Properties

Label 35.35.180...921.1
Degree $35$
Signature $[35, 0]$
Discriminant $1.803\times 10^{83}$
Root discriminant $239.19$
Ramified prime $281$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{35}$ (as 35T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^35 - x^34 - 136*x^33 + 115*x^32 + 7922*x^31 - 5864*x^30 - 262379*x^29 + 176786*x^28 + 5536174*x^27 - 3463751*x^26 - 78912031*x^25 + 45605901*x^24 + 785104244*x^23 - 406720184*x^22 - 5548252572*x^21 + 2436396552*x^20 + 28031325744*x^19 - 9529852980*x^18 - 100888672963*x^17 + 22693210859*x^16 + 254967397769*x^15 - 26498170926*x^14 - 440501997963*x^13 - 3333796454*x^12 + 500538383949*x^11 + 44640325883*x^10 - 358942724140*x^9 - 43476115800*x^8 + 159203701176*x^7 + 16418359222*x^6 - 42718375268*x^5 - 1999390430*x^4 + 6413147106*x^3 - 204719982*x^2 - 415325205*x + 48529823)
 
gp: K = bnfinit(x^35 - x^34 - 136*x^33 + 115*x^32 + 7922*x^31 - 5864*x^30 - 262379*x^29 + 176786*x^28 + 5536174*x^27 - 3463751*x^26 - 78912031*x^25 + 45605901*x^24 + 785104244*x^23 - 406720184*x^22 - 5548252572*x^21 + 2436396552*x^20 + 28031325744*x^19 - 9529852980*x^18 - 100888672963*x^17 + 22693210859*x^16 + 254967397769*x^15 - 26498170926*x^14 - 440501997963*x^13 - 3333796454*x^12 + 500538383949*x^11 + 44640325883*x^10 - 358942724140*x^9 - 43476115800*x^8 + 159203701176*x^7 + 16418359222*x^6 - 42718375268*x^5 - 1999390430*x^4 + 6413147106*x^3 - 204719982*x^2 - 415325205*x + 48529823, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![48529823, -415325205, -204719982, 6413147106, -1999390430, -42718375268, 16418359222, 159203701176, -43476115800, -358942724140, 44640325883, 500538383949, -3333796454, -440501997963, -26498170926, 254967397769, 22693210859, -100888672963, -9529852980, 28031325744, 2436396552, -5548252572, -406720184, 785104244, 45605901, -78912031, -3463751, 5536174, 176786, -262379, -5864, 7922, 115, -136, -1, 1]);
 

\(x^{35} - x^{34} - 136 x^{33} + 115 x^{32} + 7922 x^{31} - 5864 x^{30} - 262379 x^{29} + 176786 x^{28} + 5536174 x^{27} - 3463751 x^{26} - 78912031 x^{25} + 45605901 x^{24} + 785104244 x^{23} - 406720184 x^{22} - 5548252572 x^{21} + 2436396552 x^{20} + 28031325744 x^{19} - 9529852980 x^{18} - 100888672963 x^{17} + 22693210859 x^{16} + 254967397769 x^{15} - 26498170926 x^{14} - 440501997963 x^{13} - 3333796454 x^{12} + 500538383949 x^{11} + 44640325883 x^{10} - 358942724140 x^{9} - 43476115800 x^{8} + 159203701176 x^{7} + 16418359222 x^{6} - 42718375268 x^{5} - 1999390430 x^{4} + 6413147106 x^{3} - 204719982 x^{2} - 415325205 x + 48529823\)  Toggle raw display

