}-\frac{46\!\cdots\!49}{35\!\cdots\!25}a^{5}+\frac{45\!\cdots\!97}{10\!\cdots\!75}a^{4}+\frac{58\!\cdots\!17}{63\!\cdots\!25}a^{3}-\frac{10\!\cdots\!38}{10\!\cdots\!75}a^{2}+\frac{21\!\cdots\!69}{70\!\cdots\!25}a+\frac{16\!\cdots\!64}{16\!\cdots\!25}$, $\frac{47\!\cdots\!78}{64\!\cdots\!25}a^{24}-\frac{23\!\cdots\!24}{31\!\cdots\!25}a^{23}-\frac{44\!\cdots\!09}{31\!\cdots\!25}a^{22}-\frac{58\!\cdots\!76}{10\!\cdots\!75}a^{21}+\frac{72\!\cdots\!21}{63\!\cdots\!25}a^{20}+\frac{46\!\cdots\!16}{63\!\cdots\!25}a^{19}+\frac{81\!\cdots\!41}{63\!\cdots\!25}a^{18}-\frac{11\!\cdots\!46}{10\!\cdots\!75}a^{17}-\frac{40\!\cdots\!77}{31\!\cdots\!25}a^{16}+\frac{33\!\cdots\!23}{31\!\cdots\!25}a^{15}-\frac{99\!\cdots\!78}{31\!\cdots\!25}a^{14}+\frac{12\!\cdots\!19}{10\!\cdots\!75}a^{13}-\frac{79\!\cdots\!79}{31\!\cdots\!25}a^{12}-\frac{10\!\cdots\!87}{31\!\cdots\!25}a^{11}+\frac{74\!\cdots\!53}{36\!\cdots\!99}a^{10}-\frac{15\!\cdots\!39}{21\!\cdots\!75}a^{9}+\frac{76\!\cdots\!82}{31\!\cdots\!25}a^{8}-\frac{42\!\cdots\!83}{10\!\cdots\!25}a^{7}+\frac{18\!\cdots\!76}{31\!\cdots\!25}a^{6}-\frac{75\!\cdots\!38}{11\!\cdots\!75}a^{5}+\frac{12\!\cdots\!11}{31\!\cdots\!25}a^{4}-\frac{53\!\cdots\!88}{63\!\cdots\!25}a^{3}-\frac{25\!\cdots\!89}{31\!\cdots\!25}a^{2}-\frac{67\!\cdots\!67}{78\!\cdots\!25}a+\frac{22\!\cdots\!07}{50\!\cdots\!75}$, $\frac{15\!\cdots\!08}{31\!\cdots\!25}a^{24}-\frac{19\!\cdots\!16}{31\!\cdots\!25}a^{23}-\frac{10\!\cdots\!88}{11\!\cdots\!75}a^{22}-\frac{10\!\cdots\!82}{31\!\cdots\!25}a^{21}+\frac{16\!\cdots\!69}{21\!\cdots\!75}a^{20}+\frac{54\!\cdots\!04}{12\!\cdots\!65}a^{19}+\frac{15\!\cdots\!97}{21\!\cdots\!75}a^{18}-\frac{19\!\cdots\!93}{35\!\cdots\!25}a^{17}-\frac{22\!\cdots\!93}{31\!\cdots\!25}a^{16}+\frac{29\!\cdots\!47}{31\!\cdots\!25}a^{15}-\frac{29\!\cdots\!79}{10\!\cdots\!75}a^{14}+\frac{24\!\cdots\!03}{31\!\cdots\!25}a^{13}-\frac{20\!\cdots\!67}{10\!\cdots\!75}a^{12}-\frac{64\!\cdots\!89}{45\!\cdots\!75}a^{11}+\frac{28\!\cdots\!33}{21\!\cdots\!75}a^{10}-\frac{75\!\cdots\!86}{14\!\cdots\!85}a^{9}+\frac{55\!\cdots\!53}{31\!\cdots\!25}a^{8}-\frac{33\!\cdots\!17}{10\!\cdots\!25}a^{7}+\frac{52\!\cdots\!68}{10\!\cdots\!75}a^{6}-\frac{16\!\cdots\!99}{31\!\cdots\!25}a^{5}+\frac{42\!\cdots\!83}{10\!\cdots\!75}a^{4}-\frac{37\!\cdots\!37}{63\!\cdots\!25}a^{3}-\frac{14\!\cdots\!07}{10\!\cdots\!75}a^{2}+\frac{50\!\cdots\!63}{10\!\cdots\!75}a+\frac{31\!\cdots\!78}{23\!\cdots\!75}$, $\frac{68\!\cdots\!77}{31\!\cdots\!25}a^{24}-\frac{21\!\cdots\!79}{31\!\cdots\!25}a^{23}-\frac{12\!\cdots\!94}{31\!\cdots\!25}a^{22}-\frac{24\!\cdots\!58}{31\!\cdots\!25}a^{21}+\frac{45\!\cdots\!28}{63\!\cdots\!25}a^{20}+\frac{20\!\cdots\!67}{12\!\cdots\!65}a^{19}-\frac{21\!\cdots\!82}{21\!\cdots\!75}a^{18}-\frac{46\!\cdots\!67}{35\!\cdots\!25}a^{17}-\frac{10\!