↑

Properties

Label 33.33.2985626020...7681.1
Degree $33$
Signature $[33, 0]$
Discriminant $3^{44}\cdot 89^{30}$
Root discriminant $256.06$
Ramified primes $3, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{33}$ (as 33T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![394127189, 463085148, -10943386668, -20763764254, 87344653842, 217064440848, -194929659142, -693751861143, 167558013330, 1074317235504, -29098948383, -969959674314, -55951112920, 559616885679, 52038465948, -216880006359, -23073983745, 58029594807, 6321530429, -10866675378, -1144629303, 1429876464, 139517502, -131654085, -11392799, 8367087, 608835, -356924, -20259, 9687, 378, -150, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^33 - 3*x^32 - 150*x^31 + 378*x^30 + 9687*x^29 - 20259*x^28 - 356924*x^27 + 608835*x^26 + 8367087*x^25 - 11392799*x^24 - 131654085*x^23 + 139517502*x^22 + 1429876464*x^21 - 1144629303*x^20 - 10866675378*x^19 + 6321530429*x^18 + 58029594807*x^17 - 23073983745*x^16 - 216880006359*x^15 + 52038465948*x^14 + 559616885679*x^13 - 55951112920*x^12 - 969959674314*x^11 - 29098948383*x^10 + 1074317235504*x^9 + 167558013330*x^8 - 693751861143*x^7 - 194929659142*x^6 + 217064440848*x^5 + 87344653842*x^4 - 20763764254*x^3 - 10943386668*x^2 + 463085148*x + 394127189)
 
gp: K = bnfinit(x^33 - 3*x^32 - 150*x^31 + 378*x^30 + 9687*x^29 - 20259*x^28 - 356924*x^27 + 608835*x^26 + 8367087*x^25 - 11392799*x^24 - 131654085*x^23 + 139517502*x^22 + 1429876464*x^21 - 1144629303*x^20 - 10866675378*x^19 + 6321530429*x^18 + 58029594807*x^17 - 23073983745*x^16 - 216880006359*x^15 + 52038465948*x^14 + 559616885679*x^13 - 55951112920*x^12 - 969959674314*x^11 - 29098948383*x^10 + 1074317235504*x^9 + 167558013330*x^8 - 693751861143*x^7 - 194929659142*x^6 + 217064440848*x^5 + 87344653842*x^4 - 20763764254*x^3 - 10943386668*x^2 + 463085148*x + 394127189, 1)
 

Normalized defining polynomial

\( x^{33} - 3 x^{32} - 150 x^{31} + 378 x^{30} + 9687 x^{29} - 20259 x^{28} - 356924 x^{27} + 608835 x^{26} + 8367087 x^{25} - 11392799 x^{24} - 131654085 x^{23} + 139517502 x^{22} + 1429876464 x^{21} - 1144629303 x^{20} - 10866675378 x^{19} + 6321530429 x^{18} + 58029594807 x^{17} - 23073983745 x^{16} - 216880006359 x^{15} + 52038465948 x^{14} + 559616885679 x^{13} - 55951112920 x^{12} - 969959674314 x^{11} - 29098948383 x^{10} + 1074317235504 x^{9} + 167558013330 x^{8} - 693751861143 x^{7} - 194929659142 x^{6} + 217064440848 x^{5} + 87344653842 x^{4} - 20763764254 x^{3} - 10943386668 x^{2} + 463085148 x + 394127189 \)

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

Invariants

Degree:  $33$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[33, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(29856260200966476270645992674393331456331864986925653442461868091060255697367681=3^{44}\cdot 89^{30}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $256.06$
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:  $3, 89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(801=3^{2}\cdot 89\)
