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Properties

Label 35.35.145...169.1
Degree $35$
Signature $[35, 0]$
Discriminant $1.456\times 10^{78}$
Root discriminant $171.09$
Ramified primes $11, 43$
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 - 2*x^34 - 121*x^33 + 124*x^32 + 6435*x^31 - 1020*x^30 - 195036*x^29 - 112597*x^28 + 3683986*x^27 + 4452851*x^26 - 44769607*x^25 - 80359740*x^24 + 347984933*x^23 + 850835561*x^22 - 1630509423*x^21 - 5621246939*x^20 + 3615565971*x^19 + 23234081997*x^18 + 3088781711*x^17 - 57619940147*x^16 - 39347873390*x^15 + 76695393961*x^14 + 92864249975*x^13 - 38675667681*x^12 - 94158719636*x^11 - 10380297650*x^10 + 42855721796*x^9 + 16263455858*x^8 - 7226936109*x^7 - 4634581071*x^6 - 47622511*x^5 + 349101928*x^4 + 70736582*x^3 + 5060011*x^2 + 136327*x + 859)
 
gp: K = bnfinit(x^35 - 2*x^34 - 121*x^33 + 124*x^32 + 6435*x^31 - 1020*x^30 - 195036*x^29 - 112597*x^28 + 3683986*x^27 + 4452851*x^26 - 44769607*x^25 - 80359740*x^24 + 347984933*x^23 + 850835561*x^22 - 1630509423*x^21 - 5621246939*x^20 + 3615565971*x^19 + 23234081997*x^18 + 3088781711*x^17 - 57619940147*x^16 - 39347873390*x^15 + 76695393961*x^14 + 92864249975*x^13 - 38675667681*x^12 - 94158719636*x^11 - 10380297650*x^10 + 42855721796*x^9 + 16263455858*x^8 - 7226936109*x^7 - 4634581071*x^6 - 47622511*x^5 + 349101928*x^4 + 70736582*x^3 + 5060011*x^2 + 136327*x + 859, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![859, 136327, 5060011, 70736582, 349101928, -47622511, -4634581071, -7226936109, 16263455858, 42855721796, -10380297650, -94158719636, -38675667681, 92864249975, 76695393961, -39347873390, -57619940147, 3088781711, 23234081997, 3615565971, -5621246939, -1630509423, 850835561, 347984933, -80359740, -44769607, 4452851, 3683986, -112597, -195036, -1020, 6435, 124, -121, -2, 1]);
 

\( x^{35} - 2 x^{34} - 121 x^{33} + 124 x^{32} + 6435 x^{31} - 1020 x^{30} - 195036 x^{29} - 112597 x^{28} + 3683986 x^{27} + 4452851 x^{26} - 44769607 x^{25} - 80359740 x^{24} + 347984933 x^{23} + 850835561 x^{22} - 1630509423 x^{21} - 5621246939 x^{20} + 3615565971 x^{19} + 23234081997 x^{18} + 3088781711 x^{17} - 57619940147 x^{16} - 39347873390 x^{15} + 76695393961 x^{14} + 92864249975 x^{13} - 38675667681 x^{12} - 94158719636 x^{11} - 10380297650 x^{10} + 42855721796 x^{9} + 16263455858 x^{8} - 7226936109 x^{7} - 4634581071 x^{6} - 47622511 x^{5} + 349101928 x^{4} + 70736582 x^{3} + 5060011 x^{2} + 136327 x + 859 \)

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:  \(145\!\cdots\!169\)\(\medspace = 11^{28}\cdot 43^{30}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $171.09$
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:  $11, 43$
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:  \(473=11\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{473}(256,·)$, $\chi_{473}(1,·)$, $\chi_{473}(130,·)$, $\chi_{473}(4,·)$, $\chi_{473}(133,·)$, $\chi_{473}(262,·)$, $\chi_{473}(269,·)$, $\chi_{473}(16,·)$, $\chi_{473}(279,·)$, $\chi_{473}(408,·)$, $\chi_{473}(388,·)$, $\chi_{473}(422,·)$, $\chi_{473}(170,·)$, $\chi_{473}(302,·)$, $\chi_{473}(47,·)$, $\chi_{473}(434,·)$, $\chi_{473}(312,·)$, $\chi_{473}(441,·)$, $\chi_{473}(59,·)$, $\chi_{473}(188,·)$, $\chi_{473}(317,·)$, $\chi_{473}(64,·)$, $\chi_