↑

Properties

Label 34.34.1537691324...2729.1
Degree $34$
Signature $[34, 0]$
Discriminant $409^{33}$
Root discriminant $342.70$
Ramified prime $409$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{34}$ (as 34T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1452144329, -201364551450, 1666698081563, 145449028860, -23032463830036, 41638323145590, 29962714731703, -133847512932970, 71782661637932, 96040685434967, -108443692715447, -14647345379566, 59946275418054, -11538613521181, -17513024380811, 6988511541500, 2892031385664, -1865190697480, -235614212401, 299499953469, -2769204238, -31343491334, 2853676317, 2186175165, -327117700, -100755361, 20098756, 2952967, -744065, -50633, 16435, 423, -198, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^34 - x^33 - 198*x^32 + 423*x^31 + 16435*x^30 - 50633*x^29 - 744065*x^28 + 2952967*x^27 + 20098756*x^26 - 100755361*x^25 - 327117700*x^24 + 2186175165*x^23 + 2853676317*x^22 - 31343491334*x^21 - 2769204238*x^20 + 299499953469*x^19 - 235614212401*x^18 - 1865190697480*x^17 + 2892031385664*x^16 + 6988511541500*x^15 - 17513024380811*x^14 - 11538613521181*x^13 + 59946275418054*x^12 - 14647345379566*x^11 - 108443692715447*x^10 + 96040685434967*x^9 + 71782661637932*x^8 - 133847512932970*x^7 + 29962714731703*x^6 + 41638323145590*x^5 - 23032463830036*x^4 + 145449028860*x^3 + 1666698081563*x^2 - 201364551450*x + 1452144329)
 
gp: K = bnfinit(x^34 - x^33 - 198*x^32 + 423*x^31 + 16435*x^30 - 50633*x^29 - 744065*x^28 + 2952967*x^27 + 20098756*x^26 - 100755361*x^25 - 327117700*x^24 + 2186175165*x^23 + 2853676317*x^22 - 31343491334*x^21 - 2769204238*x^20 + 299499953469*x^19 - 235614212401*x^18 - 1865190697480*x^17 + 2892031385664*x^16 + 6988511541500*x^15 - 17513024380811*x^14 - 11538613521181*x^13 + 59946275418054*x^12 - 14647345379566*x^11 - 108443692715447*x^10 + 96040685434967*x^9 + 71782661637932*x^8 - 133847512932970*x^7 + 29962714731703*x^6 + 41638323145590*x^5 - 23032463830036*x^4 + 145449028860*x^3 + 1666698081563*x^2 - 201364551450*x + 1452144329, 1)
 

Normalized defining polynomial

\( x^{34} - x^{33} - 198 x^{32} + 423 x^{31} + 16435 x^{30} - 50633 x^{29} - 744065 x^{28} + 2952967 x^{27} + 20098756 x^{26} - 100755361 x^{25} - 327117700 x^{24} + 2186175165 x^{23} + 2853676317 x^{22} - 31343491334 x^{21} - 2769204238 x^{20} + 299499953469 x^{19} - 235614212401 x^{18} - 1865190697480 x^{17} + 2892031385664 x^{16} + 6988511541500 x^{15} - 17513024380811 x^{14} - 11538613521181 x^{13} + 59946275418054 x^{12} - 14647345379566 x^{11} - 108443692715447 x^{10} + 96040685434967 x^{9} + 71782661637932 x^{8} - 133847512932970 x^{7} + 29962714731703 x^{6} + 41638323145590 x^{5} - 23032463830036 x^{4} + 145449028860 x^{3} + 1666698081563 x^{2} - 201364551450 x + 1452144329 \)