Dirichlet character group:    $\lbrace$$\chi_{801}(256,·)$, $\chi_{801}(1,·)$, $\chi_{801}(4,·)$, $\chi_{801}(523,·)$, $\chi_{801}(268,·)$, $\chi_{801}(271,·)$, $\chi_{801}(16,·)$, $\chi_{801}(790,·)$, $\chi_{801}(535,·)$, $\chi_{801}(388,·)$, $\chi_{801}(538,·)$, $\chi_{801}(283,·)$, $\chi_{801}(550,·)$, $\chi_{801}(64,·)$, $\chi_{801}(67,·)$, $\chi_{801}(331,·)$, $\chi_{801}(334,·)$, $\chi_{801}(598,·)$, $\chi_{801}(601,·)$, $\chi_{801}(91,·)$, $\chi_{801}(655,·)$, $\chi_{801}(223,·)$, $\chi_{801}(97,·)$, $\chi_{801}(484,·)$, $\chi_{801}(358,·)$, $\chi_{801}(490,·)$, $\chi_{801}(364,·)$, $\chi_{801}(751,·)$, $\chi_{801}(625,·)$, $\chi_{801}(757,·)$, $\chi_{801}(631,·)$, $\chi_{801}(121,·)$, $\chi_{801}(217,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{37} a^{20} + \frac{3}{37} a^{19} - \frac{8}{37} a^{18} + \frac{18}{37} a^{17} - \frac{17}{37} a^{16} - \frac{2}{37} a^{14} + \frac{9}{37} a^{13} - \frac{11}{37} a^{12} + \frac{15}{37} a^{11} - \frac{10}{37} a^{10} - \frac{14}{37} a^{9} - \frac{18}{37} a^{7} + \frac{17}{37} a^{6} + \frac{16}{37} a^{5} + \frac{10}{37} a^{4} + \frac{9}{37} a^{3} - \frac{4}{37} a^{2} - \frac{14}{37} a + \frac{11}{37}$, $\frac{1}{37} a^{21} - \frac{17}{37} a^{19} + \frac{5}{37} a^{18} + \frac{3}{37} a^{17} + \frac{14}{37} a^{16} - \frac{2}{37} a^{15} + \frac{15}{37} a^{14} - \frac{1}{37} a^{13} + \frac{11}{37} a^{12} - \frac{18}{37} a^{11} + \frac{16}{37} a^{10} + \frac{5}{37} a^{9} - \frac{18}{37} a^{8} - \frac{3}{37} a^{7} + \frac{2}{37} a^{6} - \frac{1}{37} a^{5} + \frac{16}{37} a^{4} + \frac{6}{37} a^{3} - \frac{2}{37} a^{2} + \frac{16}{37} a + \frac{4}{37}$, $\frac{1}{37} a^{22} - \frac{18}{37} a^{19} + \frac{15}{37} a^{18} - \frac{13}{37} a^{17} + \frac{5}{37} a^{16} + \frac{15}{37} a^{15} + \frac{2}{37} a^{14} + \frac{16}{37} a^{13} + \frac{17}{37} a^{12} + \frac{12}{37} a^{11} - \frac{17}{37} a^{10} + \frac{3}{37} a^{9} - \frac{3}{37} a^{8} - \frac{8}{37} a^{7} - \frac{8}{37} a^{6} - \frac{8}{37} a^{5} - \frac{9}{37} a^{4} + \frac{3}{37} a^{3} - \frac{15}{37} a^{2} - \frac{12}{37} a + \frac{2}{37}$, $\frac{1}{37} a^{23} - \frac{5}{37} a^{19} - \frac{9}{37} a^{18} - \frac{4}{37} a^{17} + \frac{5}{37} a^{16} + \frac{2}{37} a^{15} + \frac{17}{37} a^{14} - \frac{6}{37} a^{13} - \frac{1}{37} a^{12} - \frac{6}{37} a^{11} + \frac{8}{37} a^{10} + \frac{4}{37} a^{9} - \frac{8}{37} a^{8} + \frac{1}{37} a^{7} + \frac{2}{37} a^{6} - \frac{17}{37} a^{5} - \frac{2}{37} a^{4} - \frac{1}{37} a^{3} - \frac{10}{37} a^{2} + \frac{9}{37} a + \frac{13}{37}$, $\frac{1}{37} a^{24} + \frac{6}{37} a^{19} - \frac{7}{37} a^{18} - \frac{16}{37} a^{17} - \frac{9}{37} a^{16} + \frac{17}{37} a^{15} - \frac{16}{37} a^{14} + \frac{7}{37} a^{13} + \frac{13}{37} a^{12} + \frac{9}{37} a^{11} - \frac{9}{37} a^{10} - \frac{4}{37} a^{9} + \frac{1}{37} a^{8} - \frac{14}{37} a^{7} - \frac{6}{37} a^{6} + \frac{4}{37} a^{5} + \frac{12}{37} a^{4} - \frac{2}{37} a^{3} - \frac{11}{37} a^{2} + \frac{17}{37} a + \frac{18}{37}$, $\frac{1}{37} a^{25} + \frac{12}{37} a^{19} - \frac{5}{37} a^{18} - \frac{6}{37} a^{17} + \frac{8}{37} a^{16} - \frac{16}{37} a^{15} - \frac{18}{37} a^{14} - \frac{4}{37} a^{13} + \frac{1}{37} a^{12} + \frac{12}{37} a^{11} - \frac{18}{37} a^{10} + \frac{11}{37} a^{9} - \frac{14}{37} a^{8} - \frac{9}{37} a^{7} + \frac{13}{37} a^{6} - \frac{10}{37} a^{5} + \frac{12}{37} a^{4} + \frac{9}{37} a^{3} + \frac{4}{37} a^{2} - \frac{9}{37} a + \frac{8}{37}$, $\frac{1}{37} a^{26} - \frac{4}{37} a^{19} + \frac{16}{37} a^{18} + \frac{14}{37} a^{17} + \frac{3}{37} a^{16} - \frac{18}{37} a^{15} - \frac{17}{37} a^{14} + \frac{4}{37} a^{13} - \frac{4}{37} a^{12} - \frac{13}{37} a^{11} - \frac{17}{37} a^{10} + \frac{6}{37} a^{9} - \frac{9}{37} a^{8} + \frac{7}{37} a^{7} + \frac{8}{37} a^{6} + \frac{5}{37} a^{5} + \frac{7}{37} a^{3} + \frac{2}{37} a^{2} - \frac{9}{37} a + \frac{16}{37}$, $\frac{1}{37} a^{27} - \frac{9}{37} a^{19} - \frac{18}{37} a^{18} + \frac{1}{37} a^{17} - \frac{12}{37} a^{16} - \frac{17}{37} a^{15} - \frac{4}{37} a^{14} - \frac{5}{37} a^{13} + \frac{17}{37} a^{12} + \frac{6}{37} a^{11} + \frac{3}{37} a^{10} + \frac{9}{37} a^{9} + \frac{7}{37} a^{8} + \frac{10}{37} a^{7} - \frac{1}{37} a^{6} - \frac{10}{37} a^{5} + \frac{10}{37} a^{4} + \frac{1}{37} a^{3} + \frac{12}{37} a^{2} - \frac{3}{37} a + \frac{7}{37}$, $\frac{1}{37} a^{28} + \frac{9}{37} a^{19} + \frac{3}{37} a^{18} + \frac{2}{37} a^{17} + \frac{15}{37} a^{16} - \frac{4}{37} a^{15} + \frac{14}{37} a^{14} - \frac{13}{37} a^{13} + \frac{18}{37} a^{12} - \frac{10}{37} a^{11} - \frac{7}{37} a^{10} - \frac{8}{37} a^{9} + \frac{10}{37} a^{8} - \frac{15}{37} a^{7} - \frac{5}{37} a^{6} + \frac{6}{37} a^{5} + \frac{17}{37} a^{4} - \frac{18}{37} a^{3} - \frac{2}{37} a^{2} - \frac{8}{37} a - \frac{12}{37}$, $\frac{1}{6623} a^{29} + \frac{14}{6623} a^{28} + \frac{78}{6623} a^{27} + \frac{52}{6623} a^{26} + \frac{87}{6623} a^{25} - \frac{10}{6623} a^{24} + \frac{80}{6623} a^{23} - \frac{16}{6623} a^{22} - \frac{23}{6623} a^{21} - \frac{2}{179} a^{20} + \frac{233}{6623} a^{19} + \frac{2544}{6623} a^{18} + \frac{2919}{6623} a^{17} - \frac{81}{6623} a^{16} - \frac{718}{6623} a^{15} - \frac{1270}{6623} a^{14} - \frac{1854}{6623} a^{13} - \frac{891}{6623} a^{12} + \frac{2722}{6623} a^{11} + \frac{178}{6623} a^{10} + \frac{2118}{6623} a^{9} + \frac{2867}{6623} a^{8} - \frac{570}{6623} a^{7} + \frac{32}{179} a^{6} + \frac{1610}{6623} a^{5} + \frac{3115}{6623} a^{4} - \frac{2723}{6623} a^{3} + \frac{2427}{6623} a^{2} + \frac{1777}{6623} a + \frac{249}{6623}$, $\frac{1}{677618999} a^{30} - \frac{16933}{677618999} a^{29} - \frac{167399}{18314027} a^{28} - \frac{3468382}{677618999} a^{27} + \frac{44442}{18314027} a^{26} + \frac{1564842}{677618999} a^{25} - \frac{4895076}{677618999} a^{24} + \frac{92871}{677618999} a^{23} - \frac{101728}{677618999} a^{22} + \frac{2206557}{677618999} a^{21} + \frac{8645937}{677618999} a^{20} - \frac{152042294}{677618999} a^{19} + \frac{335570516}{677618999} a^{18} + \frac{124553994}{677618999} a^{17} - \frac{247514802}{677618999} a^{16} + \frac{35841574}{677618999} a^{15} - \frac{212427510}{677618999} a^{14} - \frac{26662894}{677618999} a^{13} + \frac{929279}{677618999} a^{12} - \frac{319160583}{677618999} a^{11} + \frac{23263110}{677618999} a^{10} - \frac{192687003}{677618999} a^{9} - \frac{71439833}{677618999} a^{8} - \frac{195394803}{677618999} a^{7} + \frac{203448113}{677618999} a^{6} - \frac{310068977}{677618999} a^{5} - \frac{6481483}{18314027} a^{4} - \frac{155804599}{677618999} a^{3} - \frac{17340745}{677618999} a^{2} - \frac{25074288}{677618999} a + \frac{45639752}{677618999}$, $\frac{1}{25071902963} a^{31} + \frac{4}{25071902963} a^{30} - \frac{1600560}{25071902963} a^{29} + \frac{302248717}{25071902963} a^{28} + \frac{82824709}{25071902963} a^{27} + \frac{222567516}{25071902963} a^{26} + \frac{57471410}{25071902963} a^{25} + \frac{217615265}{25071902963} a^{24} - \frac{59856455}{25071902963} a^{23} - \frac{321016426}{25071902963} a^{22} - \frac{143316347}{25071902963} a^{21} + \frac{186139699}{25071902963} a^{20} + \frac{12402689004}{25071902963} a^{19} - \frac{10119246983}{25071902963} a^{18} + \frac{11068628854}{25071902963} a^{17} - \frac{8406690480}{25071902963} a^{16} - \frac{233781652}{25071902963} a^{15} + \frac{7763751105}{25071902963} a^{14} + \frac{3546576955}{25071902963} a^{13} + \frac{5761997579}{25071902963} a^{12} - \frac{9560243096}{25071902963} a^{11} + \frac{430944115}{25071902963} a^{10} - \frac{10404855340}{25071902963} a^{9} - \frac{9545775282}{25071902963} a^{8} - \frac{10916244523}{25071902963} a^{7} - \frac{1810553997}{25071902963} a^{6} + \frac{5622859713}{25071902963} a^{5} - \frac{5461158802}{25071902963} a^{4} + \frac{9152594145}{25071902963} a^{3} + \frac{2889834382}{25071902963} a^{2} + \frac{7393287903}{25071902963} a - \frac{10514929888}{25071902963}$, $\frac{1}{1775407938395215220716443955288324547141042231464231137252557020753255093059831989368580953756525138415647739294672260402429347547} a^{32} - \frac{22236932674998397167842214213778691929193543922646004122345496157673641622808351874835914978499292796436281029852981576}{1775407938395215220716443955288324547141042231464231137252557020753255093059831989368580953756525138415647739294672260402429347547} a^{31} - \frac{814256846560229479388711128852038919410486165128273472618757163531997385492907124855909187041461993117775719923467965726}{1775407938395215220716443955288324547141042231464231137252557020753255093059831989368580953756525138415647739294672260402429347547} a^{30} - \frac{2837991477024271364013004212134177611230804210387331475436479572317066685540257354532738631698657611611313239852401223839615}{1775407938395215220716443955288324547141042231464231137252557020753255093059831989368580953756525138415647739294672260402429347547} a^{29} + \frac{11408242823046909085626185880119763283923457587868443807589569669958645674022814156397800388915748590297145221163356273766793317}{1775407938395215220716443955288324547141042231464231137252557020753255093059831989368580953756525138415647739294672260402429347547} a^{28} + \frac{131600701803987527250232147887547839963390448382184724317428102842230814858398194904651254613717670620608722195189323530300561}{47983998335005816776120106899684447220028168417952192898717757317655543055671134847799485236662841578801290251207358389254847231} a^{27} + \frac{17860359960376877335424095582000907009544266414845057813052975830413108209012574021123954740545682942114908678626194086430543644}{1775407938395215220716443955288324547141042231464231137252557020753255093059831989368580953756525138415647739294672260402429347547} a^{26} + \frac{3754740068734648575705651613707480384754610376028039323006750483824447479581970361666065273451833220522311323374372950170202707}{1775407938395215220716443955288324547141042231464231137252557020753255093059831989368580953756525138415647739294672260402429347547} a^{25} - \frac{15623691093478919717845183552769835628393456456697394966081282186006560313666170854972673468371757225166739604144826421609456323}{1775407938395215220716443955288324547141042231464231137252557020753255093059831989368580953756525138415647739294672260402429347547} a^{24} + \frac{771045356628746989363110148333400216156293986787487979419852402008577511413716874610863397922050323485204740403420174386904976}{1775407938395215220716443955288324547141042231464231137252557020753255093059831989368580953756525138415647739294672260402429347547} a^{23} + \frac{868341367682136451695562977670074417818406300853835753481939362014776849316852591125186821604534890444336184952520377659995146}{1775407938395215220716443955288324547141042231464231137252557020753255093059831989368580953756525138415647739294672260402429347547} a^{22} + \frac{6682725019698290402035495291018884201877169075292231351533728791487043699482587300352574724239192611754018071240047071007504815}{1775407938395215220716443