Properties

Label 35.35.103...625.1
Degree $35$
Signature $[35, 0]$
Discriminant $1.033\times 10^{83}$
Root discriminant $235.42$
Ramified primes $5, 29$
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 - 5*x^34 - 120*x^33 + 560*x^32 + 6265*x^31 - 27063*x^30 - 188895*x^29 + 746800*x^28 + 3677845*x^27 - 13130310*x^26 - 48923530*x^25 + 155517810*x^24 + 458129250*x^23 - 1279983515*x^22 - 3063191465*x^21 + 7446901792*x^20 + 14674072475*x^19 - 30859715040*x^18 - 50096318770*x^17 + 91037787910*x^16 + 120155286235*x^15 - 189447334860*x^14 - 197298653000*x^13 + 272612081065*x^12 + 212356717105*x^11 - 261877354307*x^10 - 139209319745*x^9 + 158110983885*x^8 + 48766657725*x^7 - 53789658890*x^6 - 7150843552*x^5 + 8233276160*x^4 + 456884545*x^3 - 363355535*x^2 - 50594420*x - 1782107)
 
gp: K = bnfinit(x^35 - 5*x^34 - 120*x^33 + 560*x^32 + 6265*x^31 - 27063*x^30 - 188895*x^29 + 746800*x^28 + 3677845*x^27 - 13130310*x^26 - 48923530*x^25 + 155517810*x^24 + 458129250*x^23 - 1279983515*x^22 - 3063191465*x^21 + 7446901792*x^20 + 14674072475*x^19 - 30859715040*x^18 - 50096318770*x^17 + 91037787910*x^16 + 120155286235*x^15 - 189447334860*x^14 - 197298653000*x^13 + 272612081065*x^12 + 212356717105*x^11 - 261877354307*x^10 - 139209319745*x^9 + 158110983885*x^8 + 48766657725*x^7 - 53789658890*x^6 - 7150843552*x^5 + 8233276160*x^4 + 456884545*x^3 - 363355535*x^2 - 50594420*x - 1782107, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1782107, -50594420, -363355535, 456884545, 8233276160, -7150843552, -53789658890, 48766657725, 158110983885, -139209319745, -261877354307, 212356717105, 272612081065, -197298653000, -189447334860, 120155286235, 91037787910, -50096318770, -30859715040, 14674072475, 7446901792, -3063191465, -1279983515, 458129250, 155517810, -48923530, -13130310, 3677845, 746800, -188895, -27063, 6265, 560, -120, -5, 1]);
 

\(x^{35} - 5 x^{34} - 120 x^{33} + 560 x^{32} + 6265 x^{31} - 27063 x^{30} - 188895 x^{29} + 746800 x^{28} + 3677845 x^{27} - 13130310 x^{26} - 48923530 x^{25} + 155517810 x^{24} + 458129250 x^{23} - 1279983515 x^{22} - 3063191465 x^{21} + 7446901792 x^{20} + 14674072475 x^{19} - 30859715040 x^{18} - 50096318770 x^{17} + 91037787910 x^{16} + 120155286235 x^{15} - 189447334860 x^{14} - 197298653000 x^{13} + 272612081065 x^{12} + 212356717105 x^{11} - 261877354307 x^{10} - 139209319745 x^{9} + 158110983885 x^{8} + 48766657725 x^{7} - 53789658890 x^{6} - 7150843552 x^{5} + 8233276160 x^{4} + 456884545 x^{3} - 363355535 x^{2} - 50594420 x - 1782107\)  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:  \(103\!\cdots\!625\)\(\medspace = 5^{56}\cdot 29^{30}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $235.42$
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:  $5, 29$
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:  \(725=5^{2}\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{725}(256,·)$, $\chi_{725}(1,·)$, $\chi_{725}(516,·)$, $\chi_{725}(136,·)$, $\chi_{725}(141,·)$, $\chi_{725}(16,·)$, $\chi_{725}(401,·)$, $\chi_{725}(146,·)$, $\chi_{725}(661,·)$, $\chi_{725}(281,·)$, $\chi_{725}(286,·)$, $\chi_{725}(161,·)$, $\chi_{725}(546,·)$, $\chi_{725}(291,·)$, $\chi_{725}(36,·)$, $\chi_{725}(426,·)$, $\chi_{725}(431,·)$, $\chi_{725}(306,·)$, $\chi_{725}(691,·)$, $\chi_{725}(436,·)$, $\chi_{725}(181,·)$, $\chi_{725}(571,·)$, $\chi_{725}(576,·)$, $\chi_{725}(451,·)$, $\chi_{725}(581,·)$, $\chi_{725}(326,·)$, $\chi_{725}(716,·)$, $\chi_{725}(81,·)$, $\chi_{725}(596,·)$, $\chi_{725}(471,·)$, $\chi_{725}(226,·)$, $\chi_{725}(721,·)$, $\chi_{725}(616,·)$, $\chi_{725}(111,·)$, $\chi_{725}(371,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $\frac{1}{7} a^{28} + \frac{3}{7} a^{27} - \frac{3}{7} a^{26} + \frac{3}{7} a^{25} + \frac{3}{7} a^{23} + \frac{2}{7} a^{22} - \frac{2}{7} a^{21} + \frac{2}{7} a^{20} + \frac{1}{7} a^{19} - \frac{1}{7} a^{17} + \frac{1}{7} a^{16} + \frac{1}{7} a^{15} + \frac{2}{7} a^{14} - \frac{2}{7} a^{13} + \frac{3}{7} a^{12} - \frac{1}{7} a^{11} + \frac{3}{7} a^{10} + \frac{1}{7} a^{9} + \frac{1}{7} a^{8} + \frac{1}{7} a^{7} + \frac{1}{7} a^{6} + \frac{3}{7} a^{5} + \frac{1}{7} a^{3} + \frac{2}{7} a^{2} - \frac{3}{7} a - \frac{1}{7}$, $\frac{1}{7} a^{29} + \frac{2}{7} a^{27} - \frac{2}{7} a^{26} - \frac{2}{7} a^{25} + \frac{3}{7} a^{24} - \frac{1}{7} a^{22} + \frac{1}{7} a^{21} + \frac{2}{7} a^{20} - \frac{3}{7} a^{19} - \frac{1}{7} a^{18} - \frac{3}{7} a^{17} - \frac{2}{7} a^{16} - \frac{1}{7} a^{15} - \frac{1}{7} a^{14} + \frac{2}{7} a^{13} - \frac{3}{7} a^{12} - \frac{1}{7} a^{11} - \frac{1}{7} a^{10} - \frac{2}{7} a^{9} - \frac{2}{7} a^{8} - \frac{2}{7} a^{7} - \frac{2}{7} a^{5} + \frac{1}{7} a^{4} - \frac{1}{7} a^{3} - \frac{2}{7} a^{2} + \frac{1}{7} a + \frac{3}{7}$, $\frac{1}{119} a^{30} - \frac{1}{17} a^{29} + \frac{1}{119} a^{28} - \frac{40}{119} a^{27} + \frac{1}{119} a^{26} + \frac{5}{17} a^{24} + \frac{31}{119} a^{23} - \frac{36}{119} a^{22} - \frac{52}{119} a^{21} - \frac{19}{119} a^{20} - \frac{44}{119} a^{19} + \frac{18}{119} a^{18} + \frac{13}{119} a^{17} - \frac{44}{119} a^{16} - \frac{23}{119} a^{15} - \frac{5}{17} a^{14} - \frac{29}{119} a^{13} + \frac{45}{119} a^{12} - \frac{5}{17} a^{11} + \frac{44}{119} a^{10} - \frac{38}{119} a^{9} + \frac{18}{119} a^{8} - \frac{50}{119} a^{7} + \frac{4}{119} a^{6} + \frac{40}{119} a^{5} - \frac{15}{119} a^{4} - \frac{59}{119} a^{3} - \frac{8}{119} a^{2} + \frac{6}{119} a - \frac{41}{119}$, $\frac{1}{18683} a^{31} - \frac{25}{18683} a^{30} + \frac{977}{18683} a^{29} + \frac{10}{18683} a^{28} + \frac{137}{2669} a^{27} + \frac{1648}{18683} a^{26} + \frac{324}{18683} a^{25} - \frac{52}{119} a^{24} - \frac{8006}{18683} a^{23} + \frac{4880}{18683} a^{22} + \frac{403}{2669} a^{21} - \frac{4649}{18683} a^{20} - \frac{6432}{18683} a^{19} + \frac{8359}{18683} a^{18} - \frac{2301}{18683} a^{17} + \frac{7229}{18683} a^{16} + \frac{8522}{18683} a^{15} + \frac{2267}{18683} a^{14} + \frac{8557}{18683} a^{13} - \frac{6761}{18683} a^{12} - \frac{9169}{18683} a^{11} - \frac{8854}{18683} a^{10} - \frac{454}{18683} a^{9} + \frac{260}{1099} a^{8} - \frac{53}{2669} a^{7} + \frac{7533}{18683} a^{6} - \frac{5563}{18683} a^{5} + \frac{3679}{18683} a^{4} + \frac{366}{1099} a^{3} - \frac{678}{2669} a^{2} - \frac{354}{2669} a - \frac{4}{17}$, $\frac{1}{18683} a^{32} + \frac{38}{18683} a^{30} - \frac{57}{18683} a^{29} + \frac{895}{18683} a^{28} + \frac{498}{2669} a^{27} + \frac{1175}{18683} a^{26} - \frac{2733}{18683} a^{25} - \frac{4238}{18683} a^{24} + \frac{509}{18683} a^{23} - \frac{5332}{18683} a^{22} - \frac{535}{18683} a^{21} - \frac{1924}{18683} a^{20} - \frac{358}{2669} a^{19} + \frac{3516}{18683} a^{18} + \frac{7009}{18683} a^{17} - \frac{5119}{18683} a^{16} + \frac{6350}{18683} a^{15} - \frac{9186}{18683} a^{14} - \frac{751}{2669} a^{13} - \frac{156}{18683} a^{12} + \frac{5114}{18683} a^{11} - \frac{201}{1099} a^{10} + \frac{2333}{18683} a^{9} + \frac{1199}{2669} a^{8} - \frac{7394}{18683} a^{7} - \frac{5324}{18683} a^{6} - \frac{1161}{18683} a^{5} + \frac{1485}{18683} a^{4} + \frac{9190}{18683} a^{3} - \frac{9187}{18683} a^{2} - \frac{215}{2669} a - \frac{57}{119}$, $\frac{1}{6520367} a^{33} + \frac{45}{6520367} a^{32} - \frac{69}{6520367} a^{31} + \frac{3848}{931481} a^{30} - \frac{336528}{6520367} a^{29} - \frac{337720}{6520367} a^{28} - \frac{2407584}{6520367} a^{27} + \frac{427545}{6520367} a^{26} - \frac{1830}{383551} a^{25} - \frac{1600061}{6520367} a^{24} - \frac{269216}{6520367} a^{23} - \frac{1918155}{6520367} a^{22} - \frac{1049732}{6520367} a^{21} + \frac{16066}{41531} a^{20} + \frac{892028}{6520367} a^{19} + \frac{2421492}{6520367} a^{18} + \frac{1170677}{6520367} a^{17} + \frac{3241649}{6520367} a^{16} - \frac{2850089}{6520367} a^{15} - \frac{504981}{6520367} a^{14} - \frac{257135}{931481} a^{13} + \frac{1493333}{6520367} a^{12} + \frac{760817}{6520367} a^{11} + \frac{1998880}{6520367} a^{10} - \frac{1778093}{6520367} a^{9} - \frac{236393}{931481} a^{8} - \frac{2061310}{6520367} a^{7} + \frac{938650}{6520367} a^{6} - \frac{643695}{6520367} a^{5} - \frac{2186095}{6520367} a^{4} - \frac{134188}{383551} a^{3} + \frac{1267890}{6520367} a^{2} + \frac{2619460}{6520367} a - \frac{1308}{41531}$, $\frac{1}{29944507291416398156968115809474246794525188634484455248565832042360378732204109664685120173323001908220438692676033073874143023501911526762952817} a^{34} - \frac{1262032862740686860401113388195276038712809163220147028002537404432455204541146464015886863299114879220763721912817233478551774341971949184}{29944507291416398156968115809474246794525188634484455248565832042360378732204109664685120173323001908220438692676033073874143023501911526762952817} a^{33} + \frac{346679266093458275468740662050120987292424016584486456542265534554927834551396365516374223648896397023645072437905191214706146737047111792378}{29944507291416398156968115809474246794525188634484455248565832042360378732204109664685120173323001908220438692676033073874143023501911526762952817} a^{32} - \frac{4374364158650367877703983316809339179552747667612107467137844065714876630070430790892644640322598057301904018649713678797999071584729198692}{190729345805199988260943412799199024168950246079518823239272815556435533326140825889714141231356700052359482118955624674357598875808353673649381} a^{31} + \frac{51983769138173781986856712290615285764646562071852514482674897419476452539992137908090500866777168570115848845449238221778793382477115218440512}{29944507291416398156968115809474246794525188634484455248565832042360378732204109664685120173323001908220438692676033073874143023501911526762952817} a^{30} + \frac{162664348313916495176382756295136433392494770397856819786974321993643828424921368817636404757247942912027912288115249619405633803246014282708220}{29944507291416398156968115809474246794525188634484455248565832042360378732204109664685120173323001908220438692676033073874143023501911526762952817} a^{29} + \frac{1393487891653901399236658236559210645143815898692478106604631394259047019947653716418985175428095877510079841870829525758461626023351809241488258}{29944507291416398156968115809474246794525188634484455248565832042360378732204109664685120173323001908220438692676033073874143023501911526762952817} a^{28} + \frac{9224862618714008077936508845400803449508618805058593874514966519926879943452876160477797581691340352545667918671591567025251456004287293583580524}{29944507291416398156968115809474246794525188634484455248565832042360378732204109664685120173323001908220438692676033073874143023501911526762952817} a^{27} + \frac{1661187531531011826445572885920598818494168302715887083917910157322167504923784651650639280418050962026266770458870806730001241868611919036191550}{4277786755916628308138302258496320970646455519212065035509404577480054104600587094955017167617571701174348384668004724839163289071701646680421831} a^{26} - \frac{5876252727690213448328678481006487780216879001338347886968620692690067824670110465803102831574943762889482656551509109617651349336461355010911770}{29944507291416398156968115809474246794525188634484455248565832042360378732204109664685120173323001908220438692676033073874143023501911526762952817} a^{25} + \frac{13557186008875270212166031601936293111073269277358416034290543473302057218356877531828780302443106066737940037406932185492704764030636381797933318}{29944507291416398156968115809474246794525188634484455248565832042360378732204109664685120173323001908220438692676033073874143023501911526762952817} a^{24} - \frac{9840861027513888983296218162123007927471220512216535166515054611483427197625694098281227304736020269810710430908573515495825321465494949376284788}{29944507291416398156968115809474246794525188634484455248565832042360378732204109664685120173323001908220438692676033073874143023501911526762952817} a^{23} - \frac{14958938315825149746397839284995037200320776370016335392522994052701566052850345407812023810084061589968083217925794254483744051973432931434