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Properties

Label 36.36.5744476530...5625.2
Degree $36$
Signature $[36, 0]$
Discriminant $5^{27}\cdot 37^{35}$
Root discriminant $111.91$
Ramified primes $5, 37$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_{36}$ (as 36T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 73, -1109, -27048, 16170, 1081472, -1052612, -13242338, 14998099, 75196320, -86474808, -237247220, 264470377, 453279451, -485802821, -551208426, 571604935, 441080699, -447116805, -238153128, 238235194, 88382528, -87888652, -22805506, 22653732, 4104557, -4083615, -511341, 509935, 43067, -43030, -2332, 2332, 73, -73, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 - 73*x^34 + 73*x^33 + 2332*x^32 - 2332*x^31 - 43030*x^30 + 43067*x^29 + 509935*x^28 - 511341*x^27 - 4083615*x^26 + 4104557*x^25 + 22653732*x^24 - 22805506*x^23 - 87888652*x^22 + 88382528*x^21 + 238235194*x^20 - 238153128*x^19 - 447116805*x^18 + 441080699*x^17 + 571604935*x^16 - 551208426*x^15 - 485802821*x^14 + 453279451*x^13 + 264470377*x^12 - 237247220*x^11 - 86474808*x^10 + 75196320*x^9 + 14998099*x^8 - 13242338*x^7 - 1052612*x^6 + 1081472*x^5 + 16170*x^4 - 27048*x^3 - 1109*x^2 + 73*x + 1)
 
gp: K = bnfinit(x^36 - x^35 - 73*x^34 + 73*x^33 + 2332*x^32 - 2332*x^31 - 43030*x^30 + 43067*x^29 + 509935*x^28 - 511341*x^27 - 4083615*x^26 + 4104557*x^25 + 22653732*x^24 - 22805506*x^23 - 87888652*x^22 + 88382528*x^21 + 238235194*x^20 - 238153128*x^19 - 447116805*x^18 + 441080699*x^17 + 571604935*x^16 - 551208426*x^15 - 485802821*x^14 + 453279451*x^13 + 264470377*x^12 - 237247220*x^11 - 86474808*x^10 + 75196320*x^9 + 14998099*x^8 - 13242338*x^7 - 1052612*x^6 + 1081472*x^5 + 16170*x^4 - 27048*x^3 - 1109*x^2 + 73*x + 1, 1)
 

Normalized defining polynomial

\( x^{36} - x^{35} - 73 x^{34} + 73 x^{33} + 2332 x^{32} - 2332 x^{31} - 43030 x^{30} + 43067 x^{29} + 509935 x^{28} - 511341 x^{27} - 4083615 x^{26} + 4104557 x^{25} + 22653732 x^{24} - 22805506 x^{23} - 87888652 x^{22} + 88382528 x^{21} + 238235194 x^{20} - 238153128 x^{19} - 447116805 x^{18} + 441080699 x^{17} + 571604935 x^{16} - 551208426 x^{15} - 485802821 x^{14} + 453279451 x^{13} + 264470377 x^{12} - 237247220 x^{11} - 86474808 x^{10} + 75196320 x^{9} + 14998099 x^{8} - 13242338 x^{7} - 1052612 x^{6} + 1081472 x^{5} + 16170 x^{4} - 27048 x^{3} - 1109 x^{2} + 73 x + 1 \)