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

Invariants

Degree:  $35$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[35, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(180\!\cdots\!921\)\(\medspace = 281^{34}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $239.19$
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:  $281$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $35$
This field is Galois and abelian over $\Q$.
Conductor:  \(281\)
Dirichlet character group:    $\lbrace$$\chi_{281}(256,·)$, $\chi_{281}(1,·)$, $\chi_{281}(4,·)$, $\chi_{281}(140,·)$, $\chi_{281}(16,·)$, $\chi_{281}(273,·)$, $\chi_{281}(279,·)$, $\chi_{281}(153,·)$, $\chi_{281}(155,·)$, $\chi_{281}(162,·)$, $\chi_{281}(35,·)$, $\chi_{281}(165,·)$, $\chi_{281}(50,·)$, $\chi_{281}(181,·)$, $\chi_{281}(58,·)$, $\chi_{281}(59,·)$, $\chi_{281}(63,·)$, $\chi_{281}(64,·)$, $\chi_{281}(200,·)$, $\chi_{281}(79,·)$, $\chi_{281}(163,·)$, $\chi_{281}(85,·)$, $\chi_{281}(86,·)$, $\chi_{281}(90,·)$, $\chi_{281}(98,·)$, $\chi_{281}(101,·)$, $\chi_{281}(232,·)$, $\chi_{281}(236,·)$, $\chi_{281}(109,·)$, $\chi_{281}(238,·)$, $\chi_{281}(111,·)$, $\chi_{281}(211,·)$, $\chi_{281}(249,·)$, $\chi_{281}(123,·)$, $\chi_{281}(252,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $\frac{1}{53} a^{27} + \frac{10}{53} a^{26} - \frac{21}{53} a^{25} + \frac{16}{53} a^{24} + \frac{17}{53} a^{23} + \frac{16}{53} a^{22} - \frac{19}{53} a^{21} - \frac{10}{53} a^{20} + \frac{7}{53} a^{19} + \frac{23}{53} a^{18} - \frac{21}{53} a^{17} - \frac{12}{53} a^{16} + \frac{20}{53} a^{15} + \frac{18}{53} a^{14} + \frac{18}{53} a^{13} - \frac{13}{53} a^{12} - \frac{19}{53} a^{11} - \frac{7}{53} a^{10} - \frac{24}{53} a^{9} + \frac{26}{53} a^{8} - \frac{24}{53} a^{7} + \frac{11}{53} a^{6} + \frac{25}{53} a^{5} - \frac{17}{53} a^{4} + \frac{15}{53} a^{3} - \frac{17}{53} a^{2} + \frac{21}{53} a - \frac{16}{53}$, $\frac{1}{4717} a^{28} - \frac{18}{4717} a^{27} - \frac{1785}{4717} a^{26} + \frac{710}{4717} a^{25} + \frac{470}{4717} a^{24} + \frac{1713}{4717} a^{23} - \frac{1686}{4717} a^{22} - \frac{750}{4717} a^{21} - \frac{31}{4717} a^{20} + \frac{887}{4717} a^{19} + \frac{183}{4717} a^{18} + \frac{788}{4717} a^{17} - \frac{1287}{4717} a^{16} - \frac{2079}{4717} a^{15} + \frac{839}{4717} a^{14} - \frac{93}{4717} a^{13} + \frac{1458}{4717} a^{12} - \frac{2019}{4717} a^{11} + \frac{1391}{4717} a^{10} + \frac{963}{4717} a^{9} - \frac{964}{4717} a^{8} - \frac{2126}{4717} a^{7} - \frac{1608}{4717} a^{6} - \frac{1830}{4717} a^{5} - \frac{145}{4717} a^{4} + \frac{994}{4717} a^{3} + \frac{338}{4717} a^{2} + \frac{1569}{4717} a + \frac{395}{4717}$, $\frac{1}{4717} a^{29} + \frac{27}{4717} a^{27} - \frac{626}{4717} a^{26} + \frac{1413}{4717} a^{25} + \frac{1896}{4717} a^{24} - \frac{578}{4717} a^{23} - \frac{1639}{4717} a^{22} - \frac{2228}{4717} a^{21} - \frac{2163}{4717} a^{20} - \frac{1918}{4717} a^{19} + \frac{1323}{4717} a^{18} + \frac{20}{89} a^{17} + \frac{1010}{4717} a^{16} + \frac{1420}{4717} a^{15} + \frac{1570}{4717} a^{14} + \frac{496}{4717} a^{13} + \frac{1174}{4717} a^{12} - \frac{63}{4717} a^{11} + \frac{1615}{4717} a^{10} - \frac{1875}{4717} a^{9} - \frac{1678}{4717} a^{8} - \frac{1517}{4717} a^{7} + \frac{2156}{4717} a^{6} + \frac{1447}{4717} a^{5} - \frac{192}{4717} a^{4} - \frac{1617}{4717} a^{3} - \frac{357}{4717} a^{2} - \frac{1979}{4717} a + \frac{1236}{4717}$, $\frac{1}{4717} a^{30} + \frac{38}{4717} a^{27} - \frac{499}{4717} a^{26} - \frac{2144}{4717} a^{25} - \frac{986}{4717} a^{24} + \frac{2306}{4717} a^{23} - \frac{1028}{4717} a^{22} + \frac{554}{4717} a^{21} + \frac{1856}{4717} a^{20} + \frac{2205}{4717} a^{19} + \frac{213}{4717} a^{18} - \frac{419}{4717} a^{17} + \frac{1014}{4717} a^{16} - \frac{58}{4717} a^{15} - \frac{85}{4717} a^{14} + \frac{2172}{4717} a^{13} + \frac{710}{4717} a^{12} + \frac{859}{4717} a^{11} + \frac{1775}{4717} a^{10} + \frac{12}{53} a^{9} + \frac{837}{4717} a^{8} - \frac{1318}{4717} a^{7} - \frac{349}{4717} a^{6} + \frac{1781}{4717} a^{5} - \frac{728}{4717} a^{4} - \frac{940}{4717} a^{3} + \frac{20}{4717} a^{2} + \frac{347}{4717} a + \frac{638}{4717}$, $\frac{1}{4717} a^{31} + \frac{7}{4717} a^{27} - \frac{2132}{4717} a^{26} - \frac{643}{4717} a^{25} + \frac{466}{4717} a^{24} + \frac{1607}{4717} a^{23} + \frac{453}{4717} a^{22} + \frac{719}{4717} a^{21} + \frac{446}{4717} a^{20} - \frac{1720}{4717} a^{19} - \frac{2033}{4717} a^{18} - \frac{1607}{4717} a^{17} - \frac{903}{4717} a^{16} - \frac{115}{4717} a^{15} + \frac{105}{4717} a^{14} + \frac{1040}{4717} a^{13} - \frac{344}{4717} a^{12} + \frac{1690}{4717} a^{11} + \frac{1343}{4717} a^{10} + \frac{1534}{4717} a^{9} - \frac{2333}{4717} a^{8} - \frac{195}{4717} a^{7} - \frac{394}{4717} a^{6} - \frac{1676}{4717} a^{5} - \frac{1838}{4717} a^{4} + \frac{2031}{4717} a^{3} - \frac{37}{4717} a^{2} - \frac{1401}{4717} a + \frac{1989}{4717}$, $\frac{1}{4717} a^{32} + \frac{41}{4717} a^{27} - \frac{697}{4717} a^{26} - \frac{321}{4717} a^{25} - \frac{1950}{4717} a^{24} - \frac{324}{4717} a^{23} - \frac{1897}{4717} a^{22} - \frac{2}{53} a^{21} + \frac{1612}{4717} a^{20} + \frac{1370}{4717} a^{19} + \frac{1740}{4717} a^{18} - \frac{2236}{4717} a^{17} - \frac{1519}{4717} a^{16} - \frac{1006}{4717} a^{15} - \frac{1006}{4717} a^{14} - \frac{11}{89} a^{13} - \frac{2108}{4717} a^{12} + \frac{168}{4717} a^{11} + \frac{1053}{4717} a^{10} - \frac{1598}{4717} a^{9} - \frac{1546}{4717} a^{8} - \frac{1621}{4717} a^{7} - \frac{922}{4717} a^{6} + \frac{826}{4717} a^{5} + \frac{1266}{4717} a^{4} + \frac{125}{4717} a^{3} - \frac{830}{4717} a^{2} + \frac{974}{4717} a + \frac{2219}{4717}$, $\frac{1}{4717} a^{33} + \frac{41}{4717} a^{27} + \frac{2109}{4717} a^{26} + \frac{1959}{4717} a^{25} - \frac{726}{4717} a^{24} - \frac{1375}{4717} a^{23} - \frac{1807}{4717} a^{22} - \frac{657}{4717} a^{21} - \frac{2076}{4717} a^{20} - \frac{1608}{4717} a^{19} - \frac{305}{4717} a^{18} - \frac{808}{4717} a^{17} - \frac{126}{4717} a^{16} - \frac{673}{4717} a^{15} - \frac{1963}{4717} a^{14} + \frac{1705}{4717} a^{13} + \frac{1711}{4717} a^{12} - \frac{1074}{4717} a^{11} - \frac{2025}{4717} a^{10} + \frac{16}{53} a^{9} + \frac{167}{4717} a^{8} + \frac{1338}{4717} a^{7} + \frac{716}{4717} a^{6} + \frac{824}{4717} a^{5} + \frac{1353}{4717} a^{4} + \frac{869}{4717} a^{3} + \frac{1267}{4717} a^{2} - \frac{789}{4717} a - \frac{2044}{4717}$, $\frac{1}{1501146006221970135524255525492615234752326684039142579878789059923784594110989541622280918133636567212892349892960061129231539013} a^{34} + \frac{131462965814250560839983567239829925890634487361852805679362276408532073990038115417428146491830658785122335882513472129117450}{1501146006221970135524255525492615234752326684039142579878789059923784594110989541622280918133636567212892349892960061129231539013} a^{33} + \frac{24458546286814548946835036212641738038898528982638985059162581218532413014174319562691646721016177614243178361661881541367126}{1501146006221970135524255525492615234752326684039142579878789059923784594110989541622280918133636567212892349892960061129231539013} a^{32} + \frac{35428765879251032949537274248868529258544786218417145529597163775883643781547601791650932973913944815393630912637979604619167}{1501146006221970135524255525492615234752326684039142579878789059923784594110989541622280918133636567212892349892960061129231539013} a^{31} - \frac{57714284946386922495672663024829813905258257070027521562180242286886487210189065787526535906971696807321097349954487995105597}{1501146006221970135524255525492615234752326684039142579878789059923784594110989541622280918133636567212892349892960061129231539013} a^{30} + \frac{132704856988854805368639805591086583084687124388148326611957043339127190187950326091541543932191430414335755369981596564708218}{1501146006221970135524255525492615234752326684039142579878789059923784594110989541622280918133636567212892349892960061129231539013} a^{29} + \frac{103716487033563614835366829045968269489110530036062008579030185606639464974739779902282346771218064921701717226074793612982674}{1501146006221970135524255525492615234752326684039142579878789059923784594110989541622280918133636567212892349892960061129231539013} a^{28} - \frac{10380035878714491542660477002372737717511322088797607733705660429990490771684668141281315458759219200376755265256193739841410666}{1501146006221970135524255525492615234752326684039142579878789059923784594110989541622280918133636567212892349892960061129231539013} a^{27} + \frac{140794988604615897693248135168447885867842777086257137052059447195110153091369794097330636349383416665782035285931791655461025832}{1501146006221970135524255525492615234752326684039142579878789059