{473}(322,·)$, $\chi_{473}(78,·)$, $\chi_{473}(207,·)$, $\chi_{473}(465,·)$, $\chi_{473}(213,·)$, $\chi_{473}(342,·)$, $\chi_{473}(471,·)$, $\chi_{473}(345,·)$, $\chi_{473}(97,·)$, $\chi_{473}(355,·)$, $\chi_{473}(102,·)$, $\chi_{473}(236,·)$, $\chi_{473}(379,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $\frac{1}{7} a^{25} - \frac{2}{7} a^{24} - \frac{1}{7} a^{22} + \frac{3}{7} a^{21} - \frac{2}{7} a^{20} + \frac{1}{7} a^{19} - \frac{2}{7} a^{18} + \frac{3}{7} a^{17} - \frac{1}{7} a^{16} - \frac{1}{7} a^{15} + \frac{3}{7} a^{14} - \frac{3}{7} a^{13} + \frac{3}{7} a^{12} + \frac{2}{7} a^{11} - \frac{2}{7} a^{9} + \frac{3}{7} a^{8} + \frac{2}{7} a^{6} + \frac{3}{7} a^{4} - \frac{2}{7} a^{3} - \frac{1}{7} a^{2} + \frac{2}{7} a + \frac{2}{7}$, $\frac{1}{7} a^{26} + \frac{3}{7} a^{24} - \frac{1}{7} a^{23} + \frac{1}{7} a^{22} - \frac{3}{7} a^{21} - \frac{3}{7} a^{20} - \frac{1}{7} a^{18} - \frac{2}{7} a^{17} - \frac{3}{7} a^{16} + \frac{1}{7} a^{15} + \frac{3}{7} a^{14} - \frac{3}{7} a^{13} + \frac{1}{7} a^{12} - \frac{3}{7} a^{11} - \frac{2}{7} a^{10} - \frac{1}{7} a^{9} - \frac{1}{7} a^{8} + \frac{2}{7} a^{7} - \frac{3}{7} a^{6} + \frac{3}{7} a^{5} - \frac{3}{7} a^{4} + \frac{2}{7} a^{3} - \frac{1}{7} a - \frac{3}{7}$, $\frac{1}{7} a^{27} - \frac{2}{7} a^{24} + \frac{1}{7} a^{23} + \frac{2}{7} a^{21} - \frac{1}{7} a^{20} + \frac{3}{7} a^{19} - \frac{3}{7} a^{18} + \frac{2}{7} a^{17} - \frac{3}{7} a^{16} - \frac{1}{7} a^{15} + \frac{2}{7} a^{14} + \frac{3}{7} a^{13} + \frac{2}{7} a^{12} - \frac{1}{7} a^{11} - \frac{1}{7} a^{10} - \frac{2}{7} a^{9} - \frac{3}{7} a^{7} - \frac{3}{7} a^{6} - \frac{3}{7} a^{5} - \frac{1}{7} a^{3} + \frac{2}{7} a^{2} - \frac{2}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{28} - \frac{3}{7} a^{24} - \frac{2}{7} a^{21} - \frac{1}{7} a^{20} - \frac{1}{7} a^{19} - \frac{2}{7} a^{18} + \frac{3}{7} a^{17} - \frac{3}{7} a^{16} + \frac{2}{7} a^{14} + \frac{3}{7} a^{13} - \frac{2}{7} a^{12} + \frac{3}{7} a^{11} - \frac{2}{7} a^{10} + \frac{3}{7} a^{9} + \frac{3}{7} a^{8} - \frac{3}{7} a^{7} + \frac{1}{7} a^{6} - \frac{2}{7} a^{4} - \frac{2}{7} a^{3} + \frac{3}{7} a^{2} - \frac{2}{7} a - \frac{3}{7}$, $\frac{1}{7} a^{29} + \frac{1}{7} a^{24} + \frac{2}{7} a^{22} + \frac{1}{7} a^{21} + \frac{1}{7} a^{19} - \frac{3}{7} a^{18} - \frac{1}{7} a^{17} - \frac{3}{7} a^{16} - \frac{1}{7} a^{15} - \frac{2}{7} a^{14} + \frac{3}{7} a^{13} - \frac{2}{7} a^{12} - \frac{3}{7} a^{11} + \frac{3}{7} a^{10} - \frac{3}{7} a^{9} - \frac{1}{7} a^{8} + \frac{1}{7} a^{7} - \frac{1}{7} a^{6} - \frac{2}{7} a^{5} - \frac{3}{7} a^{3} + \frac{2}{7} a^{2} + \frac{3}{7} a - \frac{1}{7}$, $\frac{1}{7} a^{30} + \frac{2}{7} a^{24} + \frac{2}{7} a^{23} + \frac{2}{7} a^{22} - \frac{3}{7} a^{21} + \frac{3}{7} a^{20} + \frac{3}{7} a^{19} + \frac{1}{7} a^{18} + \frac{1}{7} a^{17} - \frac{1}{7} a^{15} + \frac{1}{7} a^{13} + \frac{1}{7} a^{12} + \frac{1}{7} a^{11} - \frac{3}{7} a^{10} + \frac{1}{7} a^{9} - \frac{2}{7} a^{8} - \frac{1}{7} a^{7} + \frac{3}{7} a^{6} + \frac{1}{7} a^{4} - \frac{3}{7} a^{3} - \frac{3}{7} a^{2} - \frac{3}{7} a - \frac{2}{7}$, $\frac{1}{7} a^{31} - \frac{1}{7} a^{24} + \frac{2}{7} a^{23} - \frac{1}{7} a^{22} - \frac{3}{7} a^{21} - \frac{1}{7} a^{19} - \frac{2}{7} a^{18} + \frac{1}{7} a^{17} + \frac{1}{7} a^{16} + \frac{2}{7} a^{15} + \frac{2}{7} a^{14} + \frac{2}{7} a^{12} + \frac{1}{7} a^{10} + \frac{2}{7} a^{9} + \frac{3}{7} a^{7} + \frac{3}{7} a^{6} + \frac{1}{7} a^{5} - \frac{2}{7} a^{4} + \frac{1}{7} a^{3} - \frac{1}{7} a^{2} + \frac{1}{7} a + \frac{3}{7}$, $\frac{1}{7} a^{32} - \frac{1}{7} a^{23} + \frac{3}{7} a^{22} + \frac{3}{7} a^{21} - \frac{3}{7} a^{20} - \frac{1}{7} a^{19} - \frac{1}{7} a^{18} - \frac{3}{7} a^{17} + \frac{1}{7} a^{16} + \frac{1}{7} a^{15} + \frac{3}{7} a^{14} - \frac{1}{7} a^{13} + \frac{3}{7} a^{12} + \frac{3}{7} a^{11} + \frac{2}{7} a^{10} - \frac{2}{7} a^{9} - \frac{1}{7} a^{8} + \frac{3}{7} a^{7} + \frac{3}{7} a^{6} - \frac{2}{7} a^{5} - \frac{3}{7} a^{4} - \frac{3}{7} a^{3} - \frac{2}{7} a + \frac{2}{7}$, $\frac{1}{7} a^{33} - \frac{1}{7} a^{24} + \frac{3}{7} a^{23} + \frac{3}{7} a^{22} - \frac{3}{7} a^{21} - \frac{1}{7} a^{20} - \frac{1}{7} a^{19} - \frac{3}{7} a^{18} + \frac{1}{7} a^{17} + \frac{1}{7} a^{16} + \frac{3}{7} a^{15} - \frac{1}{7} a^{14} + \frac{3}{7} a^{13} + \frac{3}{7} a^{12} + \frac{2}{7} a^{11} - \frac{2}{7} a^{10} - \frac{1}{7} a^{9} + \frac{3}{7} a^{8} + \frac{3}{7} a^{7} - \frac{2}{7} a^{6} - \frac{3}{7} a^{5} - \frac{3}{7} a^{4} - \frac{2}{7} a^{2} + \frac{2}{7} a$, $\frac{1}{162900046236433869357823725174307393303091954001966261515408608569297782669765426202982097226559948812782317126928745298690192072559728072319202119} a^{34} + \frac{7754017277037348987371218791517446342794604145472140781347143858778605432615817330148790740945104397793491486696048796635461271834911553635208182}{162900046236433869357823725174307393303091954001966261515408608569297782669765426202982097226559948812782317126928745298690192072559728072319202119} a^{33} - \frac{806735589745815567764915773849085505237217225640798131083640707680190651111889103340609961916348116233283760847431051033372236402120964157560044}{162900046236433869357823725174307393303091954001966261515408608569297782669765426202982097226559948812782317126928745298690192072559728072319202119} a^{32} - \frac{5451525026986057363840481420854010962791971500974581573469441764753823195290616810483011056261912034761355309932795925374787453647434578055326127}{162900046236433869357823725174307393303091954001966261515408608569297782669765426202982097226559948812782317126928745298690192072559728072319202119} a^{31} - \frac{3439235533965468831649750124840353631779600197127092439883589252712490435008567463027126482903696482374694623997800182104422542346251578935600540}{162900046236433869357823725174307393303091954001966261515408608569297782669765426202982097226559948812782317126928745298690192072559728072319202119} a^{30} + \frac{1533391814929505559363443169272695132707508072494039716324689399553936819582641402018515496187970509480687613158704227747803064495458027599738022}{162900046236433869357823725174307393303091954001966261515408608569297782669765426202982097226559948812782317126928745298690192072559728072319202119} a^{29} + \frac{10046813531471811884816745930737032479348703781231944845047995392863910832486936654192478572221831066601590041440680755782130051432883261968200163}{162900046236433869357823725174307393303091954001966261515408608569297782669765426202982097226559948812782317126928745298690