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

Invariants

Degree:  $36$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[36, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(57444765302724909954814307473256133361395843470561362005770206451416015625=5^{27}\cdot 37^{35}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $111.91$
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:  $5, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(185=5\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{185}(1,·)$, $\chi_{185}(4,·)$, $\chi_{185}(139,·)$, $\chi_{185}(143,·)$, $\chi_{185}(16,·)$, $\chi_{185}(17,·)$, $\chi_{185}(18,·)$, $\chi_{185}(22,·)$, $\chi_{185}(26,·)$, $\chi_{185}(159,·)$, $\chi_{185}(163,·)$, $\chi_{185}(167,·)$, $\chi_{185}(168,·)$, $\chi_{185}(169,·)$, $\chi_{185}(42,·)$, $\chi_{185}(46,·)$, $\chi_{185}(181,·)$, $\chi_{185}(184,·)$, $\chi_{185}(64,·)$, $\chi_{185}(68,·)$, $\chi_{185}(71,·)$, $\chi_{185}(72,·)$, $\chi_{185}(81,·)$, $\chi_{185}(82,·)$, $\chi_{185}(86,·)$, $\chi_{185}(87,·)$, $\chi_{185}(88,·)$, $\chi_{185}(97,·)$, $\chi_{185}(98,·)$, $\chi_{185}(99,·)$, $\chi_{185}(103,·)$, $\chi_{185}(104,·)$, $\chi_{185}(113,·)$, $\chi_{185}(114,·)$, $\chi_{185}(117,·)$, $\chi_{185}(121,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{43} a^{24} + \frac{4}{43} a^{23} + \frac{20}{43} a^{22} + \frac{4}{43} a^{21} + \frac{9}{43} a^{20} + \frac{20}{43} a^{19} - \frac{8}{43} a^{18} - \frac{17}{43} a^{17} - \frac{5}{43} a^{16} - \frac{2}{43} a^{15} - \frac{19}{43} a^{14} + \frac{9}{43} a^{13} + \frac{5}{43} a^{12} + \frac{8}{43} a^{11} + \frac{10}{43} a^{10} - \frac{18}{43} a^{9} + \frac{9}{43} a^{8} + \frac{20}{43} a^{7} - \frac{3}{43} a^{6} + \frac{14}{43} a^{5} - \frac{7}{43} a^{4} - \frac{4}{43} a^{2} - \frac{9}{43} a + \frac{2}{43}$, $\frac{1}{43} a^{25} + \frac{4}{43} a^{23} + \frac{10}{43} a^{22} - \frac{7}{43} a^{21} - \frac{16}{43} a^{20} - \frac{2}{43} a^{19} + \frac{15}{43} a^{18} + \frac{20}{43} a^{17} + \frac{18}{43} a^{16} - \frac{11}{43} a^{15} - \frac{1}{43} a^{14} + \frac{12}{43} a^{13} - \frac{12}{43} a^{12} + \frac{21}{43} a^{11} - \frac{15}{43} a^{10} - \frac{5}{43} a^{9} - \frac{16}{43} a^{8} + \frac{3}{43} a^{7} - \frac{17}{43} a^{6} - \frac{20}{43} a^{5} - \frac{15}{43} a^{4} - \frac{4}{43} a^{3} + \frac{7}{43} a^{2} - \frac{5}{43} a - \frac{8}{43}$, $\frac{1}{43} a^{26} - \frac{6}{43} a^{23} - \frac{1}{43} a^{22} + \frac{11}{43} a^{21} + \frac{5}{43} a^{20} + \frac{21}{43} a^{19} + \frac{9}{43} a^{18} + \frac{9}{43} a^{16} + \frac{7}{43} a^{15} + \frac{2}{43} a^{14} - \frac{5}{43} a^{13} + \frac{1}{43} a^{12} - \frac{4}{43} a^{11} - \frac{2}{43} a^{10} + \frac{13}{43} a^{9} + \frac{10}{43} a^{8} - \frac{11}{43} a^{7} - \frac{8}{43} a^{6} + \frac{15}{43} a^{5} - \frac{19}{43} a^{4} + \frac{7}{43} a^{3} + \frac{11}{43} a^{2} - \frac{15}{43} a - \frac{8}{43}$, $\frac{1}{43} a^{27} - \frac{20}{43} a^{23} + \frac{2}{43} a^{22} - \frac{14}{43} a^{21} - \frac{11}{43} a^{20} - \frac{5}{43} a^{18} - \frac{7}{43} a^{17} + \frac{20}{43} a^{16} - \frac{10}{43} a^{15} + \frac{10}{43} a^{14} + \frac{12}{43} a^{13} - \frac{17}{43} a^{12} + \frac{3}{43} a^{11} - \frac{13}{43} a^{10} - \frac{12}{43} a^{9} - \frac{17}{43} a^{7} - \frac{3}{43} a^{6} - \frac{21}{43} a^{5} + \frac{8}{43} a^{4} + \frac{11}{43} a^{3} + \frac{4}{43} a^{2} - \frac{19}{43} a + \frac{12}{43}$, $\frac{1}{43} a^{28} - \frac{4}{43} a^{23} - \frac{1}{43} a^{22} - \frac{17}{43} a^{21} + \frac{8}{43} a^{20} + \frac{8}{43} a^{19} + \frac{5}{43} a^{18} - \frac{19}{43} a^{17} + \frac{19}{43} a^{16} + \frac{13}{43} a^{15} + \frac{19}{43} a^{14} - \frac{9}{43} a^{13} + \frac{17}{43} a^{12} + \frac{18}{43} a^{11} + \frac{16}{43} a^{10} - \frac{16}{43} a^{9} - \frac{9}{43} a^{8} + \frac{10}{43} a^{7} + \frac{5}{43} a^{6} - \frac{13}{43} a^{5} + \frac{4}{43} a^{3} - \frac{13}{43} a^{2} + \frac{4}{43} a - \frac{3}{43}$, $\frac{1}{43} a^{29} + \frac{15}{43} a^{23} + \frac{20}{43} a^{22} - \frac{19}{43} a^{21} + \frac{1}{43} a^{20} - \frac{1}{43} a^{19} - \frac{8}{43} a^{18} - \frac{6}{43} a^{17} - \frac{7}{43} a^{16} + \frac{11}{43} a^{15} + \frac{1}{43} a^{14} + \frac{10}{43} a^{13} - \frac{5}{43} a^{12} + \frac{5}{43} a^{11} - \frac{19}{43} a^{10} + \frac{5}{43} a^{9} + \frac{3}{43} a^{8} - \frac{1}{43} a^{7} + \frac{18}{43} a^{6} + \frac{13}{43} a^{5} + \frac{19}{43} a^{4} - \frac{13}{43} a^{3} - \frac{12}{43} a^{2} + \frac{4}{43} a + \frac{8}{43}$, $\frac{1}{86} a^{30} - \frac{1}{86} a^{29} - \frac{1}{86} a^{24} - \frac{8}{43} a^{23} + \frac{14}{43} a^{22} - \frac{1}{86} a^{21} - \frac{17}{86} a^{20} - \frac{13}{43} a^{19} - \frac{21}{43} a^{18} - \frac{15}{43} a^{17} + \frac{6}{43} a^{16} - \frac{21}{86} a^{15} - \frac{31}{86} a^{14} + \frac{13}{86} a^{13} + \frac{8}{43} a^{12} + \frac{10}{43} a^{11} - \frac{7}{86} a^{10} - \frac{15}{86} a^{9} - \frac{19}{86} a^{8} - \frac{1}{2} a^{7} - \frac{3}{86} a^{5} + \frac{37}{86} a^{4} - \frac{21}{43} a^{3} + \frac{37}{86} a^{2} + \frac{19}{86} a + \frac{3}{86}$, $\frac{1}{86} a^{31} - \frac{1}{86} a^{29} - \frac{1}{86} a^{25} - \frac{1}{86} a^{24} - \frac{5}{43} a^{23} + \frac{3}{86} a^{22} - \frac{20}{43} a^{21} + \frac{15}{86} a^{20} - \frac{3}{43} a^{19} - \frac{14}{43} a^{18} - \frac{16}{43} a^{17} - \frac{3}{86} a^{16} + \frac{1}{43} a^{15} + \frac{11}{43} a^{14} + \frac{1}{86} a^{13} + \frac{15}{43} a^{12} - \frac{31}{86} a^{11} - \frac{17}{43} a^{10} + \frac{11}{43} a^{9} - \frac{2}{43} a^{8} + \frac{19}{86} a^{7} + \frac{35}{86} a^{6} - \frac{31}{86} a^{4} - \frac{5}{86} a^{3} - \frac{4}{43} a^{2} - \frac{18}{43} a + \frac{35}{86}$, $\frac{1}{86} a^{32} - \frac{1}{86} a^{29} - \frac{1}{86} a^{26} - \frac{1}{86} a^{25} - \frac{1}{86} a^{24} + \frac{27}{86} a^{23} + \frac{8}{43} a^{22} - \frac{16}{43} a^{21} - \frac{19}{86} a^{20} - \frac{13}{43} a^{19} + \frac{9}{43} a^{18} - \frac{31}{86} a^{17} - \frac{18}{43} a^{16} - \frac{19}{86} a^{15} + \frac{19}{43} a^{14} - \frac{39}{86} a^{13} + \frac{35}{86} a^{12} - \frac{10}{43} a^{11} + \frac{29}{86} a^{10} - \frac{27}{86} a^{9} + \frac{2}{43} a^{8} + \frac{10}{43} a^{7} - \frac{15}{43} a^{6} + \frac{10}{43} a^{5} - \frac{19}{43} a^{4} + \frac{18}{43} a^{3} - \frac{39}{86} a^{2} - \frac{18}{43} a + \frac{23}{86}$, $\frac{1}{7396} a^{33} + \frac{15}{3698} a^{32} - \frac{15}{7396} a^{31} - \frac{27}{7396} a^{30} + \frac{33}{7396} a^{29} + \frac{2}{1849} a^{28} + \frac{27}{7396} a^{27} - \frac{35}{7396} a^{26} + \frac{23}{3698} a^{25} - \frac{16}{1849} a^{24} - \frac{535}{3698} a^{23} + \frac{1069}{7396} a^{22} - \frac{121}{7396} a^{21} + \frac{1081}{7396} a^{20} + \frac{761}{1849} a^{19} + \frac{1735}{7396} a^{18} - \frac{378}{1849} a^{17} + \frac{819}{1849} a^{16} - \frac{1733}{3698} a^{15} - \frac{831}{7396} a^{14} + \frac{281}{3698} a^{13} + \frac{819}{3698} a^{12} + \frac{279}{3698} a^{11} - \frac{3297}{7396} a^{10} - \frac{788}{1849} a^{9} - \frac{1061}{3698} a^{8} + \frac{1497}{7396} a^{7} + \frac{467}{7396} a^{6} + \frac{17}{86} a^{5} + \frac{1825}{7396} a^{4} + \frac{1509}{3698} a^{3} + \frac{591}{1849} a^{2} - \frac{841}{7396} a - \frac{1641}{7396}$, $\frac{1}{237548773612} a^{34} - \frac{4173563}{237548773612} a^{33} - \frac{1375179363}{237548773612} a^{32} - \frac{379234159}{118774386806} a^{31} + \frac{604747677}{118774386806} a^{30} - \frac{1315280379}{237548773612} a^{29} + \frac{2145757199}{237548773612} a^{28} + \frac{1240962263}{118774386806} a^{27} - \frac{259756221}{237548773612} a^{26} + \frac{1201108091}{118774386806} a^{25} + \frac{35333793}{59387193403} a^{24} - \frac{23126756463}{237548773612} a^{23} + \frac{359522466}{1381097521} a^{22} - \frac{21027424671}{59387193403} a^{21} + \frac{61411927737}{237548773612} a^{20} + \frac{40916968375}{237548773612} a^{19} - \frac{63080983303}{237