\cdots\!92}{31\!\cdots\!25}a^{16}+\frac{35\!\cdots\!18}{31\!\cdots\!25}a^{15}-\frac{37\!\cdots\!53}{31\!\cdots\!25}a^{14}+\frac{16\!\cdots\!07}{31\!\cdots\!25}a^{13}-\frac{63\!\cdots\!67}{45\!\cdots\!75}a^{12}+\frac{82\!\cdots\!88}{31\!\cdots\!25}a^{11}+\frac{18\!\cdots\!92}{21\!\cdots\!75}a^{10}-\frac{15\!\cdots\!99}{46\!\cdots\!95}a^{9}+\frac{34\!\cdots\!07}{31\!\cdots\!25}a^{8}-\frac{38\!\cdots\!64}{15\!\cdots\!75}a^{7}+\frac{11\!\cdots\!51}{31\!\cdots\!25}a^{6}-\frac{20\!\cdots\!08}{45\!\cdots\!75}a^{5}+\frac{17\!\cdots\!33}{45\!\cdots\!75}a^{4}-\frac{55\!\cdots\!08}{63\!\cdots\!25}a^{3}-\frac{16\!\cdots\!79}{39\!\cdots\!25}a^{2}-\frac{19\!\cdots\!51}{70\!\cdots\!25}a+\frac{17\!\cdots\!58}{55\!\cdots\!75}$, $\frac{21\!\cdots\!13}{35\!\cdots\!25}a^{24}-\frac{67\!\cdots\!47}{45\!\cdots\!75}a^{23}-\frac{54\!\cdots\!32}{45\!\cdots\!75}a^{22}-\frac{59\!\cdots\!76}{10\!\cdots\!75}a^{21}+\frac{69\!\cdots\!92}{12\!\cdots\!65}a^{20}+\frac{14\!\cdots\!93}{21\!\cdots\!75}a^{19}+\frac{10\!\cdots\!77}{63\!\cdots\!25}a^{18}+\frac{73\!\cdots\!63}{35\!\cdots\!25}a^{17}-\frac{42\!\cdots\!82}{39\!\cdots\!25}a^{16}+\frac{12\!\cdots\!78}{31\!\cdots\!25}a^{15}-\frac{77\!\cdots\!18}{31\!\cdots\!25}a^{14}+\frac{28\!\cdots\!13}{35\!\cdots\!25}a^{13}-\frac{46\!\cdots\!54}{31\!\cdots\!25}a^{12}-\frac{42\!\cdots\!99}{10\!\cdots\!75}a^{11}+\frac{92\!\cdots\!42}{63\!\cdots\!25}a^{10}-\frac{35\!\cdots\!07}{70\!\cdots\!25}a^{9}+\frac{82\!\cdots\!84}{50\!\cdots\!75}a^{8}-\frac{21\!\cdots\!28}{10\!\cdots\!25}a^{7}+\frac{14\!\cdots\!08}{45\!\cdots\!75}a^{6}-\frac{28\!\cdots\!47}{10\!\cdots\!75}a^{5}+\frac{23\!\cdots\!31}{31\!\cdots\!25}a^{4}+\frac{19\!\cdots\!24}{21\!\cdots\!75}a^{3}+\frac{15\!\cdots\!96}{31\!\cdots\!25}a^{2}-\frac{31\!\cdots\!44}{70\!\cdots\!25}a-\frac{47\!\cdots\!53}{50\!\cdots\!75}$, $\frac{38\!\cdots\!51}{31\!\cdots\!25}a^{24}-\frac{49\!\cdots\!37}{31\!\cdots\!25}a^{23}-\frac{71\!\cdots\!22}{31\!\cdots\!25}a^{22}-\frac{38\!\cdots\!37}{45\!\cdots\!75}a^{21}+\frac{27\!\cdots\!44}{12\!\cdots\!65}a^{20}+\frac{14\!\cdots\!23}{12\!\cdots\!25}a^{19}+\frac{22\!\cdots\!54}{12\!\cdots\!25}a^{18}-\frac{88\!\cdots\!61}{35\!\cdots\!25}a^{17}-\frac{64\!\cdots\!76}{31\!\cdots\!25}a^{16}+\frac{73\!\cdots\!09}{31\!\cdots\!25}a^{15}-\frac{16\!\cdots\!29}{31\!\cdots\!25}a^{14}+\frac{91\!\cdots\!68}{45\!\cdots\!75}a^{13}-\frac{14\!\cdots\!37}{31\!\cdots\!25}a^{12}-\frac{13\!\cdots\!16}{31\!\cdots\!25}a^{11}+\frac{22\!\cdots\!11}{63\!\cdots\!25}a^{10}-\frac{43\!\cdots\!86}{33\!\cdots\!25}a^{9}+\frac{13\!\cdots\!26}{31\!\cdots\!25}a^{8}-\frac{79\!\cdots\!09}{10\!\cdots\!25}a^{7}+\frac{34\!\cdots\!43}{31\!\cdots\!25}a^{6}-\frac{38\!\cdots\!73}{31\!\cdots\!25}a^{5}+\frac{24\!\cdots\!68}{31\!\cdots\!25}a^{4}+\frac{18\!\cdots\!96}{63\!\cdots\!25}a^{3}-\frac{63\!