Properties

Label 47.47.605...769.1
Degree $47$
Signature $[47, 0]$
Discriminant $6.056\times 10^{112}$
Root discriminant $250.97$
Ramified prime $283$
Class number not computed
Class group not computed
Galois group $C_{47}$ (as 47T1)

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Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^47 - x^46 - 138*x^45 + 315*x^44 + 8338*x^43 - 29804*x^42 - 276833*x^41 + 1433626*x^40 + 5033859*x^39 - 41190458*x^38 - 30657314*x^37 + 748097961*x^36 - 742659788*x^35 - 8506344013*x^34 + 21519259357*x^33 + 52948548811*x^32 - 268879641855*x^31 - 30332528938*x^30 + 1920252236103*x^29 - 2430736233424*x^28 - 7367030656288*x^27 + 21401598866455*x^26 + 5373046913681*x^25 - 89240242581627*x^24 + 85735098102709*x^23 + 174332690558567*x^22 - 418760640969237*x^21 + 24016008538438*x^20 + 845143941649693*x^19 - 850093833789498*x^18 - 563502754038610*x^17 + 1652337279635119*x^16 - 688139958495907*x^15 - 1132046847057208*x^14 + 1399925903803443*x^13 - 129366579867529*x^12 - 739733924950208*x^11 + 464611724348759*x^10 + 61382685657965*x^9 - 169596960618209*x^8 + 46703998783537*x^7 + 17972540150505*x^6 - 11555848466611*x^5 + 591153994800*x^4 + 757301830397*x^3 - 134933083169*x^2 - 1756074840*x + 132954859)
 
gp: K = bnfinit(x^47 - x^46 - 138*x^45 + 315*x^44 + 8338*x^43 - 29804*x^42 - 276833*x^41 + 1433626*x^40 + 5033859*x^39 - 41190458*x^38 - 30657314*x^37 + 748097961*x^36 - 742659788*x^35 - 8506344013*x^34 + 21519259357*x^33 + 52948548811*x^32 - 268879641855*x^31 - 30332528938*x^30 + 1920252236103*x^29 - 2430736233424*x^28 - 7367030656288*x^27 + 21401598866455*x^26 + 5373046913681*x^25 - 89240242581627*x^24 + 85735098102709*x^23 + 174332690558567*x^22 - 418760640969237*x^21 + 24016008538438*x^20 + 845143941649693*x^19 - 850093833789498*x^18 - 563502754038610*x^17 + 1652337279635119*x^16 - 688139958495907*x^15 - 1132046847057208*x^14 + 1399925903803443*x^13 - 129366579867529*x^12 - 739733924950208*x^11 + 464611724348759*x^10 + 61382685657965*x^9 - 169596960618209*x^8 + 46703998783537*x^7 + 17972540150505*x^6 - 11555848466611*x^5 + 591153994800*x^4 + 757301830397*x^3 - 134933083169*x^2 - 1756074840*x + 132954859, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![132954859, -1756074840, -134933083169, 757301830397, 591153994800, -11555848466611, 17972540150505, 46703998783537, -169596960618209, 61382685657965, 464611724348759, -739733924950208, -129366579867529, 1399925903803443, -1132046847057208, -688139958495907, 1652337279635119, -563502754038610, -850093833789498, 845143941649693, 24016008538438, -418760640969237, 174332690558567, 85735098102709, -89240242581627, 5373046913681, 21401598866455, -7367030656288, -2430736233424, 1920252236103, -30332528938, -268879641855, 52948548811, 21519259357, -8506344013, -742659788, 748097961, -30657314, -41190458, 5033859, 1433626, -276833, -29804, 8338, 315, -138, -1, 1]);
 