203274}{29944507291416398156968115809474246794525188634484455248565832042360378732204109664685120173323001908220438692676033073874143023501911526762952817} a^{22} - \frac{5722526355682631125624244464785218393649838907745037740260105798508764322880413547357355424856631407463594864768383994998535656274635314847905660}{29944507291416398156968115809474246794525188634484455248565832042360378732204109664685120173323001908220438692676033073874143023501911526762952817} a^{21} + \frac{13882812880099261034523055686712120993461526105516349082584749676098546882744769996934403120688968837208636100153838439551719316543763288937046866}{29944507291416398156968115809474246794525188634484455248565832042360378732204109664685120173323001908220438692676033073874143023501911526762952817} a^{20} + \frac{13986149554142222088840041681941174853550834918614301058910052145272783981019430418273267530989584947001676690511319286883102427074399021587061786}{29944507291416398156968115809474246794525188634484455248565832042360378732204109664685120173323001908220438692676033073874143023501911526762952817} a^{19} - \frac{13676830033443539296157643880552526753446622736553664278888481962504997280051843897169701153391578138174242499595497223621440271349059561883471902}{29944507291416398156968115809474246794525188634484455248565832042360378732204109664685120173323001908220438692676033073874143023501911526762952817} a^{18} - \frac{1584024144596300694649068159053556797881169022651691308652828930897807279817883914297928358652529383062460260895896674151095549654873145526189960}{29944507291416398156968115809474246794525188634484455248565832042360378732204109664685120173323001908220438692676033073874143023501911526762952817} a^{17} + \frac{14534914102068239408718632847678712119635257671744420257292039129557980283283862377971287953077986876357245439032634535765410064428888019978033812}{29944507291416398156968115809474246794525188634484455248565832042360378732204109664685120173323001908220438692676033073874143023501911526762952817} a^{16} + \frac{2029968418336343361296851203446853807019519701524272944312594684722332472262150622408573080132481540291544512927336471688101481965377182830763506}{4277786755916628308138302258496320970646455519212065035509404577480054104600587094955017167617571701174348384668004724839163289071701646680421831} a^{15} + \frac{10863503162921350257495277136293364586427743301016709082054729773426040293165301084445416121733790160998996811776921605889810215793782966548110747}{29944507291416398156968115809474246794525188634484455248565832042360378732204109664685120173323001908220438692676033073874143023501911526762952817} a^{14} - \frac{3420891078379855984085493525777555798257473338071455360782763935672845710715967633575581658577347370822459992952412022739573436496658219178230192}{29944507291416398156968115809474246794525188634484455248565832042360378732204109664685120173323001908220438692676033073874143023501911526762952817} a^{13} - \frac{11927441593096875025855487604990506215328866979364780535783352746740848636289318087949733859910432469609550046418072794640050272049378546073713056}{29944507291416398156968115809474246794525188634484455248565832042360378732204109664685120173323001908220438692676033073874143023501911526762952817} a^{12} + \frac{13297739035804269006156280883681421107701227361352118949748811037941236420535092125313879928618434675859120248252579684717486300634811924173269041}{29944507291416398156968115809474246794525188634484455248565832042360378732204109664685120173323001908220438692676033073874143023501911526762952817} a^{11} - \frac{623625261515372837763282597621691422281865355683760586235572210195620378557102987647123528760629312430623347782749875776274033372778474574551391}{1761441605377435185704006812322014517325011096146144426386225414256492866600241744981477657254294229895319923098590180816126060205994795691938401} a^{10} + \frac{11467012250802581056288058200244388340663393593159741