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Properties

Label 29.1.518...696.1
Degree $29$
Signature $[1, 14]$
Discriminant $5.186\times 10^{74}$
Root discriminant \(377.03\)
Ramified primes $2,233$
Class number not computed
Class group not computed
Galois group $D_{29}$ (as 29T2)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^29 - 11*x^28 + 195*x^27 - 1437*x^26 + 15205*x^25 - 155107*x^24 + 689275*x^23 - 5195005*x^22 + 26758319*x^21 - 56551537*x^20 + 352159969*x^19 + 1341875833*x^18 - 12787919441*x^17 + 110486806247*x^16 - 277033483183*x^15 + 1655495992937*x^14 - 1367495435876*x^13 + 1228765225000*x^12 + 26218345289248*x^11 + 182179101160232*x^10 + 1253529597788960*x^9 + 1798908522252752*x^8 + 6631547125034608*x^7 + 55953424596411280*x^6 + 146264587683703680*x^5 + 405537226671593344*x^4 + 919709516139564800*x^3 + 1198381256681144576*x^2 + 1255295926519726080*x + 893279166685773824)
 
gp: K = bnfinit(y^29 - 11*y^28 + 195*y^27 - 1437*y^26 + 15205*y^25 - 155107*y^24 + 689275*y^23 - 5195005*y^22 + 26758319*y^21 - 56551537*y^20 + 352159969*y^19 + 1341875833*y^18 - 12787919441*y^17 + 110486806247*y^16 - 277033483183*y^15 + 1655495992937*y^14 - 1367495435876*y^13 + 1228765225000*y^12 + 26218345289248*y^11 + 182179101160232*y^10 + 1253529597788960*y^9 + 1798908522252752*y^8 + 6631547125034608*y^7 + 55953424596411280*y^6 + 146264587683703680*y^5 + 405537226671593344*y^4 + 919709516139564800*y^3 + 1198381256681144576*y^2 + 1255295926519726080*y + 893279166685773824, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^29 - 11*x^28 + 195*x^27 - 1437*x^26 + 15205*x^25 - 155107*x^24 + 689275*x^23 - 5195005*x^22 + 26758319*x^21 - 56551537*x^20 + 352159969*x^19 + 1341875833*x^18 - 12787919441*x^17 + 110486806247*x^16 - 277033483183*x^15 + 1655495992937*x^14 - 1367495435876*x^13 + 1228765225000*x^12 + 26218345289248*x^11 + 182179101160232*x^10 + 1253529597788960*x^9 + 1798908522252752*x^8 + 6631547125034608*x^7 + 55953424596411280*x^6 + 146264587683703680*x^5 + 405537226671593344*x^4 + 919709516139564800*x^3 + 1198381256681144576*x^2 + 1255295926519726080*x + 893279166685773824);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^29 - 11*x^28 + 195*x^27 - 1437*x^26 + 15205*x^25 - 155107*x^24 + 689275*x^23 - 5195005*x^22 + 26758319*x^21 - 56551537*x^20 + 352159969*x^19 + 1341875833*x^18 - 12787919441*x^17 + 110486806247*x^16 - 277033483183*x^15 + 1655495992937*x^14 - 1367495435876*x^13 + 1228765225000*x^12 + 26218345289248*x^11 + 182179101160232*x^10 + 1253529597788960*x^9 + 1798908522252752*x^8 + 6631547125034608*x^7 + 55953424596411280*x^6 + 146264587683703680*x^5 + 405537226671593344*x^4 + 919709516139564800*x^3 + 1198381256681144576*x^2 + 1255295926519726080*x + 893279166685773824)
 

\( x^{29} - 11 x^{28} + 195 x^{27} - 1437 x^{26} + 15205 x^{25} - 155107 x^{24} + 689275 x^{23} + \cdots + 89\!\cdots\!24 \) Copy content Toggle raw display