\( x^{47} - x^{46} - 138 x^{45} + 315 x^{44} + 8338 x^{43} - 29804 x^{42} - 276833 x^{41} + 1433626 x^{40} + 5033859 x^{39} - 41190458 x^{38} - 30657314 x^{37} + 748097961 x^{36} - 742659788 x^{35} - 8506344013 x^{34} + 21519259357 x^{33} + 52948548811 x^{32} - 268879641855 x^{31} - 30332528938 x^{30} + 1920252236103 x^{29} - 2430736233424 x^{28} - 7367030656288 x^{27} + 21401598866455 x^{26} + 5373046913681 x^{25} - 89240242581627 x^{24} + 85735098102709 x^{23} + 174332690558567 x^{22} - 418760640969237 x^{21} + 24016008538438 x^{20} + 845143941649693 x^{19} - 850093833789498 x^{18} - 563502754038610 x^{17} + 1652337279635119 x^{16} - 688139958495907 x^{15} - 1132046847057208 x^{14} + 1399925903803443 x^{13} - 129366579867529 x^{12} - 739733924950208 x^{11} + 464611724348759 x^{10} + 61382685657965 x^{9} - 169596960618209 x^{8} + 46703998783537 x^{7} + 17972540150505 x^{6} - 11555848466611 x^{5} + 591153994800 x^{4} + 757301830397 x^{3} - 134933083169 x^{2} - 1756074840 x + 132954859 \)