955288324547141042231464231137252557020753255093059831989368580953756525138415647739294672260402429347547} a^{21} - \frac{5627655041999929787389193140567390197250458843822825717803096716173076544068595396030696832314781883800790823933415035883011042}{1775407938395215220716443955288324547141042231464231137252557020753255093059831989368580953756525138415647739294672260402429347547} a^{20} + \frac{550041356039708745579127239302627066138725449817724988186015478770670177496374452235313863819197765621170878697178402765556075426}{1775407938395215220716443955288324547141042231464231137252557020753255093059831989368580953756525138415647739294672260402429347547} a^{19} - \frac{134144523790635810566513156764589546183670690672304610617545849017299543205830034876261046857716273855360812764707680601395940165}{1775407938395215220716443955288324547141042231464231137252557020753255093059831989368580953756525138415647739294672260402429347547} a^{18} - \frac{599446893631186430897861895961020988817462095982232175780501453366067649035180249526476886603858949106744491055632375380970022868}{1775407938395215220716443955288324547141042231464231137252557020753255093059831989368580953756525138415647739294672260402429347547} a^{17} - \frac{35662301411514365016269498371370836576399569121404335522172371653001608919509161080837113448516151916823219421956881259660733858}{1775407938395215220716443955288324547141042231464231137252557020753255093059831989368580953756525138415647739294672260402429347547} a^{16} + \frac{135091461205389984572380169491781869344153488310742008691437508284347133755216876243765230135178055701148442819001667347017528315}{1775407938395215220716443955288324547141042231464231137252557020753255093059831989368580953756525138415647739294672260402429347547} a^{15} + \frac{108256232039911496724651683446706496556483065001918701639771219957126092180266097189063283025454046452154767254469131380591621152}{1775407938395215220716443955288324547141042231464231137252557020753255093059831989368580953756525138415647739294672260402429347547} a^{14} - \frac{112680696763875780823942831539745493344969554236495213739382625596952377501886007787878168519710306891858789321885538937103969251}{1775407938395215220716443955288324547141042231464231137252557020753255093059831989368580953756525138415647739294672260402429347547} a^{13} + \frac{393532213492341518871769490436252344176040459543724238731802983004888689215466565771869618339650415191203161910421164986535241476}{1775407938395215220716443955288324547141042231464231137252557020753255093059831989368580953756525138415647739294672260402429347547} a^{12} + \frac{656503768661737100952522297373560570937741457404641565768903850972008478714842179979181402264935319072284724834049467102640247103}{1775407938395215220716443955288324547141042231464231137252557020753255093059831989368580953756525138415647739294672260402429347547} a^{11} + \frac{547688798625126010066779334608673903909989052518780393191594625763455996324776066632742276518057831632861107593490903199746913672}{1775407938395215220716443955288324547141042231464231137252557020753255093059831989368580953756525138415647739294672260402429347547} a^{10} - \frac{712017107434328256749338821652458640913701313255684769256115968148176171801989881341481767681015977916931639709343038464698078205}{1775407938395215220716443955288324547141042231464231137252557020753255093059831989368580953756525138415647739294672260402429347547} a^{9} + \frac{408892830889519575