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

Invariants

Degree:  $29$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[1, 14]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(518567944687111380182898551430862041255874608320223181011467052276772765696\) \(\medspace = 2^{28}\cdot 233^{28}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(377.03\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2\cdot 233^{28/29}\approx 386.147256755339$
Ramified primes:   \(2\), \(233\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $1$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{12}a^{7}-\frac{1}{12}a^{6}+\frac{1}{12}a^{5}-\frac{1}{12}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{24}a^{8}-\frac{1}{4}a^{5}-\frac{1}{8}a^{4}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}+\frac{1}{3}$, $\frac{1}{24}a^{9}+\frac{1}{8}a^{5}-\frac{1}{6}a$, $\frac{1}{48}a^{10}-\frac{1}{48}a^{9}-\frac{1}{24}a^{7}-\frac{1}{48}a^{6}+\frac{1}{48}a^{5}+\frac{1}{6}a^{4}+\frac{5}{24}a^{3}-\frac{1}{6}a^{2}+\frac{1}{6}a-\frac{1}{3}$, $\frac{1}{48}a^{11}-\frac{1}{48}a^{9}+\frac{1}{48}a^{7}-\frac{1}{12}a^{6}+\frac{1}{48}a^{5}+\frac{1}{6}a^{4}+\frac{1}{8}a^{3}-\frac{1}{12}a^{2}+\frac{1}{6}a-\frac{1}{3}$, $\frac{1}{48}a^{12}-\frac{1}{48}a^{9}-\frac{1}{48}a^{8}-\frac{1}{24}a^{7}-\frac{1}{12}a^{6}+\frac{1}{48}a^{5}-\frac{1}{6}a^{4}+\frac{5}{24}a^{3}-\frac{1}{12}a^{2}+\frac{1}{6}a$, $\frac{1}{48}a^{13}-\frac{1}{24}a^{7}-\frac{1}{12}a^{6}-\frac{3}{16}a^{5}+\frac{1}{6}a^{4}+\frac{5}{24}a^{3}-\frac{1}{12}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{576}a^{14}-\frac{5}{576}a^{13}-\frac{1}{192}a^{12}-\frac{1}{576}a^{11}-\frac{1}{576}a^{10}-\frac{1}{192}a^{9}+\frac{7}{576}a^{8}+\frac{13}{576}a^{7}-\frac{1}{18}a^{6}+\frac{1}{8}a^{5}+\frac{11}{144}a^{4}+\frac{17}{144}a^{3}-\frac{1}{12}a^{2}+\frac{1}{36}a-\frac{2}{9}$, $\frac{1}{1152}a^{15}-\frac{1}{288}a^{13}+\frac{1}{144}a^{12}-\frac{1}{192}a^{11}-\frac{1}{144}a^{10}+\frac{1}{72}a^{9}-\frac{1}{48}a^{8}+\frac{11}{384}a^{7}-\frac{11}{144}a^{6}-\frac{31}{288}a^{5}+\frac{1}{24}a^{4}+\frac{1}{288}a^{3}-\frac{4}{9}a^{2}+\frac{1}{8}a+\frac{4}{9}$, $\frac{1}{1152}a^{16}-\frac{1}{96}a^{13}+\frac{1}{192}a^{12}-\frac{1}{96}a^{11}-\frac{1}{96}a^{10}+\frac{1}{96}a^{9}-\frac{11}{1152}a^{8}-\frac{1}{32}a^{7}-\frac{1}{32}a^{6}-\frac{1}{12}a^{5}+\frac{19}{96}a^{4}-\frac{5}{24}a^{3}-\frac{5}{24}a^{2}-\frac{1}{6