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Properties

Label 40.40.2456281703...7153.1
Degree $40$
Signature $[40, 0]$
Discriminant $11^{32}\cdot 17^{35}$
Root discriminant $81.24$
Ramified primes $11, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{40}$ (as 40T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 12, -198, -796, 9801, 23602, -214738, -373700, 2547284, 3409848, -18043315, -18792058, 80705559, 64898604, -235839274, -145456360, 461405669, 219419444, -618396670, -229527510, 579752554, 170144186, -386729741, -90641480, 185791871, 34947020, -64712614, -9751872, 16351547, 1954887, -2977848, -276856, 384678, 26871, -34203, -1692, 1981, 62, -67, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - x^39 - 67*x^38 + 62*x^37 + 1981*x^36 - 1692*x^35 - 34203*x^34 + 26871*x^33 + 384678*x^32 - 276856*x^31 - 2977848*x^30 + 1954887*x^29 + 16351547*x^28 - 9751872*x^27 - 64712614*x^26 + 34947020*x^25 + 185791871*x^24 - 90641480*x^23 - 386729741*x^22 + 170144186*x^21 + 579752554*x^20 - 229527510*x^19 - 618396670*x^18 + 219419444*x^17 + 461405669*x^16 - 145456360*x^15 - 235839274*x^14 + 64898604*x^13 + 80705559*x^12 - 18792058*x^11 - 18043315*x^10 + 3409848*x^9 + 2547284*x^8 - 373700*x^7 - 214738*x^6 + 23602*x^5 + 9801*x^4 - 796*x^3 - 198*x^2 + 12*x + 1)
 
gp: K = bnfinit(x^40 - x^39 - 67*x^38 + 62*x^37 + 1981*x^36 - 1692*x^35 - 34203*x^34 + 26871*x^33 + 384678*x^32 - 276856*x^31 - 2977848*x^30 + 1954887*x^29 + 16351547*x^28 - 9751872*x^27 - 64712614*x^26 + 34947020*x^25 + 185791871*x^24 - 90641480*x^23 - 386729741*x^22 + 170144186*x^21 + 579752554*x^20 - 229527510*x^19 - 618396670*x^18 + 219419444*x^17 + 461405669*x^16 - 145456360*x^15 - 235839274*x^14 + 64898604*x^13 + 80705559*x^12 - 18792058*x^11 - 18043315*x^10 + 3409848*x^9 + 2547284*x^8 - 373700*x^7 - 214738*x^6 + 23602*x^5 + 9801*x^4 - 796*x^3 - 198*x^2 + 12*x + 1, 1)
 

Normalized defining polynomial

\( x^{40} - x^{39} - 67 x^{38} + 62 x^{37} + 1981 x^{36} - 1692 x^{35} - 34203 x^{34} + 26871 x^{33} + 384678 x^{32} - 276856 x^{31} - 2977848 x^{30} + 1954887 x^{29} + 16351547 x^{28} - 9751872 x^{27} - 64712614 x^{26} + 34947020 x^{25} + 185791871 x^{24} - 90641480 x^{23} - 386729741 x^{22} + 170144186 x^{21} + 579752554 x^{20} - 229527510 x^{19} - 618396670 x^{18} + 219419444 x^{17} + 461405669 x^{16} - 145456360 x^{15} - 235839274 x^{14} + 64898604 x^{13} + 80705559 x^{12} - 18792058 x^{11} - 18043315 x^{10} + 3409848 x^{9} + 2547284 x^{8} - 373700 x^{7} - 214738 x^{6} + 23602 x^{5} + 9801 x^{4} - 796 x^{3} - 198 x^{2} + 12 x + 1 \)