192072559728072319202119} a^{28} + \frac{295300441784795239796062140380672013448564328682801314798356752176399444126268658106374900585663555356515573198262277831944676888595615153490330}{23271435176633409908260532167758199043298850571709465930772658367042540381395060886140299603794278401826045303846963614098598867508532581759886017} a^{27} - \frac{5539207175429769464810815484449423212341524242437826506889363563829709070597856616394290987088101527286207329937765315866182248065151408390490451}{162900046236433869357823725174307393303091954001966261515408608569297782669765426202982097226559948812782317126928745298690192072559728072319202119} a^{26} - \frac{2802199464569138089225557933077413053011156698654629586540736071496642224812785183885605225866208263437219501814549896322907973147721231839968869}{162900046236433869357823725174307393303091954001966261515408608569297782669765426202982097226559948812782317126928745298690192072559728072319202119} a^{25} - \frac{49768599890691493319642209786043424120797018028086202468632159944979053677659126458549236934200169733024704957173089599953424261284397655695022055}{162900046236433869357823725174307393303091954001966261515408608569297782669765426202982097226559948812782317126928745298690192072559728072319202119} a^{24} + \frac{48598033049147974404734921643743686782546072857251892741021303990284786922646066569938034143542806282683674050884649510864114825017130394903185049}{162900046236433869357823725174307393303091954001966261515408608569297782669765426202982097226559948812782317126928745298690192072559728072319202119} a^{23} + \frac{47019792990867065052328521792482004156675981004429538421258874021716442879638880399410027329060799601631523370007718020575731603070157264923709816}{162900046236433869357823725174307393303091954001966261515408608569297782669765426202982097226559948812782317126928745298690192072559728072319202119} a^{22} - \frac{31400886296680547143018397726586632334329184509392072809966233760583359723694993962977371178191256583607983411769284449011738081876146978167847871}{162900046236433869357823725174307393303091954001966261515408608569297782669765426202982097226559948812782317126928745298690192072559728072319202119} a^{21} + \frac{48393877027353479889013995318800345922414051512523105428413079912873516731662034539969084975974311231273113188367805384971562333000299698863093826}{162900046236433869357823725174307393303091954001966261515408608569297782669765426202982097226559948812782317126928745298690192072559728072319202119} a^{20} - \frac{81080270653856386320665058479328657413038742619901581183313363964817437651857691074999650533721769892717059458920192203852634486883428404524054191}{162900046236433869357823725174307393303091954001966261515408608569297782669765426202982097226559948812782317126928745298690192072559728072319202119} a^{19} - \frac{2657677125311006792113907270811653901206786762378538324508761001137803994527421347361350561537808238580791944429508910713637168758107673651882624}{23271435176633409908260532167758199043298850571709465930772658367042540381395060886140299603794278401826045303846963614098598867508532581759886017} a^{18} - \frac{69006997353928011362667190349097546069805444570995996805560494610423320299367737431354377144087754136463839