548773612} a^{18} - \frac{56005790249}{118774386806} a^{17} - \frac{21302874079}{59387193403} a^{16} - \frac{2117819981}{237548773612} a^{15} + \frac{13011043815}{237548773612} a^{14} - \frac{7319651131}{118774386806} a^{13} - \frac{56985601225}{118774386806} a^{12} - \frac{32160256737}{237548773612} a^{11} - \frac{94338737355}{237548773612} a^{10} - \frac{52225212957}{118774386806} a^{9} + \frac{69814374401}{237548773612} a^{8} + \frac{23668516881}{118774386806} a^{7} - \frac{76791787067}{237548773612} a^{6} - \frac{47819902231}{237548773612} a^{5} + \frac{41866906589}{237548773612} a^{4} + \frac{13221967666}{59387193403} a^{3} - \frac{48000895453}{237548773612} a^{2} - \frac{33378006657}{118774386806} a - \frac{24882760373}{237548773612}$, $\frac{1}{3702369690887563977224033573658909904581024696349226195802764203016233427252} a^{35} - \frac{1049397623160563165564427174943043406682762409429096787174987329}{925592422721890994306008393414727476145256174087306548950691050754058356813} a^{34} + \frac{31247686017260761837799912393524359509095255980868168734760328347221230}{925592422721890994306008393414727476145256174087306548950691050754058356813} a^{33} + \frac{10409746766998987217334126600382277287313226412246436451141650958550127695}{3702369690887563977224033573658909904581024696349226195802764203016233427252} a^{32} + \frac{2885272349537952362371867816434968407627987152791624998996518405363700215}{1851184845443781988612016786829454952290512348174613097901382101508116713626} a^{31} + \frac{15113048532474722081879218430336333414026441661435988884863669301495459573}{3702369690887563977224033573658909904581024696349226195802764203016233427252} a^{30} + \frac{482314166414173648736224829411796964155701502889087683743401747980342879}{925592422721890994306008393414727476145256174087306548950691050754058356813} a^{29} - \frac{16172756147545924467181273660131374704772386953900897438819354566876035873}{3702369690887563977224033573658909904581024696349226195802764203016233427252} a^{28} - \frac{42983510414731541900617378968711861210348165311836807926363140668345674015}{3702369690887563977224033573658909904581024696349226195802764203016233427252} a^{27} + \frac{12123478783514705878620521820786339079289542746898092566537543985642870317}{3702369690887563977224033573658909904581024696349226195802764203016233427252} a^{26} - \frac{14269335162058158949209983787582829361947215555783549970609958135766691921}{1851184845443781988612016786829454952290512348174613097901382101508116713626} a^{25} + \frac{36324078460125377970322493271879098755184599368683262243053802871316441999}{3702369690887563977224033573658909904581024696349226195802764203016233427252} a^{24} - \frac{614748519741320932791685113631881826012826318496393417969538782513730223247}{3702369690887563977224033573658909904581024696349226195802764203016233427252} a^{23} + \frac{603360782565369078643344176324108616780096640538639303605232747018465521353}{1851184845443781988612016786829454952290512348174613097901382101508116713626} a^{22} - \frac{1333597377747172598880496866171750211165409218510218024416325657067216463405}{3702369690887563977224033573658909904581024696349226195802764203016233427252} a^{21} + \frac{126019824125804711851971518576693604249707695836984557143414035704540817768}{925592422721890994306008393414727476145256174087306548950691050754058356813} a^{20} + \frac{104750502185600078444663271389712220300181987926361754813713951682721831901}{1851184845443781988612016786829454952290512348174613097901382101508116713626} a^{19} - \frac{782165222145656153360940060738383169836078440033506102802468940914042956147}{3702369690887563977224033573658909904581024696349226195802764203016233427252} a^{18} + \frac{399851707998882287829449499033357884615782483231012154928090582644223007772}{925592422721890994306008393414727476145256174087306548950691050754058356813} a^{17