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

Invariants

Degree:  $40$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[40, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(24562817038400928776197921239227357886542077974183334844678041435576602047153=11^{32}\cdot 17^{35}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $81.24$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $11, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(187=11\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{187}(1,·)$, $\chi_{187}(4,·)$, $\chi_{187}(135,·)$, $\chi_{187}(9,·)$, $\chi_{187}(15,·)$, $\chi_{187}(16,·)$, $\chi_{187}(152,·)$, $\chi_{187}(25,·)$, $\chi_{187}(26,·)$, $\chi_{187}(155,·)$, $\chi_{187}(157,·)$, $\chi_{187}(36,·)$, $\chi_{187}(38,·)$, $\chi_{187}(168,·)$, $\chi_{187}(169,·)$, $\chi_{187}(42,·)$, $\chi_{187}(174,·)$, $\chi_{187}(47,·)$, $\chi_{187}(49,·)$, $\chi_{187}(179,·)$, $\chi_{187}(53,·)$, $\chi_{187}(137,·)$, $\chi_{187}(185,·)$, $\chi_{187}(59,·)$, $\chi_{187}(60,·)$, $\chi_{187}(64,·)$, $\chi_{187}(67,·)$, $\chi_{187}(69,·)$, $\chi_{187}(70,·)$, $\chi_{187}(81,·)$, $\chi_{187}(86,·)$, $\chi_{187}(89,·)$, $\chi_{187}(93,·)$, $\chi_{187}(144,·)$, $\chi_{187}(100,·)$, $\chi_{187}(166,·)$, $\chi_{187}(103,·)$, $\chi_{187}(104,·)$, $\chi_{187}(111,·)$, $\chi_{187}(115,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $\frac{1}{67} a^{32} - \frac{32}{67} a^{31} + \frac{5}{67} a^{30} + \frac{26}{67} a^{29} + \frac{8}{67} a^{28} + \frac{13}{67} a^{27} + \frac{12}{67} a^{26} - \frac{15}{67} a^{25} - \frac{19}{67} a^{24} + \frac{17}{67} a^{23} - \frac{15}{67} a^{22} - \frac{26}{67} a^{21} + \frac{16}{67} a^{20} - \frac{27}{67} a^{19} + \frac{29}{67} a^{18} + \frac{13}{67} a^{17} - \frac{29}{67} a^{16} + \frac{5}{67} a^{15} - \frac{17}{67} a^{14} - \frac{13}{67} a^{13} - \frac{15}{67} a^{12} + \frac{16}{67} a^{11} - \frac{17}{67} a^{10} - \frac{11}{67} a^{9} - \frac{28}{67} a^{8} - \frac{32}{67} a^{7} + \frac{10}{67} a^{6} + \frac{18}{67} a^{5} + \frac{24}{67} a^{4} + \frac{6}{67} a^{3} + \frac{24}{67} a^{2} - \frac{25}{67} a + \frac{11}{67}$, $\frac{1}{67} a^{33} - \frac{14}{67} a^{31} - \frac{15}{67} a^{30} - \frac{31}{67} a^{29} + \frac{1}{67} a^{28} + \frac{26}{67} a^{27} - \frac{33}{67} a^{26} - \frac{30}{67} a^{25} + \frac{12}{67} a^{24} - \frac{7}{67} a^{23} + \frac{30}{67} a^{22} - \frac{12}{67} a^{21} + \frac{16}{67} a^{20} - \frac{31}{67} a^{19} + \frac{3}{67} a^{18} - \frac{15}{67} a^{17} + \frac{15}{67} a^{16} + \frac{9}{67} a^{15} - \frac{21}{67} a^{14} - \frac{29}{67} a^{13} + \frac{5}{67} a^{12} + \frac{26}{67} a^{11} - \frac{19}{67} a^{10} + \frac{22}{67} a^{9} + \frac{10}{67} a^{8} - \frac{9}{67} a^{7} + \frac{3}{67} a^{6} - \frac{3}{67} a^{5} - \frac{30}{67} a^{4} + \frac{15}{67} a^{3} + \frac{6}{67} a^{2} + \frac{15}{67} a + \frac{17}{67}$, $\frac{1}{67} a^{34} + \frac{6}{67} a^{31} - \frac{28}{67} a^{30} + \frac{30}{67} a^{29} + \frac{4}{67} a^{28} + \frac{15}{67} a^{27} + \frac{4}{67} a^{26} + \frac{3}{67} a^{25} - \frac{5}{67} a^{24} - \frac{21}{67} a^{22} - \frac{13}{67} a^{21} - \frac{8}{67} a^{20} + \frac{27}{67} a^{19} - \frac{11}{67} a^{18} - \frac{4}{67} a^{17} + \frac{5}{67} a^{16} - \frac{18}{67} a^{15} + \frac{1}{67} a^{14} + \frac{24}{67} a^{13} + \frac{17}{67} a^{12} + \frac{4}{67} a^{11} - \frac{15}{67} a^{10} - \frac{10}{67} a^{9} + \frac{1}{67} a^{8} + \frac{24}{67} a^{7} + \frac{3}{67} a^{6} + \frac{21}{67} a^{5} + \frac{16}{67} a^{4} + \frac{23}{67} a^{3} + \frac{16}{67} a^{2} + \frac{2}{67} a + \frac{20}{67}$, $\frac{1}{67} a^{35} + \frac{30}{67} a^{31} - \frac{18}{67} a^{29} - \frac{33}{67} a^{28} - \frac{7}{67} a^{27} - \frac{2}{67} a^{26} + \frac{18}{67} a^{25} - \frac{20}{67} a^{24} + \frac{11}{67} a^{23} + \frac{10}{67} a^{22} + \frac{14}{67} a^{21} - \frac{2}{67} a^{20} + \frac{17}{67} a^{19} + \frac{23}{67} a^{18} - \frac{6}{67} a^{17} + \frac{22}{67} a^{16} - \frac{29}{67} a^{15} - \frac{8}{67} a^{14} + \frac{28}{67} a^{13} + \frac{27}{67} a^{12} + \frac{23}{67} a^{11} + \frac{25}{67} a^{10} - \frac{9}{67} a^{8} - \frac{6}{67} a^{7} + \frac{28}{67} a^{6} - \frac{25}{67} a^{5} + \frac{13}{67} a^{4} - \frac{20}{67} a^{3} - \frac{8}{67} a^{2} - \frac{31}{67} a + \frac{1}{67}$, $\frac{1}{67} a^{36} + \frac{22}{67} a^{31} + \frac{33}{67} a^{30} - \frac{9}{67} a^{29} + \frac{21}{67} a^{28} + \frac{10}{67} a^{27} - \frac{7}{67} a^{26} + \frac{28}{67} a^{25} - \frac{22}{67} a^{24} - \frac{31}{67} a^{23} - \frac{5}{67} a^{22} - \frac{26}{67} a^{21} + \frac{6}{67} a^{20} + \frac{29}{67} a^{19} - \frac{5}{67} a^{18} - \frac{33}{67} a^{17} - \frac{30}{67} a^{16} - \frac{24}{67} a^{15} + \frac{2}{67} a^{14} + \frac{15}{67} a^{13} + \frac{4}{67} a^{12} + \frac{14}{67} a^{11} - \frac{26}{67} a^{10} - \frac{14}{67} a^{9} + \frac{30}{67} a^{8} - \frac{17}{67} a^{7} + \frac{10}{67} a^{6} + \frac{9}{67} a^{5} - \frac{3}{67} a^{4} + \frac{13}{67} a^{3} - \frac{14}{67} a^{2} + \frac{14}{67} a + \frac{5}{67}$, $\frac{1}{20569} a^{37} - \frac{83}{20569} a^{36} - \frac{89}{20569} a^{35} + \frac{32}{20569} a^{34} + \frac{26}{20569} a^{33} - \frac{57}{20569} a^{32} + \frac{6603}{20569} a^{31} + \frac{4549}{20569} a^{30} - \frac{9111}{20569} a^{29} + \frac{8766}{20569} a^{28} + \frac{8893}{20569} a^{27} - \frac{4710}{20569} a^{26} + \frac{573}{20569} a^{25} - \frac{3080}{20569} a^{24} + \frac{2543}{20569} a^{23} + \frac{6152}{20569} a^{22} - \frac{972}{20569} a^{21} - \frac{7559}{20569} a^{20} + \frac{9455}{20569} a^{19} - \frac{4029}{20569} a^{18} - \frac{2992}{20569} a^{17} + \frac{6766}{20569} a^{16} - \frac{10165}{20569} a^{15} - \frac{3635}{20569} a^{14} + \frac{3271}{20569} a^{13} - \frac{10108}{20569} a^{12} + \frac{727}{20569} a^{11} - \frac{8824}{20569} a^{10} + \frac{6467}{20569} a^{9} - \frac{2552}{20569} a^{8} - \frac{6574}{20569} a^{7} + \frac{1699}{20569} a^{6} - \frac{3909}{20569} a^{5} - \frac{1049}{20569} a^{4} - \frac{537}{20569} a^{3} + \frac{3474}{20569} a^{2} - \frac{5952}{20569} a - \frac{1430}{20569}$, $\frac{1}{20569} a^{38} + \frac{83}{20569} a^{36} + \frac{13}{20569} a^{35} - \frac{81}{20569} a^{34} - \frac{48}{20569} a^{33} + \frac{30}{20569} a^{32} - \frac{6449}{20569} a^{31} + \frac{2512}{20569} a^{30} + \frac{98}{20569} a^{29} - \frac{3399}{20569} a^{28} - \frac{6461}{20569} a^{27} - \frac{5379}{20569} a^{26} + \frac{6104}{20569} a^{25} + \frac{2020}{20569} a^{24} + \frac{1400}{20569} a^{23} - \frac{6730}{20569} a^{22} + \frac{2637}{20569} a^{21} + \frac{5575}{20569} a^{20} + \frac{649}{20569} a^{19} - \frac{6453}{20569} a^{18} - \frac{5794}{20569} a^{17} - \frac{5178}{20569} a^{16} + \frac{64}{307} a^{15} - \frac{2179}{20569} a^{14} - \frac{2942}{20569} a^{13} - \frac{3504}{20569} a^{12} - \frac{2822}{20569} a^{11} + \frac{5349}{20569} a^{10} + \frac{60}{307} a^{9} - \frac{4411}{20569} a^{8} + \frac{3131}{20569} a^{7} - \frac{9945}{20569} a^{6} + \frac{3915}{20569} a^{5} + \frac{4189}{20569} a^{4} + \frac{2190}{20569} a^{3} + \frac{2713}{20569} a^{2} + \frac{359}{20569} a - \frac{2951}{20569}$, $\frac{1}{5829314843710619797203883913757122957762993355091534380381363925809253235007601523} a^{39} + \frac{18405364058598086739725227218799394714914715090714644255626814184982341640755}{5829314843710619797203883913757122957762993355091534380381363925809253235007601523} a^{38} + \frac{141108958545698043829186478243331041385287212191839553473592485304064048576968}{5829314843710619797203883913757122957762993355091534380381363925809253235007601523} a^{37} - \frac{40081078965622438334323565525671804069837339466604159474310693463264283545745061}{5829314843710619797203883913757122957762993355091534380381363925809253235007601523} a^{36} + \frac{12646536639715979191