\cdots\!37}{31\!\cdots\!25}a^{2}-\frac{21\!\cdots\!98}{78\!\cdots\!25}a+\frac{29\!\cdots\!91}{50\!\cdots\!75}$, $\frac{10\!\cdots\!38}{13\!\cdots\!75}a^{24}-\frac{37\!\cdots\!16}{13\!\cdots\!75}a^{23}-\frac{16\!\cdots\!36}{13\!\cdots\!75}a^{22}-\frac{30\!\cdots\!47}{13\!\cdots\!75}a^{21}+\frac{22\!\cdots\!97}{91\!\cdots\!25}a^{20}+\frac{98\!\cdots\!23}{27\!\cdots\!75}a^{19}-\frac{15\!\cdots\!18}{27\!\cdots\!75}a^{18}-\frac{13\!\cdots\!24}{45\!\cdots\!25}a^{17}-\frac{82\!\cdots\!43}{13\!\cdots\!75}a^{16}+\frac{62\!\cdots\!17}{13\!\cdots\!75}a^{15}-\frac{12\!\cdots\!42}{13\!\cdots\!75}a^{14}+\frac{23\!\cdots\!73}{13\!\cdots\!75}a^{13}-\frac{23\!\cdots\!82}{45\!\cdots\!25}a^{12}+\frac{61\!\cdots\!32}{13\!\cdots\!75}a^{11}+\frac{80\!\cdots\!26}{27\!\cdots\!75}a^{10}-\frac{13\!\cdots\!52}{91\!\cdots\!25}a^{9}+\frac{64\!\cdots\!48}{13\!\cdots\!75}a^{8}-\frac{48\!\cdots\!52}{47\!\cdots\!75}a^{7}+\frac{23\!\cdots\!89}{13\!\cdots\!75}a^{6}-\frac{27\!\cdots\!69}{13\!\cdots\!75}a^{5}+\frac{81\!\cdots\!58}{45\!\cdots\!25}a^{4}-\frac{28\!\cdots\!41}{39\!\cdots\!25}a^{3}-\frac{11\!\cdots\!08}{19\!\cdots\!25}a^{2}+\frac{58\!\cdots\!59}{61\!\cdots\!95}a+\frac{91\!\cdots\!63}{21\!\cdots\!25}$, $\frac{20\!\cdots\!92}{31\!\cdots\!25}a^{24}-\frac{70\!\cdots\!84}{31\!\cdots\!25}a^{23}-\frac{42\!\cdots\!36}{35\!\cdots\!25}a^{22}-\frac{18\!\cdots\!18}{31\!\cdots\!25}a^{21}+\frac{35\!\cdots\!83}{63\!\cdots\!25}a^{20}+\frac{27\!\cdots\!32}{42\!\cdots\!55}a^{19}+\frac{10\!\cdots\!24}{63\!\cdots\!25}a^{18}+\frac{41\!\cdots\!29}{10\!\cdots\!75}a^{17}-\frac{32\!\cdots\!32}{31\!\cdots\!25}a^{16}+\frac{49\!\cdots\!28}{31\!\cdots\!25}a^{15}-\frac{32\!\cdots\!46}{10\!\cdots\!75}a^{14}+\frac{27\!\cdots\!72}{31\!\cdots\!25}a^{13}-\frac{55\!\cdots\!74}{31\!\cdots\!25}a^{12}-\frac{36\!\cdots\!34}{10\!\cdots\!75}a^{11}+\frac{91\!\cdots\!66}{63\!\cdots\!25}a^{10}-\frac{23\!\cdots\!18}{42\!\cdots\!55}a^{9}+\frac{57\!\cdots\!22}{31\!\cdots\!25}a^{8}-\frac{26\!\cdots\!33}{10\!\cdots\!25}a^{7}+\frac{47\!\cdots\!07}{10\!\cdots\!75}a^{6}-\frac{14\!\cdots\!76}{31\!\cdots\!25}a^{5}+\frac{10\!\cdots\!01}{31\!\cdots\!25}a^{4}-\frac{36\!\cdots\!03}{30\!\cdots\!25}a^{3}+\frac{65\!\cdots\!53}{45\!\cdots\!75}a^{2}-\frac{12\!\cdots\!01}{70\!\cdots\!25}a-\frac{47\!\cdots\!13}{50\!\cdots\!75}$, $\frac{20\!\cdots\!06}{11\!\cdots\!75}a^{24}+\frac{36\!\cdots\!81}{31\!\cdots\!25}a^{23}-\frac{99\!\cdots\!14}{31\!\cdots\!25}a^{22}-\frac{19\!\cdots\!04}{11\!\cdots\!75}a^{21}+\frac{35\!\cdots\!32}{46\!\cdots\!95}a^{20}+\frac{10\!\cdots\!29}{63\!\cdots\!25}a^{19}+\frac{31\!\cdots\!07}{63\!\cdots\!25}a^{18}+\frac{40\!\cdots\!56}{11\!\cdots\!75}a^{17}-\frac{36\!\cdots\!97}{15\!\cdots\!25}a^{16}-\frac{86\!\cdots\!17}{31\!\cdots\!25}a^{15}-\frac{39\!\cdots\!64}{45\!\cdots\!75}a^{14}+\frac{62\!