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $47$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[47, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(605\!\cdots\!769\)\(\medspace = 283^{46}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $250.97$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $283$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $47$
This field is Galois and abelian over $\Q$.
Conductor:  \(283\)
Dirichlet character group:    $\lbrace$$\chi_{283}(256,·)$, $\chi_{283}(1,·)$, $\chi_{283}(4,·)$, $\chi_{283}(262,·)$, $\chi_{283}(264,·)$, $\chi_{283}(141,·)$, $\chi_{283}(15,·)$, $\chi_{283}(16,·)$, $\chi_{283}(275,·)$, $\chi_{283}(151,·)$, $\chi_{283}(152,·)$, $\chi_{283}(281,·)$, $\chi_{283}(155,·)$, $\chi_{283}(29,·)$, $\chi_{283}(158,·)$, $\chi_{283}(161,·)$, $\chi_{283}(163,·)$, $\chi_{283}(134,·)$, $\chi_{283}(38,·)$, $\chi_{283}(168,·)$, $\chi_{283}(42,·)$, $\chi_{283}(71,·)$, $\chi_{283}(175,·)$, $\chi_{283}(51,·)$, $\chi_{283}(181,·)$, $\chi_{283}(54,·)$, $\chi_{283}(116,·)$, $\chi_{283}(60,·)$, $\chi_{283}(61,·)$, $\chi_{283}(64,·)$, $\chi_{283}(66,·)$, $\chi_{283}(199,·)$, $\chi_{283}(204,·)$, $\chi_{283}(78,·)$, $\chi_{283}(207,·)$, $\chi_{283}(86,·)$, $\chi_{283}(216,·)$, $\chi_{283}(225,·)$, $\chi_{283}(230,·)$, $\chi_{283}(106,·)$, $\chi_{283}(111,·)$, $\chi_{283}(240,·)$, $\chi_{283}(244,·)$, $\chi_{283}(250,·)$, $\chi_{283}(251,·)$, $\chi_{283}(253,·)$, $\chi_{283}(127,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$, $a^{42}$, $a^{43}$, $\frac{1}{521} a^{44} - \frac{18}{521} a^{43} - \frac{205}{521} a^{42} - \frac{101}{521} a^{41} - \frac{103}{521} a^{40} + \frac{100}{521} a^{39} - \frac{36}{521} a^{38} - \frac{97}{521} a^{37} + \frac{223}{521} a^{36} + \frac{137}{521} a^{35} - \frac{16}{521} a^{34} - \frac{221}{521} a^{33} - \frac{135}{521} a^{32} - \frac{138}{521} a^{31} - \frac{192}{521} a^{30} + \frac{66}{521} a^{29} + \frac{208}{521} a^{28} - \frac{205}{521} a^{27} - \frac{87}{521} a^{26} + \frac{90}{521} a^{25} - \frac{164}{521} a^{24} + \frac{57}{521} a^{23} + \frac{82}{521} a^{22} + \frac{252}{521} a^{21} + \frac{249}{521} a^{20} + \frac{196}{521} a^{19} - \frac{239}{521} a^{18} - \frac{175}{521} a^{17} + \frac{13}{521} a^{16} + \frac{93}{521} a^{15} + \frac{208}{521} a^{14} + \frac{204}{521} a^{13} + \frac{229}{521} a^{12} + \frac{184}{521} a^{11} - \frac{212}{521} a^{10} + \frac{64}{521} a^{9} - \frac{89}{521} a^{8} - \frac{120}{521} a^{7} + \frac{193}{521} a^{6} + \frac{60}{521} a^{5} - \frac{124}{521} a^{4} - \frac{224}{521} a^{3} - \frac{111}{521} a^{2} - \frac{213}{521} a + \frac{114}{521}$, $\frac{1}{1755160951} a^{45} - \frac{221972}{1755160951} a^{44} - \frac{767272072}{1755160951} a^{43} - \frac{380323251}{1755160951} a^{42} - \frac{405534754}{1755160951} a^{41} - \frac{612802923}{1755160951} a^{40} + \frac{822122576}{1755160951} a^{39} + \frac{822573226}{1755160951} a^{38} - \frac{780080318}{1755160951} a^{37} + \frac{331184507}{1755160951} a^{36} - \frac{617826878}{1755160951} a^{35} - \frac{855917649}{1755160951} a^{34} + \frac{626061283}{1755160951} a^{33} + \frac{50085193}{1755160951} a^{32} + \frac{293941818}{1755160951} a^{31} - \frac{227595685}{1755160951} a^{30} + \frac{747046992}{1755160951} a^{29} + \frac{238672920}{1755160951} a^{28} - \frac{358112486}{1755160951} a^{27} + \frac{702536463}{1755160951} a^{26} + \frac{447900732}{1755160951} a^{25} - \frac{106511871}{1755160951} a^{24} + \frac{6812222}{1755160951} a^{23} + \frac{315011305}{1755160951} a^{22} + \frac{23191590}{1755160951} a^{21} - \frac{542054885}{1755160951} a^{20} - \frac{798428576}{1755160951} a^{19} + \frac{799868029}{1755160951} a^{18} - \frac{779269749}{1755160951} a^{17} - \frac{223599665}{1755160951} a^{16} + \frac{517602544}{1755160951} a^{15} + \frac{512096213}{1755160951} a^{14} - \frac{787059952}{1755160951} a^{13} + \frac{35918176}{1755160951} a^{12} - \frac{230887523}{1755160951} a^{11} - \frac{538690358}{1755160951} a^{10} + \frac{612784490}{1755160951} a^{9} - \frac{706604095}{1755160951} a^{8} + \frac{239437123}{1755160951} a^{7} + \frac{396669681}{1755160951} a^{6} + \frac{293021258}{1755160951} a^{5} - \frac{802565346}{1755160951} a^{4} - \frac{685475414}{1755160951} a^{3} - \frac{347636054}{1755160951} a^{2} - \frac{665668941}{1755160951} a - \frac{315747231}{1755160951}$, $\frac{1}{2289738063810103873618634940433406924239786439850810129920488753292222165285544810406744531754416248392457411461061319364542354890036581989149354883966550751834349420690128508585426561} a^{46} + \frac{61611035999860839778054328737745447210943433745309987724460617335227101808910281452488553486491476770193166161023290188816277324895230770909465729741081148478209076050982745}{2289738063810103873618634940433406924239786439850810129920488753292222165285544810406744531754416248392457411461061319364542354890036581989149354883966550751834349420690128508585426561} a^{45} - \frac{530917729564159386428651026437675989580975785412124247802204723113711081721494742461601742656741282224028026870025613339869494468577614153063570239717990777260416112381728768657038}{2289738063810103873618634940433406924239786439850810129920488753292222165285544810406744531754416248392457411461061319364542354890036581989149354883966550751834349420690128508585426561} a^{44} - \frac{320271011286767983177149821269506346566348933740764678190730980494752863275802945550698624909546057655383919675772267506929012931767388644345130643103961477481400882244783503515641368}{2289738063810103873618634940433406924239786439850810129920488753292222165285544810406744531754416248392457411461061319364542354890036581989149354883966550751834349420690128508585426561} a^{43} + \frac{1047818149940688878567712800145104701250238304844883921502014001302447391797410012192554238714979109521640730728148441907793696360979179026739217627407617337774846760168438301809876120}{2289738063810103873618634940433406924239786439850810129920488753292222165285544810406744531754416248392457411461061319364542354890036581989149354883966550751834349420690128508585426561} a^{42} + \frac{1142641867900027586210058793424415780438213914932958495122031137503553739489147565537936626444234628063502139739060817115271036220143537920163940433513361830947810337532711374427727727}{2289738063810103873618634940433406924239786439850810129920488753292222165285544810406744531754416248392457411461061319364542354890036581989149354883966550751834349420690128508585426561} a^{41} - \frac{1013713182607949586790911066718755229032596061224102094552935995573231787303837101000698398513269205251917195127867352414153234258932904292421905166416319797186643332864910586530014078}{2289738063810103873618634940433406924239786439850810129920488753292222165285544810406744531754416248392457411461061319364542354890036581989149354883966550751834349420690128508585426561} a^{40} - \frac{184740903637651720112827763741236626438298172558122428995530417771492345780481199201814351786860813184316318278522630972352986075064852068888956973661732289931571988497220804210379515}{2289738063810103873618634940433406924239786439850810129920488753292222165285544810406744531754416248392457411461061319364542354890036581989149354883966550751834349420690128508585426561} a^{39} + \frac{456824845088312979777485331812156613436154900198231491504458942808635443672042183435634972510123395901404103074621107274171417563016063720910203665979222969930108477212629766823946522}{2289738063810103873618634940433406924239786439850810129920488753292222165285544810406744531754416248392457411461061319364542354890036581989149354883966550751834349420690128508585426561} a^{38} + \frac{696237727134564879367043490852600775967364120985491405107354415369679227466995731664276401245461432218945296783345594593584108489693872316063237809556730310533583405048378415248467748}{2289738063810103873618634940433406924239786439850810129920488753292222165285544810406744531754416248392457411461061319364542354890036581989149354883966550751834349420690128508585426561} a^{37} - \frac{455192716963205929916690258937049064635778889712921771851742906237702334048392682500816629626730536940570663091667913113914046407638229409273409748047648009402167597907960922007388135}{2289738063810103873618634940433406924239786439850810129920488753292222165285544810406744531754416248392457411461061319364542354890036581989149354883966550751834349420690128508585426561} a^{36} + \frac{225398426211495991732393369939746110181708263578212630619095924882292251055890569034862222238985550967813262261892615292509525545121119458440615582397842770270983082719620556408817302}{2289738063810103873618634940433406924239786439850810129920488753292222165285544810406744531754416248392457411461061319364542354890036581989149354883966550751834349420690128508585426561} a^{35} - \frac{895638127092465039999564532118577099057047349636437691697188539782519862301821258222936054893175155679842717561835629791074367069514712943780127270382524900253156291710748365766863835}{2289738063810103873618634940433406924239786439850810129920488753292222165285544810406744531754416248392457411461061319364542354890036581989149354883966550751834349420690128508585426561} a^{34} - \frac{831956535535479903685540557532912252973370595944785238860