387073323140624353281332158645842691386888287380901910843189}{5829314843710619797203883913757122957762993355091534380381363925809253235007601523} a^{35} + \frac{2281848019471094913027984206717458508479679651465287997533621422052812228169712}{5829314843710619797203883913757122957762993355091534380381363925809253235007601523} a^{34} + \frac{19108136723107050841733065083522835333111762972382066973493910475307933556010320}{5829314843710619797203883913757122957762993355091534380381363925809253235007601523} a^{33} - \frac{2767386143728787819588427101207593215310308924178858043392501793093466925202947}{5829314843710619797203883913757122957762993355091534380381363925809253235007601523} a^{32} - \frac{275160334860464755861080124608386282307049061334819801923605828347937602575949262}{5829314843710619797203883913757122957762993355091534380381363925809253235007601523} a^{31} + \frac{966167257189207974261912906361172933370190523137426953704896890786113562316569107}{5829314843710619797203883913757122957762993355091534380381363925809253235007601523} a^{30} + \frac{1120388307908862971960233861223719946462447164835821223636054299215718963359795678}{5829314843710619797203883913757122957762993355091534380381363925809253235007601523} a^{29} + \frac{635424744834651904586469527754183654210555559115939931393655919148154147243207941}{5829314843710619797203883913757122957762993355091534380381363925809253235007601523} a^{28} - \frac{978821769363787026732851587768523428396502337160089903301530199969559200521571396}{5829314843710619797203883913757122957762993355091534380381363925809253235007601523} a^{27} - \frac{1693310791666608653154947584053424704982918492778952288753492787391975015899111633}{5829314843710619797203883913757122957762993355091534380381363925809253235007601523} a^{26} + \frac{685348012976260593493378835382230195530959617898640550870000020321694094846112462}{5829314843710619797203883913757122957762993355091534380381363925809253235007601523} a^{25} - \frac{364379555992479045576747716547688755610300544707674236518116587680947618399776379}{5829314843710619797203883913757122957762993355091534380381363925809253235007601523} a^{24} + \frac{866470638190436795975249011986751733528916879042232165734951323676848971399303150}{5829314843710619797203883913757122957762993355091534380381363925809253235007601523} a^{23} - \frac{1241020366361484475544447255352358843617172421817459877976978797764231786014119541}{5829314843710619797203883913757122957762993355091534380381363925809253235007601523} a^{22} + \frac{246950794998851606526931047359116065205041347206346846336138041013665387147670086}{5829314843710619797203883913757122957762993355091534380381363925809253235007601523} a^{21} + \frac{2651092584741993805839395693670811817811685414538182305839171758952141826720647263}{5829314843710619797203883913757122957762993355091534380381363925809253235007601523} a^{20} - \frac{40153396576392605924266431337341318240445059488808454754267612190503060213238841}{5829314843710619797203883913757122957762993355091534380381363925809253235007601523} a^{19} - \frac{1060386429522465233733275934925686609457884309838562377623365686310324357922362703}{5829314843710619797203883913757122957762993355091534380381363925809253235007601523} a^{18} + \frac{2439120238778027098446484495272480609225365934077829696029471460914509508694153994}{5829314843710619797203883913757122957762993355091534380381363925809253235007601523} a^{17} - \frac{542379113510122139543989213983007549152491116985276336311682494728632583028692984}{5829314843710619797203883913757122957762993355091534380381363925809253235007601523} a^{16} - \frac{372174573055815438877754883553075974299160435349502285290920259786489832006743052}{5829314843710619797203883913757122957762993355091534380381363925809253235007601523} a^{15} + \frac{385689921181881671593733820907299972