\cdots\!68}{35\!\cdots\!25}a^{13}-\frac{42\!\cdots\!48}{10\!\cdots\!75}a^{12}-\frac{35\!\cdots\!67}{31\!\cdots\!25}a^{11}+\frac{21\!\cdots\!67}{63\!\cdots\!25}a^{10}-\frac{31\!\cdots\!99}{23\!\cdots\!75}a^{9}+\frac{45\!\cdots\!54}{10\!\cdots\!75}a^{8}-\frac{52\!\cdots\!83}{10\!\cdots\!25}a^{7}+\frac{31\!\cdots\!91}{31\!\cdots\!25}a^{6}-\frac{20\!\cdots\!89}{35\!\cdots\!25}a^{5}+\frac{52\!\cdots\!72}{10\!\cdots\!75}a^{4}+\frac{17\!\cdots\!67}{63\!\cdots\!25}a^{3}+\frac{58\!\cdots\!06}{31\!\cdots\!25}a^{2}+\frac{49\!\cdots\!56}{70\!\cdots\!25}a+\frac{57\!\cdots\!92}{50\!\cdots\!75}$ 14960813345581243372707783438157240544549042948944425831107749386277723009/58378139333271748895869655737860038692715967406376107913778042036843425760718875a^24 - 12615521201440327230189295511871961362344915919170948661844390497031437643/58378139333271748895869655737860038692715967406376107913778042036843425760718875a^23 - 284262883069400125219719752498244970916279262609941341252924362167986516048/58378139333271748895869655737860038692715967406376107913778042036843425760718875a^22 - 131373330949096725371210844290081107845704092088385058861627474583007232754/6486459925919083210652183970873337632523996378486234212642004670760380640079875a^21 + 8611897482184695330394488491695288989661034220121519788073692112457005219/238278119727639791411712880562694035480473336352555542505216498109565003104975a^20 + 607800354280529978471051338748229604106954521607061178190607731799124622512/2335125573330869955834786229514401547708638696255044316551121681473737030428755a^19 + 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588502708958664118943679706452050317782636707396708243089642816652286703448455749706/31699329657966559650457223065658001010144770301662226597181476826005980188070349125a^2 + 491484220045174963304518675801328056865633975327523827362105566658659698927226256/704429547954812436676827179236844466892106006703605035492921707244577337512674425a + 575740726483491671597193192170827885607087991766407797766161461765124750698976692/503163962824866026197733699454888904922932861931146453923515505174698098223338875 (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:	$ 14739079523494.871 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{12}\cdot 14739079523494.871 \cdot 5}{2\cdot\sqrt{11843998797823466973474682749162211402125799921}}\cr\approx \mathstrut & 2.56359614224321 \end{aligned}\] (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^25 - x^24 - 19*x^23 - 76*x^22 + 156*x^21 + 1005*x^20 + 1755*x^19 - 1564*x^18 - 17480*x^17 + 14003*x^16 - 40555*x^15 + 158147*x^14 - 336486*x^13 - 456783*x^12 + 2807964*x^11 - 9825280*x^10 + 32382901*x^9 - 51222985*x^8 + 76250162*x^7 - 82760080*x^6 + 46691010*x^5 + 4604688*x^4 - 3014982*x^3 - 12748972*x^2 + 5156073*x + 1384173)