133112437391197162704186251525351539951}{162900046236433869357823725174307393303091954001966261515408608569297782669765426202982097226559948812782317126928745298690192072559728072319202119} a^{17} + \frac{34060920712569808859184636889676467249919427844311685500850727293859143057369501782597048431277470399939209388369926861075065468813875257317290}{75802720445059967127884469601818237926054887855731159383624294355187427952426908423909770696398300983146727374094344019865142890907272253289531} a^{16} - \frac{52845503909553474086547078620824805992703459380652647858995950868239610510844029366598321349523816881351536998358185066909834877350473868696560672}{162900046236433869357823725174307393303091954001966261515408608569297782669765426202982097226559948812782317126928745298690192072559728072319202119} a^{15} - \frac{38523262659723272043227545008380290458206355657034735922965112856862964086532826071722307483057848191034281267618150457479413841135114993313061369}{162900046236433869357823725174307393303091954001966261515408608569297782669765426202982097226559948812782317126928745298690192072559728072319202119} a^{14} + \frac{5843457041166557633684634320673342890886455117370178830577554216735991332666816064900391319458971290667191882871797341742081644145759974243230658}{23271435176633409908260532167758199043298850571709465930772658367042540381395060886140299603794278401826045303846963614098598867508532581759886017} a^{13} - \frac{53851435093661012560325819895742987245332943049071665761416062720352231787456534604259037726924892809884490199203420276640907562800063480056721447}{162900046236433869357823725174307393303091954001966261515408608569297782669765426202982097226559948812782317126928745298690192072559728072319202119} a^{12} - \frac{1153019048898709307400942263430725056610993608719082887720569218514849269659284625780212223771463624722327700612847461666141705225335802281084506}{23271435176633409908260532167758199043298850571709465930772658367042540381395060886140299603794278401826045303846963614098598867508532581759886017} a^{11} + \frac{78467591709576938021887202195328029287554244425064051852338974774263786709660584808361540236217605360231448296780390074673854104257329636630440911}{162900046236433869357823725174307393303091954001966261515408608569297782669765426202982097226559948812782317126928745298690192072559728072319202119} a^{10} + \frac{5603140024425835696155047936367966888626956121387516407530618697995991491171564857788718504282178308890040588097728021737550218995754049621724998}{23271435176633409908260532167758199043298850571709465930772658367042540381395060886140299603794278401826045303846963614098598867508532581759886017} a^{9} - \frac{478074400187247974983776135651390265719071681471289955347124861300249236120375794311996730199589305692900737189590200750754060997887978344366359}{23271435176633409908260532167758199043298850571709465930772658367042540381395060886140299603794278401826045303846963614098598867508532581759886017} a^{8} + \frac{73110075580174303919373278936521928369206888874091780411361884351920685914098091725639378401815004842097498640335695431609507173321099521197900307}{162900046236433869357823725174307393303091954001966261515408608569297782669765426202982097226559948812782317126928745