326644928458625367881842996567995420795440153}{5829314843710619797203883913757122957762993355091534380381363925809253235007601523} a^{14} + \frac{2139386429463284574952194649466005266560706818369273118396301744533039345145964087}{5829314843710619797203883913757122957762993355091534380381363925809253235007601523} a^{13} - \frac{1669553980035723979963596856170311719077397175677839880701615164790377024164284056}{5829314843710619797203883913757122957762993355091534380381363925809253235007601523} a^{12} + \frac{639179817242572664792229924638924564803786759893763467750888184580320159638821958}{5829314843710619797203883913757122957762993355091534380381363925809253235007601523} a^{11} - \frac{231466973636868681408531505722979730608009056184971592769010165099158332717406962}{5829314843710619797203883913757122957762993355091534380381363925809253235007601523} a^{10} + \frac{1600807714449101013966785263579808051854432005533829644556778629502308859978085778}{5829314843710619797203883913757122957762993355091534380381363925809253235007601523} a^{9} + \frac{1197584136604394166077677595190422966329655468751560052565403984239968453616303182}{5829314843710619797203883913757122957762993355091534380381363925809253235007601523} a^{8} + \frac{266081520486341938828913417111497259282956387872544958775252746876202453484700879}{5829314843710619797203883913757122957762993355091534380381363925809253235007601523} a^{7} + \frac{2845982900765467083650123013424444055951598011372409457185220182386124223540643787}{5829314843710619797203883913757122957762993355091534380381363925809253235007601523} a^{6} + \frac{960485285600379251441425481434795110677704261553937674047993021260216025218318083}{5829314843710619797203883913757122957762993355091534380381363925809253235007601523} a^{5} + \frac{181396156888451604185456499176698308154868023737637118612039347131151470859236210}{5829314843710619797203883913757122957762993355091534380381363925809253235007601523} a^{4} - \frac{1974382977375930745978592489176193782512132527181467329946449960084250174067133485}{5829314843710619797203883913757122957762993355091534380381363925809253235007601523} a^{3} + \frac{106129887823462525067440952133558437824303511333626330069165935788952799849164428}{5829314843710619797203883913757122957762993355091534380381363925809253235007601523} a^{2} + \frac{2622021742084761589995575826434393142933303786591657025412883462764893714647233884}{5829314843710619797203883913757122957762993355091534380381363925809253235007601523} a + \frac{1062013049673661607765579398733970733031756231451739631598224353351313269735177581}{5829314843710619797203883913757122957762993355091534380381363925809253235007601523}$

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

Class group and class number

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

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $39$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 53402945065867880000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{40}$ (as 40T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 40
The 40 conjugacy class representatives for $C_{40}$
Character table for $C_{40}$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{17})^+\), 10.10.304358957700017.1, 20.20.131527565972137936816816034072938673.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $20^{2}$ $40$ $40$ $40$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{4}$ R $20^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{5}$ $40$ $40$ $40$ $40$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{4}$ $20^{2}$ $20^{2}$

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

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];
 
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]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
11Data not computed
17Data not computed