026020815210458455266715173972549025064578939491463570743744798649341317598105364828516539026597269983247162550795763456126204}{2289738063810103873618634940433406924239786439850810129920488753292222165285544810406744531754416248392457411461061319364542354890036581989149354883966550751834349420690128508585426561} a^{33} - \frac{512639524281990412292461573219906594699265995467158294262890164148645026347596684146684596601090866968593691081634730842065559792981936454267441137934225560049624559808801988618490742}{2289738063810103873618634940433406924239786439850810129920488753292222165285544810406744531754416248392457411461061319364542354890036581989149354883966550751834349420690128508585426561} a^{32} + \frac{849968872242396749527361600127699969785677264672921555031870918815129095516497132463361248400266373331376543291446138678623641515241182316489300285384582658477361780747652740761133329}{2289738063810103873618634940433406924239786439850810129920488753292222165285544810406744531754416248392457411461061319364542354890036581989149354883966550751834349420690128508585426561} a^{31} - \frac{92174299472435120014077331423398710667509772875334973934495622263345149725378284584670944981135866861579906126725682440473884482876476361902961636566314267583496221737859692328743339}{2289738063810103873618634940433406924239786439850810129920488753292222165285544810406744531754416248392457411461061319364542354890036581989149354883966550751834349420690128508585426561} a^{30} + \frac{444812823826250917365722608803893967959852590969454968172867525391429396281663693990063435708305982909946367653474293344989295386381615327407871317141511036522506772910164921054371096}{2289738063810103873618634940433406924239786439850810129920488753292222165285544810406744531754416248392457411461061319364542354890036581989149354883966550751834349420690128508585426561} a^{29} + \frac{330924340795820447015310043300807175843225834503815404102567770672020880188881832921011322764288542470101869967332041065419948642058006636613309882716494975973132979587901989816641068}{2289738063810103873618634940433406924239786439850810129920488753292222165285544810406744531754416248392457411461061319364542354890036581989149354883966550751834349420690128508585426561} a^{28} - \frac{907194527126347631889838666456526545971963588152940890708654616429940666628636822103078891231809163482150735902729656209851319336512208434319393362787369013092614848974379465320499158}{2289738063810103873618634940433406924239786439850810129920488753292222165285544810406744531754416248392457411461061319364542354890036581989149354883966550751834349420690128508585426561} a^{27} - \frac{927897922206015967664723033755772759814747886951638660293555946861451985530955245973572470124559637491630326387299390015159558150012401422910617683146215576265205062851028514057236652}{2289738063810103873618634940433406924239786439850810129920488753292222165285544810406744531754416248392457411461061319364542354890036581989149354883966550751834349420690128508585426561} a^{26} + \frac{413290278046005221935845205051266593883323401948044890984892325793892929658863627496925377729454984570970632252164933346935233929670995816114856745399736270208350766372540336529338471}{2289738063810103873618634940433406924239786439850810129920488753292222165285544810406744531754416248392457411461061319364542354890036581989149354883966550751834349420690128508585426561} a^{25} - \frac{22239474041134177225299766190897800298800105815166488031901344973379749155117858108372664494500087808896703581829018523374596140374400346100510709840490721401987580117878546393097830}{2289738063810103873618634940433406924239786439850810129920488753292222165285544810406744531754416248392457411461061319364542354890036581989149354883966550751834349420690128508585426561} a^{24} + \frac{303194505091211345911347173088664939466448733630300167301341205307951787316314690653074734961776973599690174888902265637140654905230839019644032680507197927557339286797115441164612366}{2289738063810103873618634940433406924239786439850810129920488753292222165285544810406744531754416248392457411461061319364542354890036581989149354883966550751834349420690128508585426561} a^{23} - \frac{100148183145120688091946017354998181921825731785097469215817583079682931554362727540242323656654422256823370612963104025485403727107388432364755391711035798008267364947484297010346942}{2289738063810103873618634940433406924239786439850810129920488753292222165285544810406744531754416248392457411461061319364542354890036581989149354883966550751834349420690128508585426561} a^{22} - \frac{194958153881169603692437963607020180