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

Invariants

Degree:  $34$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[34, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(153769132491311397559694601557065864078493667954677172237435543232728059696605988632729=409^{33}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $342.70$
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:  $409$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(409\)
Dirichlet character group:    $\lbrace$$\chi_{409}(384,·)$, $\chi_{409}(1,·)$, $\chi_{409}(259,·)$, $\chi_{409}(5,·)$, $\chi_{409}(262,·)$, $\chi_{409}(64,·)$, $\chi_{409}(147,·)$, $\chi_{409}(404,·)$, $\chi_{409}(150,·)$, $\chi_{409}(89,·)$, $\chi_{409}(408,·)$, $\chi_{409}(25,·)$, $\chi_{409}(284,·)$, $\chi_{409}(30,·)$, $\chi_{409}(36,·)$, $\chi_{409}(6,·)$, $\chi_{409}(180,·)$, $\chi_{409}(320,·)$, $\chi_{409}(193,·)$, $\chi_{409}(68,·)$, $\chi_{409}(69,·)$, $\chi_{409}(326,·)$, $\chi_{409}(327,·)$, $\chi_{409}(82,·)$, $\chi_{409}(83,·)$, $\chi_{409}(340,·)$, $\chi_{409}(341,·)$, $\chi_{409}(216,·)$, $\chi_{409}(345,·)$, $\chi_{409}(229,·)$, $\chi_{409}(403,·)$, $\chi_{409}(373,·)$, $\chi_{409}(379,·)$, $\chi_{409}(125,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{7} a^{18} + \frac{2}{7} a^{17} - \frac{1}{7} a^{16} - \frac{1}{7} a^{15} - \frac{3}{7} a^{14} - \frac{1}{7} a^{13} + \frac{2}{7} a^{12} - \frac{2}{7} a^{11} + \frac{1}{7} a^{9} - \frac{2}{7} a^{7} - \frac{2}{7} a^{6} - \frac{1}{7} a^{5} + \frac{3}{7} a^{4} - \frac{1}{7} a^{3} + \frac{1}{7} a^{2} + \frac{2}{7} a - \frac{1}{7}$, $\frac{1}{7} a^{19} + \frac{2}{7} a^{17} + \frac{1}{7} a^{16} - \frac{1}{7} a^{15} - \frac{2}{7} a^{14} - \frac{3}{7} a^{13} + \frac{1}{7} a^{12} - \frac{3}{7} a^{11} + \frac{1}{7} a^{10} - \frac{2}{7} a^{9} - \frac{2}{7} a^{8} + \frac{2}{7} a^{7} + \frac{3}{7} a^{6} - \frac{2}{7} a^{5} + \frac{3}{7} a^{3} + \frac{2}{7} a + \frac{2}{7}$, $\frac{1}{7} a^{20} - \frac{3}{7} a^{17} + \frac{1}{7} a^{16} + \frac{3}{7} a^{14} + \frac{3}{7} a^{13} - \frac{2}{7} a^{11} - \frac{2}{7} a^{10} + \frac{3}{7} a^{9} + \frac{2}{7} a^{8} + \frac{2}{7} a^{6} + \frac{2}{7} a^{5} - \frac{3}{7} a^{4} + \frac{2}{7} a^{3} - \frac{2}{7} a + \frac{2}{7}$, $\frac{1}{7} a^{21} - \frac{3}{7} a^{16} + \frac{1}{7} a^{14} - \frac{3}{7} a^{13} - \frac{3}{7} a^{12} - \frac{1}{7} a^{11} + \frac{3}{7} a^{10} - \frac{2}{7} a^{9} + \frac{3}{7} a^{7} + \frac{3}{7} a^{6} + \frac{1}{7} a^{5} - \frac{3}{7} a^{4} - \frac{3}{7} a^{3} + \frac{1}{7} a^{2} + \frac{1}{7} a - \frac{3}{7}$, $\frac{1}{7} a^{22} - \frac{3}{7} a^{17} + \frac{1}{7} a^{15} - \frac{3}{7} a^{14} - \frac{3}{7} a^{13} - \frac{1}{7} a^{12} + \frac{3}{7} a^{11} - \frac{2}{7} a^{10} + \frac{3}{7} a^{8} + \frac{3}{7} a^{7} + \frac{1}{7} a^{6} - \frac{3}{7} a^{5} - \frac{3}{7} a^{4} + \frac{1}{7} a^{3} + \frac{1}{7} a^{2} - \frac{3}{7} a$, $\frac{1}{7} a^{23} - \frac{1}{7} a^{17} - \frac{2}{7} a^{16} + \frac{1}{7} a^{15} + \frac{2}{7} a^{14} + \frac{3}{7} a^{13} + \frac{2}{7} a^{12} - \frac{1}{7} a^{11} - \frac{1}{7} a^{9} + \frac{3}{7} a^{8} + \frac{2}{7} a^{7} - \frac{2}{7} a^{6} + \frac{1}{7} a^{5} + \frac{3}{7} a^{4} - \frac{2}{7} a^{3} - \frac{1}{7} a - \frac{3}{7}$, $\frac{1}{7} a^{24} + \frac{1}{7} a^{15} + \frac{1}{7} a^{13} + \frac{1}{7} a^{12} - \frac{2}{7} a^{11} - \frac{1}{7} a^{10} - \frac{3}{7} a^{9} + \frac{2}{7} a^{8} + \frac{3}{7} a^{7} - \frac{1}{7} a^{6} + \frac{2}{7} a^{5} + \frac{1}{7} a^{4} - \frac{1}{7} a^{3} - \frac{1}{7} a - \frac{1}{7}$, $\frac{1}{7} a^{25} + \frac{1}{7} a^{16} + \frac{1}{7} a^{14} + \frac{1}{7} a^{13} - \frac{2}{7} a^{12} - \frac{1}{7} a^{11} - \frac{3}{7} a^{10} + \frac{2}{7} a^{9} + \frac{3}{7} a^{8} - \frac{1}{7} a^{7} + \frac{2}{7} a^{6} + \frac{1}{7} a^{5} - \frac{1}{7} a^{4} - \frac{1}{7} a^{2} - \frac{1}{7} a$, $\frac{1}{7} a^{26} + \frac{1}{7} a^{17} + \frac{1}{7} a^{15} + \frac{1}{7} a^{14} - \frac{2}{7} a^{13} - \frac{1}{7} a^{12} - \frac{3}{7} a^{11} + \frac{2}{7} a^{10} + \frac{3}{7} a^{9} - \frac{1}{7} a^{8} + \frac{2}{7} a^{7} + \frac{1}{7} a^{6} - \frac{1}{7} a^{5} - \frac{1}{7} a^{3} - \frac{1}{7} a^{2}$, $\frac{1}{371} a^{27} - \frac{1}{371} a^{26} - \frac{26}{371} a^{25} + \frac{24}{371} a^{24} + \frac{4}{371} a^{23} - \frac{3}{53} a^{22} - \frac{11}{371} a^{21} - \frac{20}{371} a^{20} - \frac{4}{371} a^{19} + \frac{26}{371} a^{18} + \frac{125}{371} a^{17} + \frac{21}{53} a^{16} - \frac{12}{53} a^{15} - \frac{5}{371} a^{14} - \frac{183}{371} a^{13} - \frac{1}{53} a^{12} - \frac{43}{371} a^{11} + \frac{107}{371} a^{10} + \frac{94}{371} a^{9} - \frac{89}{371} a^{8} + \frac{5}{53} a^{7} - \frac{172}{371} a^{6} - \frac{146}{371} a^{5} + \frac{5}{371} a^{4} - \frac{118}{371} a^{3} + \frac{69}{371} a^{2} + \frac{97}{371} a + \frac{155}{371}$, $\frac{1}{2597} a^{28} - \frac{2}{2597} a^{27} - \frac{25}{2597} a^{26} - \frac{3}{2597} a^{25} + \frac{33}{2597} a^{24} + \frac{4}{371} a^{23} - \frac{149}{2597} a^{22} - \frac{62}{2597} a^{21} + \frac{122}{2597} a^{20} - \frac{129}{2597} a^{19} - \frac{60}{2597} a^{18} + \frac{234}{2597} a^{17} + \frac{617}{2597} a^{16} + \frac{715}{2597} a^{15} - \frac{72}{2597} a^{14} - \frac{149}{371} a^{13} + \frac{116}{371} a^{12} + \frac{29}{371} a^{11} - \frac{10}{53} a^{10} + \frac{1195}{2597} a^{9} - \frac{830}{2597} a^{8} + \frac{137}{371} a^{7} + \frac{34}{371} a^{6} - \frac{856}{2597} a^{5} - \frac{1130}{2597} a^{4} + \frac{293}{2597} a^{3} - \frac{661}{2597} a^{2} - \frac{790}{2597} a - \frac{526}{2597}$, $\frac{1}{2597} a^{29} - \frac{1}{2597} a^{27} - \frac{81}{2597} a^{26} + \frac{41}{2597} a^{25} + \frac{24}{2597} a^{24} + \frac{19}{2597} a^{23} + \frac{165}{2597} a^{22} + \frac{61}{2597} a^{21} - \frac{74}{2597} a^{20} - \frac{59}{2597} a^{19} + \frac{100}{2597} a^{18} - \frac{87}{371} a^{17} - \frac{613}{2597} a^{16} - \frac{36}{371} a^{15} - \frac{956}{2597} a^{14} + \frac{40}{371} a^{13} + \frac{180}{371} a^{12} - \frac{78}{371} a^{11} + \frac{243}{2597} a^{10} - \frac{1002}{2597} a^{9} + \frac{888}{2597} a^{8} - \frac{135}{371} a^{7} - \frac{2}{2597} a^{6} + \frac{10}{53} a^{5} - \frac{102}{371} a^{4} - \frac{40}{2597} a^{3} - \frac{180}{2597} a^{2} - \frac{1245}{2597} a - \frac{51}{2597}$, $\frac{1}{2597} a^{30} + \frac{1}{2597} a^{27} - \frac{68}{2597} a^{26} + \frac{9}{371} a^{25} - \frac{158}{2597} a^{24} + \frac{158}{2597} a^{23} + \frac{3}{2597} a^{22} + \frac{1}{49} a^{21} - \frac{19}{371} a^{20} + \frac{6}{2597} a^{19} + \frac{31}{2597} a^{18} + \frac{846}{2597} a^{17} + \frac{99}{2597} a^{16} + \frac{865}{2597} a^{15} - \frac{4}{49} a^{14} + \frac{61}{371} a^{13} + \frac{1}{53} a^{12} + \frac{173}{2597} a^{11} + \frac{76}{2597} a^{10} + \frac{704}{2597} a^{9} - \frac{347}{2597} a^{8} - \frac{926}{2597} a^{7} + \frac{107}{371} a^{6} + \frac{1006}{2597} a^{5} - \frac{750}{2597} a^{4} - \frac{153}{2597} a^{3} + \frac{551}{2597} a^{2} + \frac{629}{2597} a - \frac{491}{2597}$, $\frac{1}{2597} a^{31} - \frac{3}{2597} a^{27} + \frac{25}{2597} a^{26} + \frac{62}{2597} a^{25} + \frac{153}{2597} a^{24} - \frac{144}{2597} a^{23} - \frac{8}{2597} a^{22} - \frac{22}{2597} a^{21} + \frac{108}{2597} a^{20} - \frac{92}{2597} a^{19} - \frac{1}{49} a^{18} + \frac{320}{2597} a^{17} + \frac{976}{2597} a^{16} + \frac{830}{2597} a^{15} + \frac{555}{2597} a^{14} + \frac{99}{371} a^{13} + \frac{33}{2597} a^{12} + \frac{874}{2597} a^{11} + \frac{886}{2597} a^{10} + \frac{299}{2597} a^{9} - \frac{509}{2597} a^{8} + \frac{73}{371} a^{7} - \frac{422}{2597} a^{6} + \frac{183}{2597} a^{5} - \frac{192}{2597} a^{4} - \frac{498}{2597} a^{3} + \frac{443}{2597} a^{2} + \frac{845}{2597} a + \frac{645}{2597}$, $\frac{1}{1351221697} a^{32} + \frac{125553}{1351221697} a^{31} - \frac{4713}{27575953} a^{30} + \frac{43251}{1351221697} a^{29} - \frac{226649}{1351221697} a^{28} + \frac{1028331}{1351221697} a^{27} - \frac{84651745}{1351221697} a^{26} + \frac{5236701}{193031671} a^{25} + \frac{27103429}{1351221697} a^{24} + \frac{64090907}{1351221697} a^{23} + \frac{92186966}{1351221697} a^{22} + \frac{8513084}{1351221697} a^{21} - \frac{34677339}{1351221697} a^{20} - \frac{23807293}{1351221697} a^{19} + \frac{72300771}{1351221697} a^{18} + \frac{550999688}{1351221697} a^{17} + \frac{290702532}{1351221697} a^{16} + \frac{625242673}{1351221697} a^{15} + \frac{617066824}{1351221697} a^{14} + \frac{12640003}{1351221697} a^{13} - \frac{28412898}{1351221697} a^{12} + \frac{499793668}{1351221697} a^{11} - \frac{337881972}{1351221697} a^{10} + \frac{255729945}{1351221697} a^{9} + \frac{668611836}{1351221697} a^{8} - \frac{336940256}{1351221697} a^{7} + \frac{380935558}{1351221697} a^{6} - \frac{366517832}{1351221697} a^{5} + \frac{46714419}{1351221697} a^{4} - \frac{98211760}{1351221697} a^{3} - \frac{352093060}{1351221697} a^{2} + \frac{97509255}{1351221697} a + \frac{39039985}{1351221697}$, $\frac{1}{113223479828156167821332854024873823750866571420071838669311259715008754253750734277792556793463786658719418341030841265185578061132757124330562581775606157033369094289} a^{33} + \frac{145320330943290076680598930288400931212420605773478538849628038571219299882039020999177224226497017340970786952481042033538571370253486757855974590720042218}{2310683261799105465741486816834159668385032069797384462639005300306301107219402740363113403948240544055498333490425331942562817574137900496542093505624615449660593761} a^{32} + \frac{15329396561520875846419669555472709615625352729432051041273304627557368522524490018007706032773389970511112665849910623970269375460853791243681665604128159991951372}{113223479828156167821332854024873823750866571420071838669311259715008754253750734277792556793463786658719418341030841265185578061132757124330562581775606157033369094289} a^{31} - \frac{6537302888258961116329325847296937445526622893187378004384939651857400453302291976343900195312026504195109507269890117608351380970201321335628010056108045092837661}{113223479828156167821332854024873823750866571420071838669311259715008754253750734277792556793463786658719418341030841265185578061132757124330562581775606157033369094289} a^{30} + \frac{10189376531302268923438060582809661147969860903097968208826602235552714007113239795454225910462583478432388062938601036610380730658094721789762283623223922440942080}{113223479828156167821332854024873823750866571420071838669311259715008754253750734277792556793463786658719418341030841265185578061132757124330562581775606157033369094289} a^{29} - \frac{1069118149537156879101303101428760867038272461445910015929800273672620929819744152769537956767968604013231623700706799427607361021953309293804815920379707837033867}{113223479828156167821332854024873823750866571420071838669311259715008754253750734277792556793463786658719418341030841265185578061132757124330562581775606157033369094289} a^{28} - \frac{113226584342551959958654490757825621107802728256350705441625631301481403169125330551500790437588754893125791061422311973857130359377294794432106288546110316165624773}{113223479828156167821332854024873823750866571420071838669311259715008754253750734277792556793463786658719418341030841265185578061132757124330562581775606157033369094289} a^{27} + \frac{5623295236866717178229324608500672805003557046286228801229555712620008750042037886800170863681879379419828824633900509495619265599862897119899675786120966620660091127}{113223479828156167821332854024873823750866571420071838669311259715008754253750734277792556793463786658719418341030841265185578061132757124330562581775606157033369094289} a^{26} + \frac{5982292597496110097427549074763833708790915934918556877727516600040904267417389862266062802971853637143455061636767550936036809453662117436621210515508080363799179009}{113223479828156167821332854024873823750866571420071838669311259715008754253750734277792556793463786658719418341030841265185578061132757124330562581775606157033369094289} a^{25} + \frac{140972645481709664609318526773679400699505894076285642595414242553138398164132133956920418231949372862485727656495908720672298290123128246385539761967410115690615560}{16174782832593738260190407717839117678695224488581691238473037102144107750535819182541793827637683808388488334432977323597939723018965303475794654539372308147624156327} a^{24} - \frac{4983807693013958105367674986010622782586281534369485260799189110548456077652180691761686824243307387258836162069503812856271014500773745541605885315407877343636738780}{113223479828156167821332854024873823750866571420071838669311259715008754253750734277792556793463786658719418341030841265185578061132757124330562581775606157033369094289} a^{23} - \frac{3836460429287050932713720351163211395247220119328293008314957840153618453009640343198002481876448853784520133942895329827187038072462838820812782818185087088814186834}{113223479828156167821332854024873823750866571420071838669311259715008754253750734277792556793463786658719418341030841265185578061132757124330562581775606157033369094289} a^{22} - \frac{5758599573623147760752258670224011926857850147066707383451037157721892457640076280016447459441628472923109994054949560169915054195786473593593463255687626229947004568}{113223479828156167821332854024873823750866571420071838669311259715008754253750734277792556793463786658719418341030841265185578061132757124330562581775606157033369094289} a^{21} - \frac{2001637178144326714166706251908654679490216247957943739330481438108540817446150592598748030846549847413067497201194133783368029726071799887721256334362414971943790672}{113223479828156167821332854024873823750866571420071838669311259715008754253750734277792556793463786658719418341030841265185578061132757124330562581775606157033369094289} a^{20} + \frac{235330503644835400282715926147464517767383776037990717069159513195853899221762817978384548158325039496506797604738975231535945120036442907692179690878290836208182732}{113223479828156167821332854024873823750866571420071838669311259715008754253750734277792556793463786658719418341030841265185578061132757124330562581775606157033369094289} a^{19} - \frac{3601201923262169603098549701902586507586661587365997312812475879597413152979164875422174624430672436408160764150655138476626208003428162839156959659484637133728693351}{113223479828156167821332854024873823750866571420071838669311259715008754253750734277792556793463786658719418341030841265185578061132757124330562581775606157033369094289} a^{18} - \frac{201775021972811837138723829406232182423