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^25 - x^24 - 19*x^23 - 76*x^22 + 156*x^21 + 1005*x^20 + 1755*x^19 - 1564*x^18 - 17480*x^17 + 14003*x^16 - 40555*x^15 + 158147*x^14 - 336486*x^13 - 456783*x^12 + 2807964*x^11 - 9825280*x^10 + 32382901*x^9 - 51222985*x^8 + 76250162*x^7 - 82760080*x^6 + 46691010*x^5 + 4604688*x^4 - 3014982*x^3 - 12748972*x^2 + 5156073*x + 1384173, 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^25 - x^24 - 19*x^23 - 76*x^22 + 156*x^21 + 1005*x^20 + 1755*x^19 - 1564*x^18 - 17480*x^17 + 14003*x^16 - 40555*x^15 + 158147*x^14 - 336486*x^13 - 456783*x^12 + 2807964*x^11 - 9825280*x^10 + 32382901*x^9 - 51222985*x^8 + 76250162*x^7 - 82760080*x^6 + 46691010*x^5 + 4604688*x^4 - 3014982*x^3 - 12748972*x^2 + 5156073*x + 1384173);

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^25 - x^24 - 19*x^23 - 76*x^22 + 156*x^21 + 1005*x^20 + 1755*x^19 - 1564*x^18 - 17480*x^17 + 14003*x^16 - 40555*x^15 + 158147*x^14 - 336486*x^13 - 456783*x^12 + 2807964*x^11 - 9825280*x^10 + 32382901*x^9 - 51222985*x^8 + 76250162*x^7 - 82760080*x^6 + 46691010*x^5 + 4604688*x^4 - 3014982*x^3 - 12748972*x^2 + 5156073*x + 1384173);