}a-\frac{4}{9}$, $\frac{1}{2304}a^{17}-\frac{1}{1152}a^{14}+\frac{1}{576}a^{13}-\frac{1}{128}a^{12}+\frac{1}{1152}a^{11}+\frac{1}{1152}a^{10}+\frac{7}{2304}a^{9}+\frac{5}{1152}a^{8}+\frac{35}{1152}a^{7}+\frac{1}{9}a^{6}+\frac{31}{192}a^{5}+\frac{61}{288}a^{4}+\frac{139}{288}a^{3}+\frac{3}{8}a^{2}-\frac{11}{72}a-\frac{2}{9}$, $\frac{1}{2304}a^{18}-\frac{1}{384}a^{13}-\frac{1}{128}a^{12}-\frac{1}{384}a^{11}-\frac{5}{2304}a^{10}+\frac{1}{384}a^{9}+\frac{7}{384}a^{8}-\frac{1}{128}a^{7}-\frac{7}{64}a^{6}+\frac{1}{6}a^{5}-\frac{7}{32}a^{4}-\frac{11}{96}a^{3}-\frac{19}{72}a^{2}+\frac{5}{24}a+\frac{1}{3}$, $\frac{1}{13824}a^{19}-\frac{1}{6912}a^{18}-\frac{1}{13824}a^{17}+\frac{1}{6912}a^{16}+\frac{1}{3456}a^{15}-\frac{11}{6912}a^{13}+\frac{5}{864}a^{12}-\frac{95}{13824}a^{11}+\frac{1}{6912}a^{10}-\frac{7}{512}a^{9}-\frac{53}{6912}a^{8}+\frac{3}{256}a^{7}-\frac{53}{432}a^{6}-\frac{835}{3456}a^{5}-\frac{23}{576}a^{4}+\frac{127}{1728}a^{3}+\frac{19}{432}a^{2}+\frac{145}{432}a-\frac{1}{27}$, $\frac{1}{27648}a^{20}+\frac{1}{27648}a^{18}-\frac{1}{4608}a^{17}+\frac{1}{3456}a^{16}-\frac{1}{6912}a^{15}+\frac{7}{13824}a^{14}+\frac{1}{144}a^{13}+\frac{185}{27648}a^{12}-\frac{47}{6912}a^{11}-\frac{155}{27648}a^{10}+\frac{55}{13824}a^{9}+\frac{155}{13824}a^{8}+\frac{5}{432}a^{7}-\frac{119}{2304}a^{6}+\frac{83}{3456}a^{5}+\frac{601}{3456}a^{4}-\frac{7}{16}a^{3}-\frac{1}{96}a^{2}-\frac{101}{216}a-\frac{7}{27}$, $\frac{1}{27648}a^{21}-\frac{1}{27648}a^{19}-\frac{1}{13824}a^{18}-\frac{1}{13824}a^{17}-\frac{1}{3456}a^{16}+\frac{1}{4608}a^{15}-\frac{1}{1152}a^{14}+\frac{229}{27648}a^{13}+\frac{1}{2304}a^{12}+\frac{251}{27648}a^{11}-\frac{127}{13824}a^{10}+\frac{43}{6912}a^{9}+\frac{115}{6912}a^{8}-\frac{23}{576}a^{7}+\frac{97}{1152}a^{6}-\frac{65}{1728}a^{5}+\frac{29}{576}a^{4}+\frac{449}{1728}a^{3}+\frac{85}{432}a^{2}-\frac{107}{432}a-\frac{8}{27}$, $\frac{1}{55296}a^{22}-\frac{1}{55296}a^{21}-\frac{1}{55296}a^{20}+\frac{1}{55296}a^{19}-\frac{1}{13824}a^{18}-\frac{1}{6912}a^{17}-\frac{1}{9216}a^{16}-\frac{11}{27648}a^{15}+\frac{13}{55296}a^{14}+\frac{25}{18432}a^{13}-\frac{59}{18432}a^{12}-\frac{167}{55296}a^{11}+\frac{191}{27648}a^{10}+\frac{17}{1024}a^{9}-\frac{37}{2304}a^{8}-\frac{163}{4608}a^{7}+\frac{7}{6912}a^{6}-\frac{131}{768}a^{5}+\frac{659}{3456}a^{4}+\frac{73}{576}a^{3}+\frac{349}{864}a^{2}-\frac{7}{27}a-\frac{7}{27}$, $\frac{1}{110592}a^{23}-\frac{1}{55