923784594110989541622280918133636567212892349892960061129231539013} a^{26} - \frac{566366863890913930337824769965203995060406053795417954533277523708435172342525484187248574888115548212447464995113126499033477472}{1501146006221970135524255525492615234752326684039142579878789059923784594110989541622280918133636567212892349892960061129231539013} a^{25} - \frac{328921101358541900323881276778627816502705269943444022321074165046848060926651869443710734417089552379584015769838032007278315841}{1501146006221970135524255525492615234752326684039142579878789059923784594110989541622280918133636567212892349892960061129231539013} a^{24} + \frac{653932141404621495130058069727340037449327139538491496979408800740759761433467782635922233178030874653330359248294513709645068246}{1501146006221970135524255525492615234752326684039142579878789059923784594110989541622280918133636567212892349892960061129231539013} a^{23} - \frac{555661196808635270269578609646155743830736599312998607877630255523740301190393243858513559541832862548797965679969174068358209961}{1501146006221970135524255525492615234752326684039142579878789059923784594110989541622280918133636567212892349892960061129231539013} a^{22} - \frac{222405021602652106708827461415854053306427267912160610926391801271808941608459607103175658092173230268254656489498736556428889342}{1501146006221970135524255525492615234752326684039142579878789059923784594110989541622280918133636567212892349892960061129231539013} a^{21} + \frac{342894846639900193048749349470132114277754874640364012723176580977671191839193307794912589681402944972842903854914581057958642030}{1501146006221970135524255525492615234752326684039142579878789059923784594110989541622280918133636567212892349892960061129231539013} a^{20} - \frac{640059320343626847394030661293793703782981130071184061800766226570277076526405221171163388808493853984148124014024722031637634650}{1501146006221970135524255525492615234752326684039142579878789059923784594110989541622280918133636567212892349892960061129231539013} a^{19} + \frac{114368205793825538515834287357196335861101548394555903578850315959484756733576029573908887123043423299871908214506735534136522947}{1501146006221970135524255525492615234752326684039142579878789059923784594110989541622280918133636567212892349892960061129231539013} a^{18} - \frac{366170434132042580020552996192118434233777825618732696365797268567669560807428808599918906098372663925983806857383831567509273666}{1501146006221970135524255525492615234752326684039142579878789059923784594110989541622280918133636567212892349892960061129231539013} a^{17} - \frac{299387723451409662547388058897880398590648505492278650849129507080326576607158301289751194646486937203493044057660813443954738782}{1501146006221970135524255525492615234752326684039142579878789059923784594110989541622280918133636567212892349892960061129231539013} a^{16} - \frac{154158443193399782789621113187955796993511246719066379097988742490293335597068302264144499359058723223409455884536946765236743081}{1501146006221970135524255525492615234752326684039142579878789059923784594110989541622280918133636567212892349892960061129231539013} a^{15} - \frac{623237181786719274631686293517179863063531911608038943052261288519036028026706761443128987797606568689128266763478987100092847705}{1501146006221970135524255525492615234752326684039142579878789059923784594110989541622280918133636567212892349892960061129231539013} a^{14} + \frac{116250708647151132400805001545645235204741745581410384531875237960201027736188019029544083393782937339541654978428760366951785486}{1501146006221970135524255525492615234752326684039142579878789059923784594110989541622280918133636567212892349892960061129231539013} a^{13} + \frac{147982956044277441124586713076086799600957979315205920729389491692159746679800325081965301085358373819193966198628923424833267576}{1501146006221970135524255525492615234752326684039142579878789059923784594110989541622280918133636567212892349892960061129231539013} a^{12} + \frac{205433328922515757036474617685709178528096193911402120864243404569532065971619375982793074276333800296897621916628848536251251298}{1501146006221970135524255525492615234752326684039142579878789059923784594110989541622280918133636567212892349892960061129231539013} a^{11} + \frac{414144578803612942280048248194425596114120030683297302006631128