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

$D_{25}$ (as 25T4):

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 50

The 14 conjugacy class representatives for $D_{25}$

Character table for $D_{25}$

Intermediate fields

5.1.16129.1

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	$25$	${\href{/padicField/3.2.0.1}{2} }^{12}{,}\,{\href{/padicField/3.1.0.1}{1} }$	${\href{/padicField/5.2.0.1}{2} }^{12}{,}\,{\href{/padicField/5.1.0.1}{1} }$	${\href{/padicField/7.2.0.1}{2} }^{12}{,}\,{\href{/padicField/7.1.0.1}{1} }$	R	$25$	$25$	$25$	${\href{/padicField/23.2.0.1}{2} }^{12}{,}\,{\href{/padicField/23.1.0.1}{1} }$	${\href{/padicField/29.2.0.1}{2} }^{12}{,}\,{\href{/padicField/29.1.0.1}{1} }$	$25$	$25$	$25$	${\href{/padicField/43.2.0.1}{2} }^{12}{,}\,{\href{/padicField/43.1.0.1}{1} }$	$25$	${\href{/padicField/53.2.0.1}{2} }^{12}{,}\,{\href{/padicField/53.1.0.1}{1} }$	${\href{/padicField/59.2.0.1}{2} }^{12}{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
$11$ 11		Deg $25$	$5$	$5$	$20$
$127$ 127	$\Q_{127}$	$x + 124$	$1$	$1$	$0$	Trivial	$[\ ]$
	127.2.1.2	$x^{2} + 127$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	127.2.1.2	$x^{2} + 127$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	127.2.1.2	$x^{2} + 127$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	127.2.1.2	$x^{2} + 127$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	127.2.1.2	$x^{2} + 127$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	127.2.1.2	$x^{2} + 127$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	127.2.1.2	$x^{2} + 127$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	127.2.1.2	$x^{2} + 127$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	127.2.1.2	$x^{2} + 127$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	127.2.1.2	$x^{2} + 127$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	127.2.1.2	$x^{2} + 127$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	127.2.1.2	$x^{2} + 127$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

Artin representations

	Label	Dimension	Conductor	Artin stem field	$G$	Ind	$\chi(c)$
*	1.1.1t1.a.a	$1$	$1$	$\Q$	$C_1$	$1$	$1$
	1.127.2t1.a.a	$1$	$ 127 $	$\Q(\sqrt{-127}) $	$C_2$ (as 2T1)	$1$	$-1$
*	2.127.5t2.a.a	$2$	$ 127 $	5.1.16129.1	$D_{5}$ (as 5T2)	$1$	$0$
*	2.127.5t2.a.b	$2$	$ 127 $	5.1.16129.1	$D_{5}$ (as 5T2)	$1$	$0$
*	2.15367.25t4.a.b	$2$	$ 11^{2} \cdot 127 $	25.1.11843998797823466973474682749162211402125799921.1	$D_{25}$ (as 25T4)	$1$	$0$
*	2.15367.25t4.a.i	$2$	$ 11^{2} \cdot 127 $	25.1.11843998797823466973474682749162211402125799921.1	$D_{25}$ (as 25T4)	$1$	$0$
*	2.15367.25t4.a.e	$2$	$ 11^{2} \cdot 127 $	25.1.11843998797823466973474682749162211402125799921.1	$D_{25}$ (as 25T4)	$1$	$0$
*	2.15367.25t4.a.g	$2$	$ 11^{2} \cdot 127 $	25.1.11843998797823466973474682749162211402125799921.1	$D_{25}$ (as 25T4)	$1$	$0$
*	2.15367.25t4.a.a	$2$	$ 11^{2} \cdot 127 $	25.1.11843998797823466973474682749162211402125799921.1	$D_{25}$ (as 25T4)	$1$	$0$
*	2.15367.25t4.a.j	$2$	$ 11^{2} \cdot 127 $	25.1.11843998797823466973474682749162211402125799921.1	$D_{25}$ (as 25T4)	$1$	$0$
*	2.15367.25t4.a.h	$2$	$ 11^{2} \cdot 127 $	25.1.11843998797823466973474682749162211402125799921.1	$D_{25}$ (as 25T4)	$1$	$0$
*	2.15367.25t4.a.f	$2$	$ 11^{2} \cdot 127 $	25.1.11843998797823466973474682749162211402125799921.1	$D_{25}$ (as 25T4)	$1$	$0$
*	2.15367.25t4.a.d	$2$	$ 11^{2} \cdot 127 $	25.1.11843998797823466973474682749162211402125799921.1	$D_{25}$ (as 25T4)	$1$	$0$
*	2.15367.25t4.a.c	$2$	$ 11^{2} \cdot 127 $	25.1.11843998797823466973474682749162211402125799921.1	$D_{25}$ (as 25T4)	$1$	$0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.

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