296}a^{21}-\frac{1}{55296}a^{20}+\frac{1}{110592}a^{19}-\frac{11}{55296}a^{18}-\frac{1}{18432}a^{17}+\frac{1}{9216}a^{16}+\frac{5}{36864}a^{15}-\frac{1}{3072}a^{14}+\frac{241}{55296}a^{13}-\frac{485}{55296}a^{12}-\frac{269}{110592}a^{11}-\frac{7}{55296}a^{10}-\frac{41}{55296}a^{9}+\frac{7}{1152}a^{8}+\frac{511}{27648}a^{7}-\frac{199}{13824}a^{6}+\frac{175}{13824}a^{5}+\frac{533}{3456}a^{4}-\frac{131}{288}a^{3}+\frac{11}{54}a^{2}-\frac{131}{288}a+\frac{1}{27}$, $\frac{1}{2322432}a^{24}+\frac{1}{258048}a^{23}-\frac{1}{1161216}a^{22}+\frac{1}{193536}a^{21}+\frac{13}{774144}a^{20}+\frac{11}{2322432}a^{19}+\frac{1}{21504}a^{18}+\frac{5}{43008}a^{17}-\frac{341}{2322432}a^{16}-\frac{41}{110592}a^{15}-\frac{65}{129024}a^{14}-\frac{989}{290304}a^{13}-\frac{979}{110592}a^{12}-\frac{1529}{774144}a^{11}-\frac{2215}{290304}a^{10}+\frac{241}{43008}a^{9}+\frac{61}{193536}a^{8}-\frac{2179}{580608}a^{7}+\frac{725}{24192}a^{6}-\frac{18299}{96768}a^{5}+\frac{4133}{72576}a^{4}+\frac{1783}{12096}a^{3}-\frac{617}{2016}a^{2}+\frac{295}{2592}a+\frac{176}{567}$, $\frac{1}{25546752}a^{25}+\frac{1}{25546752}a^{24}+\frac{47}{12773376}a^{23}-\frac{1}{456192}a^{22}-\frac{19}{8515584}a^{21}+\frac{41}{3649536}a^{20}-\frac{163}{6386688}a^{19}-\frac{145}{1419264}a^{18}+\frac{2203}{25546752}a^{17}-\frac{1997}{25546752}a^{16}-\frac{1651}{4257792}a^{15}+\frac{367}{580608}a^{14}-\frac{3965}{2322432}a^{13}-\frac{1247}{2838528}a^{12}-\frac{79}{798336}a^{11}+\frac{6911}{12773376}a^{10}-\frac{27683}{2128896}a^{9}-\frac{11119}{6386688}a^{8}+\frac{1223}{57024}a^{7}+\frac{32227}{354816}a^{6}-\frac{99307}{798336}a^{5}-\frac{8401}{36288}a^{4}-\frac{1397}{6048}a^{3}-\frac{61955}{199584}a^{2}-\frac{524}{6237}a-\frac{65}{567}$, $\frac{1}{102187008}a^{26}+\frac{1}{102187008}a^{25}+\frac{1}{17031168}a^{24}-\frac{193}{51093504}a^{23}-\frac{115}{14598144}a^{22}-\frac{1693}{102187008}a^{21}-\frac{97}{25546752}a^{20}-\frac{317}{25546752}a^{19}+\frac{3049}{14598144}a^{18}-\frac{21137}{102187008}a^{17}-\frac{19517}{51093504}a^{16}-\frac{211}{4644864}a^{15}+\frac{1447}{9289728}a^{14}+\frac{1051861}{102187008}a^{13}-\frac{75853}{12773376}a^{12}-\frac{45817}{12773376}a^{11}-\frac{65015}{12773376}a^{10}-\frac{67055}{12773376}a^{9}+\frac{46063}{3193344}a^{8}+\frac{95}{3193344}a^{7}-\frac{445355}{6386688}a^{6}-\frac{93001}{580608}a^{5}-\frac{11995}{72576}a^{4}-\frac{35015}{114048}a^{3}+\frac{123913}{399