805487509567665332423020202118361799386618580489120797490047178165}{1501146006221970135524255525492615234752326684039142579878789059923784594110989541622280918133636567212892349892960061129231539013} a^{10} - \frac{694549888739221384428679326701349075866085609133640635319757326323814990839969913082721032002708751504433322993322422762064869676}{1501146006221970135524255525492615234752326684039142579878789059923784594110989541622280918133636567212892349892960061129231539013} a^{9} + \frac{285768851772257575073302920049664590968314933139952276599173873943925718310155675456380776429691067514214543823871409616228271387}{1501146006221970135524255525492615234752326684039142579878789059923784594110989541622280918133636567212892349892960061129231539013} a^{8} + \frac{8056888519938528692537849002268281605343579569264337655043264932209015192540598599640826149904072921729683509905630106053343451}{1501146006221970135524255525492615234752326684039142579878789059923784594110989541622280918133636567212892349892960061129231539013} a^{7} + \frac{395575766346991022478092131229552391077874325996061211265600877440676870996310658052677379236265961842856388304124251811557668733}{1501146006221970135524255525492615234752326684039142579878789059923784594110989541622280918133636567212892349892960061129231539013} a^{6} + \frac{322385739260946103465844676205525830883497967151174459809089566888681769327391106942971426789170115964924311438170443468109222816}{1501146006221970135524255525492615234752326684039142579878789059923784594110989541622280918133636567212892349892960061129231539013} a^{5} + \frac{7931701817924172803648655687178629692941809525446651104346926197683371488850342439328773641963794979197622385143167072668392441}{28323509551357927085363311801747457259477861963002690186392246413656313096433764936269451285540312588922497167791699266589274321} a^{4} + \frac{514902875546004711101977268778379766056736641897076451226505088456087335456984269509679106769024048201614163651424724588769579693}{1501146006221970135524255525492615234752326684039142579878789059923784594110989541622280918133636567212892349892960061129231539013} a^{3} - \frac{153624118473764370301295314129631143274460267494545082995876464798415012108699885199163837895904038798298229973012753149090811590}{1501146006221970135524255525492615234752326684039142579878789059923784594110989541622280918133636567212892349892960061129231539013} a^{2} + \frac{91215308188638000314612462195882318352005677735077989723074374260121403011210017578093056276339246001586080002007377125830116280}{1501146006221970135524255525492615234752326684039142579878789059923784594110989541622280918133636567212892349892960061129231539013} a + \frac{446576710224988152824389141998418082123111404093077613580041380916138525371535107707750485427981663487963706820044242531403298292}{1501146006221970135524255525492615234752326684039142579878789059923784594110989541622280918133636567212892349892960061129231539013}$  Toggle raw display

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

Class group and class number

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

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $34$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)  Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 6612930761186456000000000000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{35}\cdot(2\pi)^{0}\cdot 6612930761186456000000000000000 \cdot 1}{2\sqrt{180307950435339664981987743016246180767325053322928227488471312870141585227228119921}}\approx 0.267550883703587$ (assuming GRH)

Galois group

$C_{35}$ (as 35T1):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A cyclic group of order 35
The 35 conjugacy class representatives for $C_{35}$
Character table for $C_{35}$ is not computed

Intermediate fields

5.5.6234839521.1, 7.7.492309163417681.1

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 $35$ $35$ $35$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{7}$ $35$ $35$ $35$ $35$ $35$ $35$ $35$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{5}$ $35$ $35$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{5}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{35}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{5}$

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
281Data not computed