678444044586657389223428087428641116893454097554239850701468850736916108894715704266892871446}{29944507291416398156968115809474246794525188634484455248565832042360378732204109664685120173323001908220438692676033073874143023501911526762952817} a^{9} - \frac{14852373071868092236362047473139464084316463603674486415382833084496026144496809803160985126852556310030251896270747668634335848477234357991089198}{29944507291416398156968115809474246794525188634484455248565832042360378732204109664685120173323001908220438692676033073874143023501911526762952817} a^{8} + \frac{6309663583400951505613230732190537177705353784926028608974333595014226928341480160589153706430441458361494520774821444239797940519248093618497359}{29944507291416398156968115809474246794525188634484455248565832042360378732204109664685120173323001908220438692676033073874143023501911526762952817} a^{7} - \frac{1001287748538052721787219281019995209990160024904532946108025065786123885784032388348690959793401248406010891364673375734900952708687069936418110}{29944507291416398156968115809474246794525188634484455248565832042360378732204109664685120173323001908220438692676033073874143023501911526762952817} a^{6} - \frac{304947665106007591323952987781688049914406192367340131624880250195226817062490758282161088416210297384655481210758152278002679353630768293763656}{4277786755916628308138302258496320970646455519212065035509404577480054104600587094955017167617571701174348384668004724839163289071701646680421831} a^{5} - \frac{4057986597083355178696853777280387426965118611968585029892719857840224569062331261361890195800106662697118353653401583981895916337026303772570191}{29944507291416398156968115809474246794525188634484455248565832042360378732204109664685120173323001908220438692676033073874143023501911526762952817} a^{4} - \frac{13093443759764880056496526175298377736693868682138597615213984680016102065921128352366039709083730130971957575448498834927764953381729213043474331}{29944507291416398156968115809474246794525188634484455248565832042360378732204109664685120173323001908220438692676033073874143023501911526762952817} a^{3} + \frac{11047929026355854025903451993126121149475259209526942468074852578400766599482890614275986054923917305327686295216313217081237923419112052222243728}{29944507291416398156968115809474246794525188634484455248565832042360378732204109664685120173323001908220438692676033073874143023501911526762952817} a^{2} + \frac{14060406493421577670227525244696114753500470095510253725367752926161075008624831279972133269388510487342223477695249231413189886197509790501236728}{29944507291416398156968115809474246794525188634484455248565832042360378732204109664685120173323001908220438692676033073874143023501911526762952817} a + \frac{12220054082702455582831023927641550693913309254162752200928623969905348788028490089690932448899196170195013922524339548430072071194020335649254}{190729345805199988260943412799199024168950246079518823239272815556435533326140825889714141231356700052359482118955624674357598875808353673649381}$  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:  \( 2479808589614732200000000000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{35}\cdot(2\pi)^{0}\cdot 2479808589614732200000000000000 \cdot 1}{2\sqrt{103338030412840513192336580932106187652481378569380154885948286391794681549072265625}}\approx 0.132528085033318$ (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

5.5.390625.1, 7.7.594823321.1

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 $35$ $35$ R ${\href{/LocalNumberField/7.7.0.1}{7} }^{5}$ $35$ $35$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{7}$ $35$ $35$ R $35$ $35$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{7}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{5}$ $35$ $35$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{7}$

In the table, R denotes a ramified prime. 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
5Data not computed
29Data not computed