250243882962444439963136326442622128524726578620012390437244494473457694814090450108251541827412172398244423248}{1775407938395215220716443955288324547141042231464231137252557020753255093059831989368580953756525138415647739294672260402429347547} a^{8} - \frac{269883229790475833237223990177283262606577486061949165634358581598568228388188087231319791281815208118661967139470264073884529371}{1775407938395215220716443955288324547141042231464231137252557020753255093059831989368580953756525138415647739294672260402429347547} a^{7} - \frac{650246970140790997069153105843790893269700414075532759541878880322062051381988745095224990216923808263418436805955042842678199349}{1775407938395215220716443955288324547141042231464231137252557020753255093059831989368580953756525138415647739294672260402429347547} a^{6} - \frac{535085435098199314792535825418871175758798971801097615913024317533375235451774625588903492492887672337883011145277803382287363630}{1775407938395215220716443955288324547141042231464231137252557020753255093059831989368580953756525138415647739294672260402429347547} a^{5} - \frac{669321651773543283869984775119094354772753361122099832458983446045636639491756985269641055850990775375066043807146006948400019656}{1775407938395215220716443955288324547141042231464231137252557020753255093059831989368580953756525138415647739294672260402429347547} a^{4} - \frac{412330640061641542154997766757604977320976912261703577669061699704499892730195433840803658510779738751807415390779212277338380867}{1775407938395215220716443955288324547141042231464231137252557020753255093059831989368580953756525138415647739294672260402429347547} a^{3} + \frac{668071644477511733556965905585242006010688748245474230473169217502418297224767180339037549670235884977056172598908109687999126117}{1775407938395215220716443955288324547141042231464231137252557020753255093059831989368580953756525138415647739294672260402429347547} a^{2} - \frac{353570263615142522785615258891239601649111416936144414314999950266681356385531808650484717208568899006403111180751455578637194608}{1775407938395215220716443955288324547141042231464231137252557020753255093059831989368580953756525138415647739294672260402429347547} a + \frac{368862167943863986391410233627678545371782812900173395537939920118967781336586991708640323473920900319041818536911046229619480181}{1775407938395215220716443955288324547141042231464231137252557020753255093059831989368580953756525138415647739294672260402429347547}$

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:  $32$
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:  \( 391930822437712850000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{33}$ (as 33T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 33
The 33 conjugacy class representatives for $C_{33}$
Character table for $C_{33}$ is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 11.11.31181719929966183601.1

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 $33$ R $33$ $33$ $33$ $33$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{3}$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{3}$ $33$ $33$ $33$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{33}$ $33$ $33$ $33$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{3}$ $33$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
3Data not computed
$89$89.11.10.1$x^{11} - 89$$11$$1$$10$$C_{11}$$[\ ]_{11}$
89.11.10.1$x^{11} - 89$$11$$1$$10$$C_{11}$$[\ ]_{11}$
89.11.10.1$x^{11} - 89$$11$$1$$10$$C_{11}$$[\ ]_{11}$