168}a^{2}-\frac{1045}{4032}a-\frac{79}{567}$, $\frac{1}{254805552562176}a^{27}-\frac{119935}{28311728062464}a^{26}+\frac{273913}{127402776281088}a^{25}+\frac{22191131}{127402776281088}a^{24}+\frac{265158331}{254805552562176}a^{23}+\frac{587267}{95325683712}a^{22}-\frac{16250977}{10616898023424}a^{21}-\frac{889912393}{63701388140544}a^{20}-\frac{19424891}{1048582520832}a^{19}+\frac{3622666687}{28311728062464}a^{18}-\frac{796351757}{5539251142656}a^{17}+\frac{279959209}{42467592093696}a^{16}-\frac{3027266939}{28311728062464}a^{15}-\frac{165012559627}{254805552562176}a^{14}+\frac{71878582901}{21233796046848}a^{13}+\frac{9838666625}{1179655335936}a^{12}+\frac{281067073}{197830397952}a^{11}+\frac{2865856061}{3538966007808}a^{10}-\frac{348704803}{57700532736}a^{9}+\frac{50203254133}{7962673517568}a^{8}-\frac{178560239273}{5308449011712}a^{7}-\frac{455467109177}{5308449011712}a^{6}+\frac{172361952667}{3981336758784}a^{5}-\frac{684623087}{3205585152}a^{4}+\frac{266037230777}{995334189696}a^{3}+\frac{457562233685}{995334189696}a^{2}+\frac{101769083}{1545549984}a+\frac{18729152}{706913487}$, $\frac{1}{30\!\cdots\!48}a^{28}+\frac{16\!\cdots\!47}{37\!\cdots\!56}a^{27}+\frac{36\!\cdots\!17}{30\!\cdots\!48}a^{26}-\frac{45\!\cdots\!37}{41\!\cdots\!68}a^{25}+\frac{35\!\cdots\!87}{20\!\cdots\!84}a^{24}-\frac{92\!\cdots\!09}{37\!\cdots\!56}a^{23}-\frac{27\!\cdots\!33}{33\!\cdots\!16}a^{22}+\frac{15\!\cdots\!81}{37\!\cdots\!56}a^{21}+\frac{29\!\cdots\!63}{19\!\cdots\!12}a^{20}-\frac{14\!\cdots\!55}{91\!\cdots\!04}a^{19}-\frac{36\!\cdots\!17}{30\!\cdots\!48}a^{18}-\frac{10\!\cdots\!15}{48\!\cdots\!84}a^{17}-\frac{38\!\cdots\!13}{91\!\cdots\!56}a^{16}+\frac{16\!\cdots\!39}{37\!\cdots\!56}a^{15}+\frac{72\!\cdots\!89}{27\!\cdots\!68}a^{14}-\frac{54\!\cdots\!03}{17\!\cdots\!36}a^{13}+\frac{40\!\cdots\!89}{47\!\cdots\!32}a^{12}+\frac{39\!\cdots\!61}{37\!\cdots\!56}a^{11}-\frac{25\!\cdots\!41}{12\!\cdots\!52}a^{10}+\frac{21\!\cdots\!57}{18\!\cdots\!28}a^{9}-\frac{46\!\cdots\!21}{24\!\cdots\!64}a^{8}+\frac{53\!\cdots\!47}{10\!\cdots\!04}a^{7}-\frac{20\!\cdots\!49}{18\!\cdots\!28}a^{6}+\frac{95\!\cdots\!63}{47\!\cdots\!32}a^{5}-\frac{50\!\cdots\!01}{33\!\cdots\!88}a^{4}+\frac{91\!\cdots\!03}{21\!\cdots\!04}a^{3}-\frac{65\!\cdots\!19}{16\!\cdots\!44}a^{2}+\frac{12\!\cdots\!71}{29\!\cdots\!52}a+\frac{10\!\cdots\!78}{41\!\cdots\!13}$ Copy content Toggle raw display