875191805363423857392253825788015477793298441498763015866473342438758888183309171907894529631312149741761212710447320339311627417658729898811088329}{2289738063810103873618634940433406924239786439850810129920488753292222165285544810406744531754416248392457411461061319364542354890036581989149354883966550751834349420690128508585426561} a^{21} - \frac{250595629415947741270967028011257002711663570686970733855550086772348037122647805257473601839941958900098305960671697847397071732917431678451683151543367267155048511756541014545459192}{2289738063810103873618634940433406924239786439850810129920488753292222165285544810406744531754416248392457411461061319364542354890036581989149354883966550751834349420690128508585426561} a^{20} - \frac{981726539762834426643152712179263300148065102075558563377375079361052216235416984618612537432547098607863545125608858001071185717556449518704434654076141757908563929557631260539353257}{2289738063810103873618634940433406924239786439850810129920488753292222165285544810406744531754416248392457411461061319364542354890036581989149354883966550751834349420690128508585426561} a^{19} + \frac{1115290839870714467376078745650371920025082965184356895102107140244259145450867779596255508401537284520381798767711784988598046927404636501545643712598548130481610803896641995459921602}{2289738063810103873618634940433406924239786439850810129920488753292222165285544810406744531754416248392457411461061319364542354890036581989149354883966550751834349420690128508585426561} a^{18} - \frac{684456495839791850176730317364600099812078466861970776388761604961056191006320819258697783883311464676604247375374762232710549180172689192243400327927532160524846915735899106759679920}{2289738063810103873618634940433406924239786439850810129920488753292222165285544810406744531754416248392457411461061319364542354890036581989149354883966550751834349420690128508585426561} a^{17} - \frac{367167549432805715796136765857241073510542416797313550015515019207217003120605741581241127520504470575498289058887599198926358267463603477670822822407855521728542660742877072350649311}{2289738063810103873618634940433406924239786439850810129920488753292222165285544810406744531754416248392457411461061319364542354890036581989149354883966550751834349420690128508585426561} a^{16} + \frac{582347308312363284085704710580508103242863652026670345339010720435530022857169998465392674631608186149659094312881596291752426942513144509267850732316179795480898465927412327739048631}{2289738063810103873618634940433406924239786439850810129920488753292222165285544810406744531754416248392457411461061319364542354890036581989149354883966550751834349420690128508585426561} a^{15} + \frac{772546075309405479336269193510292234421832887934742458367977325364200672091094596105682488409836545791802835423799531668494834798611450854230176491424760991649741932719123642929945449}{2289738063810103873618634940433406924239786439850810129920488753292222165285544810406744531754416248392457411461061319364542354890036581989149354883966550751834349420690128508585426561} a^{14} + \frac{586692383813597759272988908004757663285344085113341281052485124212939012465174844966394961394963689444414737744114151199133728054628429369579038421527602891776223750891569105656054319}{2289738063810103873618634940433406924239786439850810129920488753292222165285544810406744531754416248392457411461061319364542354890036581989149354883966550751834349420690128508585426561} a^{13} + \frac{165865425247456905587837804589588263278726158989358930208670929461814932830898965975461255767951606647884151704733237307852114309181831388752145664383439040005037993178557223636596463}{2289738063810103873618634940433406924239786439850810129920488753292222165285544810406744531754416248392457411461061319364542354890036581989149354883966550751834349420690128508585426561} a^{12} - \frac{605862940819161072181099241748017092202575124446996607305710911704034120506640822376921137151737171744830448657146564152901254417011242983745103347474997807684625287796818112138827995}{2289738063810103873618634940433406924239786439850810129920488753292222165285544810406744531754416248392457411461061319364542354890036581989149354883966550751834349420690128508585426561} a^{11} - \frac{152545144832336486616171914550507427410106009943725691486139469234443472597290021141734167208874957352616313733019669365067601645316553417978532310055065348219009406049306059999404074}{2289738063810103873618634940433406924239786439850810129920488753292222165285544810406744531754416248392457411461061319364542354890036581989149354883966550751834349420690128508585426561} a^{10} - \frac{616531981570