298690192072559728072319202119} a^{7} + \frac{58441019166208981778147649431171930428630019458256798862246405785212439057394125053869437336033698503012584124237746400880091505001385094000012486}{162900046236433869357823725174307393303091954001966261515408608569297782669765426202982097226559948812782317126928745298690192072559728072319202119} a^{6} - \frac{10617473955731756728524698775621524629456537484508532507332812462570881534357652448201236324155975076259519724345620360099032447183366294606986434}{23271435176633409908260532167758199043298850571709465930772658367042540381395060886140299603794278401826045303846963614098598867508532581759886017} a^{5} - \frac{23260020499119468582927224580466754218596716174275084135823455073811880469891780767856730787596482847410448916087656512264068464794438862494499530}{162900046236433869357823725174307393303091954001966261515408608569297782669765426202982097226559948812782317126928745298690192072559728072319202119} a^{4} - \frac{37684898329023128479704744315778822388501822177812007597435874868951634906852920840975743813522179542873906683228424404991546673410478856325970569}{162900046236433869357823725174307393303091954001966261515408608569297782669765426202982097226559948812782317126928745298690192072559728072319202119} a^{3} - \frac{18799983302501407287667015061040413955402304408473507430163297703943402287568410607116893211347490011292966819407278343709533855082646982089362469}{162900046236433869357823725174307393303091954001966261515408608569297782669765426202982097226559948812782317126928745298690192072559728072319202119} a^{2} + \frac{41861624842261251931969879092133075438924114818880184102637177743395777150046583766648745177956745671888376402552510256951006298061260515046493592}{162900046236433869357823725174307393303091954001966261515408608569297782669765426202982097226559948812782317126928745298690192072559728072319202119} a - \frac{10538669938163022416865532701768479198505215727868826961812528419979896524631656902436780734856467411665918520778541585665412465815019532938923}{27091309868024924223819013000882653135388650258101822969467588320189220467281793813900232367630126195373743077819515266703840357984321981094163}$

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 $2$)
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:  \( 8063096359873239000000000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{35}\cdot(2\pi)^{0}\cdot 8063096359873239000000000000 \cdot 1}{2\sqrt{1455622807785591094953547155658149343464416905925495778881639129313848614446169}}\approx 0.114814642169343$ (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

\(\Q(\zeta_{11})^+\), 7.7.6321363049.1

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 $35$ $35$ $35$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{7}$ R $35$ $35$ $35$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{5}$ $35$ $35$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{7}$ $35$ R $35$ $35$ $35$

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
11Data not computed
$43$43.7.6.1$x^{7} - 43$$7$$1$$6$$C_7$$[\ ]_{7}$
43.7.6.1$x^{7} - 43$$7$$1$$6$$C_7$$[\ ]_{7}$
43.7.6.1$x^{7} - 43$$7$$1$$6$$C_7$$[\ ]_{7}$
43.7.6.1$x^{7} - 43$$7$$1$$6$$C_7$$[\ ]_{7}$
43.7.6.1$x^{7} - 43$$7$$1$$6$$C_7$$[\ ]_{7}$