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

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$, $3$

Class group and class number

not computed

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $14$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^29 - 11*x^28 + 195*x^27 - 1437*x^26 + 15205*x^25 - 155107*x^24 + 689275*x^23 - 5195005*x^22 + 26758319*x^21 - 56551537*x^20 + 352159969*x^19 + 1341875833*x^18 - 12787919441*x^17 + 110486806247*x^16 - 277033483183*x^15 + 1655495992937*x^14 - 1367495435876*x^13 + 1228765225000*x^12 + 26218345289248*x^11 + 182179101160232*x^10 + 1253529597788960*x^9 + 1798908522252752*x^8 + 6631547125034608*x^7 + 55953424596411280*x^6 + 146264587683703680*x^5 + 405537226671593344*x^4 + 919709516139564800*x^3 + 1198381256681144576*x^2 + 1255295926519726080*x + 893279166685773824)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^29 - 11*x^28 + 195*x^27 - 1437*x^26 + 15205*x^25 - 155107*x^24 + 689275*x^23 - 5195005*x^22 + 26758319*x^21 - 56551537*x^20 + 352159969*x^19 + 1341875833*x^18 - 12787919441*x^17 + 110486806247*x^16 - 277033483183*x^15 + 1655495992937*x^14 - 1367495435876*x^13 + 1228765225000*x^12 + 26218345289248*x^11 + 182179101160232*x^10 + 1253529597788960*x^9 + 1798908522252752*x^8 + 6631547125034608*x^7 + 55953424596411280*x^6 + 146264587683703680*x^5 + 405537226671593344*x^4 + 919709516139564800*x^3 + 1198381256681144576*x^2 + 1255295926519726080*x + 893279166685773824, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^29 - 11*x^28 + 195*x^27 - 1437*x^26 + 15205*x^25 - 155107*x^24 + 689275*x^23 - 5195005*x^22 + 26758319*x^21 - 56551537*x^20 + 352159969*x^19 + 1341875833*x^18 - 12787919441*x^17 + 110486806247*x^16 - 277033483183*x^15 + 1655495992937*x^14 - 1367495435876*x^13 + 1228765225000*x^12 + 26218345289248*x^11 + 182179101160232*x^10 + 1253529597788960*x^9 + 1798908522252752*x^8 + 6631547125034608*x^7 + 55953424596411280*x^6 + 146264587683703680*x^5 + 405537226671593344*x^4 + 919709516139564800*x^3 + 1198381256681144576*x^2 + 1255295926519726080*x + 893279166685773824);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^29 - 11*x^28 + 195*x^27 - 1437*x^26 + 15205*x^25 - 155107*x^24 + 689275*x^23 - 5195005*x^22 + 26758319*x^21 - 56551537*x^20 + 352159969*x^19 + 1341875833*x^18 - 12787919441*x^17 + 110486806247*x^16 - 277033483183*x^15 + 1655495992937*x^14 - 1367495435876*x^13 + 1228765225000*x^12 + 26218345289248*x^11 + 182179101160232*x^10 + 1253529597788960*x^9 + 1798908522252752*x^8 + 6631547125034608*x^7 + 55953424596411280*x^6 + 146264587683703680*x^5 + 405537226671593344*x^4 + 919709516139564800*x^3 + 1198381256681144576*x^2 + 1255295926519726080*x + 893279166685773824);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$D_{29}$ (as 29T2):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 58
The 16 conjugacy class representatives for $D_{29}$
Character table for $D_{29}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.2.0.1}{2} }^{14}{,}\,{\href{/padicField/3.1.0.1}{1} }$ $29$ ${\href{/padicField/7.2.0.1}{2} }^{14}{,}\,{\href{/padicField/7.1.0.1}{1} }$ ${\href{/padicField/11.2.0.1}{2} }^{14}{,}\,{\href{/padicField/11.1.0.1}{1} }$ $29$ $29$ ${\href{/padicField/19.2.0.1}{2} }^{14}{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.2.0.1}{2} }^{14}{,}\,{\href{/padicField/23.1.0.1}{1} }$ $29$ ${\href{/padicField/31.2.0.1}{2} }^{14}{,}\,{\href{/padicField/31.1.0.1}{1} }$ $29$ $29$ ${\href{/padicField/43.2.0.1}{2} }^{14}{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.2.0.1}{2} }^{14}{,}\,{\href{/padicField/47.1.0.1}{1} }$ $29$ ${\href{/padicField/59.2.0.1}{2} }^{14}{,}\,{\href{/padicField/59.1.0.1}{1} }$