052628661278947629353922414160427433949921228464381976907526317391953410480413979055573651077630263314171927149003608303217089}{2310683261799105465741486816834159668385032069797384462639005300306301107219402740363113403948240544055498333490425331942562817574137900496542093505624615449660593761} a^{17} - \frac{48754396340608760543428064576053705907816667514320978505158284414351201390830087028618955381864798917790794485597060130836729299620934651803677359030011214569565119072}{113223479828156167821332854024873823750866571420071838669311259715008754253750734277792556793463786658719418341030841265185578061132757124330562581775606157033369094289} a^{16} - \frac{42740778631579145816096919404050037630607436238827672679126311420967967360056950275206330377647999405538837389351447883499201215851666372721580366275227828054734489011}{113223479828156167821332854024873823750866571420071838669311259715008754253750734277792556793463786658719418341030841265185578061132757124330562581775606157033369094289} a^{15} + \frac{50620068339097339207988769630266565919314902485755398577839948273090877959824170785872162428280587203661028207710505915635640205941336263834712742212956026886578597488}{113223479828156167821332854024873823750866571420071838669311259715008754253750734277792556793463786658719418341030841265185578061132757124330562581775606157033369094289} a^{14} - \frac{27557512199946435067809590593978477135779259700385125281476443840845037045833257223230881075671113713125150949098677384218787239623059755832399654987271597849746820228}{113223479828156167821332854024873823750866571420071838669311259715008754253750734277792556793463786658719418341030841265185578061132757124330562581775606157033369094289} a^{13} + \frac{16556132810947745583757412906442004267210208203555446286285225124489099664955495735158610823301364403613872405580241411110977364313313900395450443472698316986970695888}{113223479828156167821332854024873823750866571420071838669311259715008754253750734277792556793463786658719418341030841265185578061132757124330562581775606157033369094289} a^{12} + \frac{31393137736648698649921726389208768291125340799118671460511550700392537216096812716482936899089249240632807860524355172535442992914093346567878113568911493677589283446}{113223479828156167821332854024873823750866571420071838669311259715008754253750734277792556793463786658719418341030841265185578061132757124330562581775606157033369094289} a^{11} - \frac{9814371695408718962324194596151776648771387738510514985599565809406720369290790807620763884613936167798000309385320585840569158546533083341372202007205645904935758558}{113223479828156167821332854024873823750866571420071838669311259715008754253750734277792556793463786658719418341030841265185578061132757124330562581775606157033369094289} a^{10} - \frac{48624345997943479044695001001429230341431067643938611390521623502393888782199807181570932656752632245424313313281182504878744289409959028249463585139768131022618100209}{113223479828156167821332854024873823750866571420071838669311259715008754253750734277792556793463786658719418341030841265185578061132757124330562581775606157033369094289} a^{9} - \frac{818050622145772578764969850261413980570771474090207545703719379151925874014936292254047482125606999278680015581946463934954236902926945624434829920745857966009612422}{16174782832593738260190407717839117678695224488581691238473037102144107750535819182541793827637683808388488334432977323597939723018965303475794654539372308147624156327} a^{8} + \frac{37564666826484517519009916870574716444907360640248595436339253228892632213448541165