} + \frac{4362822575263861555013487512870623990137290748463708401717683372913781}{56534222402045595095726512447264577327200364891038589622726934340366068} a^{16} - \frac{58210986206484875285633414256825989698629791405244737395351651329943873199}{925592422721890994306008393414727476145256174087306548950691050754058356813} a^{15} - \frac{112196435094351371421607447293715220960022995033609082026852316910549632151}{3702369690887563977224033573658909904581024696349226195802764203016233427252} a^{14} - \frac{738205600080958328399827168658443197929126370575474818866988898246570705419}{1851184845443781988612016786829454952290512348174613097901382101508116713626} a^{13} + \frac{740107467195577808305077432231500764878824377576725061690673988122404033815}{3702369690887563977224033573658909904581024696349226195802764203016233427252} a^{12} - \frac{395005886983716913004821116757454618263987254826307321735098437357339938158}{925592422721890994306008393414727476145256174087306548950691050754058356813} a^{11} - \frac{820480813414101187415811044709556921223801894524798326016381307275297649007}{3702369690887563977224033573658909904581024696349226195802764203016233427252} a^{10} - \frac{324671729656715987448666737443246368597510250116578101458214528010466400989}{3702369690887563977224033573658909904581024696349226195802764203016233427252} a^{9} - \frac{569438761482379679249595001135437954331915089267625680706384342719970043321}{3702369690887563977224033573658909904581024696349226195802764203016233427252} a^{8} - \frac{1774182194404154467181940084632058074026000484453349358587037918343068828601}{3702369690887563977224033573658909904581024696349226195802764203016233427252} a^{7} - \frac{600533128122520603974292910221228607761216787787541968567657804929725768237}{1851184845443781988612016786829454952290512348174613097901382101508116713626} a^{6} - \frac{895554513530964244373530954397609404667654294742942098718912576626168369299}{1851184845443781988612016786829454952290512348174613097901382101508116713626} a^{5} + \frac{428402922999855354403017734402314803332319086541415802947331891935187408707}{3702369690887563977224033573658909904581024696349226195802764203016233427252} a^{4} - \frac{395024464582748365054166695737065339422190861223858520067046668260918697691}{3702369690887563977224033573658909904581024696349226195802764203016233427252} a^{3} - \frac{741442182756251226653297927718462855710620694215186017427814174516762013841}{3702369690887563977224033573658909904581024696349226195802764203016233427252} a^{2} + \frac{1373596173526204736507101019495322887616435456361902303589052660904965822999}{3702369690887563977224033573658909904581024696349226195802764203016233427252} a - \frac{1703977995360294830433759814695277354659390405165798809195147290194005030625}{3702369690887563977224033573658909904581024696349226195802764203016233427252}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $35$
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:  \( 23167749867944500000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{36}$ (as 36T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 36
The 36 conjugacy class representatives for $C_{36}$
Character table for $C_{36}$ is not computed

Intermediate fields

\(\Q(\sqrt{185}) \), 3.3.1369.1, 4.4.6331625.2, 6.6.8667994625.1, 9.9.3512479453921.1, 12.12.347495355038008619140625.1, 18.18.891578009425849912898724447265625.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{4}$ $36$ R $36$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{4}$ $18^{2}$ $36$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/29.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{9}$ R $18^{2}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{36}$ ${\href{/LocalNumberField/47.12.0.1}{12} }^{3}$ $36$ $36$

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
5Data not computed
37Data not computed