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

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

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display $\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
\(233\) Copy content Toggle raw display Deg $29$$29$$1$$28$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.4.2t1.a.a$1$ $ 2^{2}$ \(\Q(\sqrt{-1}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.217156.29t2.a.j$2$ $ 2^{2} \cdot 233^{2}$ 29.1.518567944687111380182898551430862041255874608320223181011467052276772765696.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.217156.29t2.a.e$2$ $ 2^{2} \cdot 233^{2}$ 29.1.518567944687111380182898551430862041255874608320223181011467052276772765696.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.217156.29t2.a.i$2$ $ 2^{2} \cdot 233^{2}$ 29.1.518567944687111380182898551430862041255874608320223181011467052276772765696.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.217156.29t2.a.a$2$ $ 2^{2} \cdot 233^{2}$ 29.1.518567944687111380182898551430862041255874608320223181011467052276772765696.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.217156.29t2.a.n$2$ $ 2^{2} \cdot 233^{2}$ 29.1.518567944687111380182898551430862041255874608320223181011467052276772765696.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.217156.29t2.a.d$2$ $ 2^{2} \cdot 233^{2}$ 29.1.518567944687111380182898551430862041255874608320223181011467052276772765696.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.217156.29t2.a.f$2$ $ 2^{2} \cdot 233^{2}$ 29.1.518567944687111380182898551430862041255874608320223181011467052276772765696.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.217156.29t2.a.g$2$ $ 2^{2} \cdot 233^{2}$ 29.1.518567944687111380182898551430862041255874608320223181011467052276772765696.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.217156.29t2.a.m$2$ $ 2^{2} \cdot 233^{2}$ 29.1.518567944687111380182898551430862041255874608320223181011467052276772765696.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.217156.29t2.a.k$2$ $ 2^{2} \cdot 233^{2}$ 29.1.518567944687111380182898551430862041255874608320223181011467052276772765696.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.217156.29t2.a.h$2$ $ 2^{2} \cdot 233^{2}$ 29.1.518567944687111380182898551430862041255874608320223181011467052276772765696.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.217156.29t2.a.b$2$ $ 2^{2} \cdot 233^{2}$ 29.1.518567944687111380182898551430862041255874608320223181011467052276772765696.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.217156.29t2.a.l$2$ $ 2^{2} \cdot 233^{2}$ 29.1.518567944687111380182898551430862041255874608320223181011467052276772765696.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.217156.29t2.a.c$2$ $ 2^{2} \cdot 233^{2}$ 29.1.518567944687111380182898551430862041255874608320223181011467052276772765696.1 $D_{29}$ (as 29T2) $1$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.