510871494093429402209631310117376436609927877272975833189607877277347489124923868273}{113223479828156167821332854024873823750866571420071838669311259715008754253750734277792556793463786658719418341030841265185578061132757124330562581775606157033369094289} a^{7} + \frac{6955696983182504988807390257190198029749440616590454871488980463507635985769515817477071784818794723093282431936013440480764568079555345829521653218476337194664928744}{113223479828156167821332854024873823750866571420071838669311259715008754253750734277792556793463786658719418341030841265185578061132757124330562581775606157033369094289} a^{6} - \frac{681990054807609921521618592473699804547020597922566457274871496597913974041702096461869897033327887771847556209266213470306613559431353537976132509837220362902456256}{113223479828156167821332854024873823750866571420071838669311259715008754253750734277792556793463786658719418341030841265185578061132757124330562581775606157033369094289} a^{5} - \frac{17754955187044569099742175854093303963741658771584701388590911066094826499967929208615145670573319726407257816045717249722372473132395475276497587409393547534332656470}{113223479828156167821332854024873823750866571420071838669311259715008754253750734277792556793463786658719418341030841265185578061132757124330562581775606157033369094289} a^{4} - \frac{28121832696107793356807922093436374201016171536639312619091612751960499543508252172096166275869750699300284543408210753279994944384763275836355887701977709241990385620}{113223479828156167821332854024873823750866571420071838669311259715008754253750734277792556793463786658719418341030841265185578061132757124330562581775606157033369094289} a^{3} + \frac{30052456998324240421364474263056413839098383095577086567911902533816386206879728595587402137017488875251704939948225479322816019415137154128764880977451678869600861613}{113223479828156167821332854024873823750866571420071838669311259715008754253750734277792556793463786658719418341030841265185578061132757124330562581775606157033369094289} a^{2} - \frac{33566088299636105091295816715981130883804722996112491760133192554254451248924725062864796085710521048274709943185177604676461112063816288317006107176555317098969908891}{113223479828156167821332854024873823750866571420071838669311259715008754253750734277792556793463786658719418341030841265185578061132757124330562581775606157033369094289} a - \frac{48811478992787511288995965767020246429233627196298320714398153675590038142568094871766193634732876301875158333857465785615911709012085814119820370156661817145321666061}{113223479828156167821332854024873823750866571420071838669311259715008754253750734277792556793463786658719418341030841265185578061132757124330562581775606157033369094289}$

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:  $33$
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:  \( 379231148308800400000000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{34}$ (as 34T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 34
The 34 conjugacy class representatives for $C_{34}$
Character table for $C_{34}$ is not computed

Intermediate fields

\(\Q(\sqrt{409}) \), 17.17.613158747081871736694796283376344205805441.1

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 $17^{2}$ $17^{2}$ $17^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{17}$ $34$ $34$ $17^{2}$ $34$ $17^{2}$ $34$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{17}$ $34$ $17^{2}$ $34$ $34$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{34}$ $34$

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
409Data not computed