417130840680750864340460617438381679783324126540147835867193105767687396246074269379724490263913583310791887132581247255159725846862291849968556166647578468019642627761718}{2289738063810103873618634940433406924239786439850810129920488753292222165285544810406744531754416248392457411461061319364542354890036581989149354883966550751834349420690128508585426561} a^{9} - \frac{904605518134979150608388224552034108833333716491695678275812327042607913192485532346125611806891369675095881486463265028225013995705851964366110158777484626423046119305730529621334421}{2289738063810103873618634940433406924239786439850810129920488753292222165285544810406744531754416248392457411461061319364542354890036581989149354883966550751834349420690128508585426561} a^{8} + \frac{775204854338496879677580489468924537456667747871448500755846537170428650490956139222508597676594221744358256642238330154959759759634314440865155771866567945939267759397258198397941196}{2289738063810103873618634940433406924239786439850810129920488753292222165285544810406744531754416248392457411461061319364542354890036581989149354883966550751834349420690128508585426561} a^{7} + \frac{870561960139527659876055822699213771258298074851910294653614589060944647659985273193801390442694385606380267580968806855061771983795848616809316693067192719416726919062817383763647910}{2289738063810103873618634940433406924239786439850810129920488753292222165285544810406744531754416248392457411461061319364542354890036581989149354883966550751834349420690128508585426561} a^{6} - \frac{307640476784039341156595846450243484170801043042916644056192368597287235754642466817850766022296869431950044529614968227286695635174140564907856040265917995089176556075370446101351610}{2289738063810103873618634940433406924239786439850810129920488753292222165285544810406744531754416248392457411461061319364542354890036581989149354883966550751834349420690128508585426561} a^{5} - \frac{317168477209353235997539382647215421539629211448511520668721172718144880238407921495161446169799373527527305496992906189625078464634428763901385251338209041895110840272542768985391675}{2289738063810103873618634940433406924239786439850810129920488753292222165285544810406744531754416248392457411461061319364542354890036581989149354883966550751834349420690128508585426561} a^{4} - \frac{103505543796467361335284071695185604566049152056358447995435732568417318613648478039783249287914663819593274501058247912242123163672608461725528309103625200564504319386785443700798724}{2289738063810103873618634940433406924239786439850810129920488753292222165285544810406744531754416248392457411461061319364542354890036581989149354883966550751834349420690128508585426561} a^{3} - \frac{1105922248919953518611922991121044824310634159356076690322580636626419446904779026931232825739195749941897649933941121313111024458586760246961442374841439627922529209958545745948785423}{2289738063810103873618634940433406924239786439850810129920488753292222165285544810406744531754416248392457411461061319364542354890036581989149354883966550751834349420690128508585426561} a^{2} + \frac{79722220888514135452055778240771440552172613352464363410839066503514968442934207618561233752191845045780526197826599794603267400551282200665821630882949912377886118268863560427641185}{2289738063810103873618634940433406924239786439850810129920488753292222165285544810406744531754416248392457411461061319364542354890036581989149354883966550751834349420690128508585426561} a + \frac{56813762543117164143737074783060148715042982130245348360308965359415559996606049297985352363139530246083465709458006140779979432148684296399942806283196944314651043868596515494793858}{2289738063810103873618634940433406924239786439850810129920488753292222165285544810406744531754416248392457411461061319364542354890036581989149354883966550751834349420690128508585426561}$

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 

Class group and class number

not computed

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $46$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) $ not computed

Galois group

$C_{47}$ (as 47T1):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A cyclic group of order 47
The 47 conjugacy class representatives for $C_{47}$
Character table for $C_{47}$ is not computed

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $47$ $47$ $47$ $47$ $47$ $47$ $47$ $47$ $47$ $47$ $47$ $47$ $47$ $47$ $47$ $47$ $47$

Cycle lengths which are repeated in a cycle type are indicated by exponents.

sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 
magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
283Data not computed