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Properties

Label 29.1.131...809.1
Degree $29$
Signature $[1, 14]$
Discriminant $1.310\times 10^{78}$
Root discriminant \(493.97\)
Ramified primes $7,233$
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 - 13*x^28 - 55*x^27 + 712*x^26 + 11351*x^25 - 127986*x^24 + 66127*x^23 + 4301081*x^22 - 11864947*x^21 - 126534305*x^20 + 781815928*x^19 + 394279608*x^18 - 18262666572*x^17 + 39703600309*x^16 + 333330481344*x^15 - 1935249895004*x^14 + 1288520615067*x^13 + 24287341394015*x^12 - 64823993632866*x^11 - 202566749188801*x^10 + 1440509151901048*x^9 - 2780340418135933*x^8 - 4068418180429144*x^7 + 3309165651131592*x^6 + 104817181048914519*x^5 - 418797024905501732*x^4 + 871709514773509500*x^3 - 878388606425292930*x^2 + 5204661351059478078*x - 13446098969951310345)
 
gp: K = bnfinit(y^29 - 13*y^28 - 55*y^27 + 712*y^26 + 11351*y^25 - 127986*y^24 + 66127*y^23 + 4301081*y^22 - 11864947*y^21 - 126534305*y^20 + 781815928*y^19 + 394279608*y^18 - 18262666572*y^17 + 39703600309*y^16 + 333330481344*y^15 - 1935249895004*y^14 + 1288520615067*y^13 + 24287341394015*y^12 - 64823993632866*y^11 - 202566749188801*y^10 + 1440509151901048*y^9 - 2780340418135933*y^8 - 4068418180429144*y^7 + 3309165651131592*y^6 + 104817181048914519*y^5 - 418797024905501732*y^4 + 871709514773509500*y^3 - 878388606425292930*y^2 + 5204661351059478078*y - 13446098969951310345, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^29 - 13*x^28 - 55*x^27 + 712*x^26 + 11351*x^25 - 127986*x^24 + 66127*x^23 + 4301081*x^22 - 11864947*x^21 - 126534305*x^20 + 781815928*x^19 + 394279608*x^18 - 18262666572*x^17 + 39703600309*x^16 + 333330481344*x^15 - 1935249895004*x^14 + 1288520615067*x^13 + 24287341394015*x^12 - 64823993632866*x^11 - 202566749188801*x^10 + 1440509151901048*x^9 - 2780340418135933*x^8 - 4068418180429144*x^7 + 3309165651131592*x^6 + 104817181048914519*x^5 - 418797024905501732*x^4 + 871709514773509500*x^3 - 878388606425292930*x^2 + 5204661351059478078*x - 13446098969951310345);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^29 - 13*x^28 - 55*x^27 + 712*x^26 + 11351*x^25 - 127986*x^24 + 66127*x^23 + 4301081*x^22 - 11864947*x^21 - 126534305*x^20 + 781815928*x^19 + 394279608*x^18 - 18262666572*x^17 + 39703600309*x^16 + 333330481344*x^15 - 1935249895004*x^14 + 1288520615067*x^13 + 24287341394015*x^12 - 64823993632866*x^11 - 202566749188801*x^10 + 1440509151901048*x^9 - 2780340418135933*x^8 - 4068418180429144*x^7 + 3309165651131592*x^6 + 104817181048914519*x^5 - 418797024905501732*x^4 + 871709514773509500*x^3 - 878388606425292930*x^2 + 5204661351059478078*x - 13446098969951310345)
 

\( x^{29} - 13 x^{28} - 55 x^{27} + 712 x^{26} + 11351 x^{25} - 127986 x^{24} + 66127 x^{23} + \cdots - 13\!\cdots\!45 \) 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:   \(1310202274198394060556452494826429036980688154429108157810117051380399416309809\) \(\medspace = 7^{14}\cdot 233^{28}\) 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:  \(493.97\)
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:  $7^{1/2}233^{28/29}\approx 510.8248054122166$
Ramified primes:   \(7\), \(233\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{3}$, $\frac{1}{21}a^{12}-\frac{3}{7}a^{6}-\frac{1}{3}a^{4}-\frac{2}{7}$, $\frac{1}{21}a^{13}-\frac{3}{7}a^{7}-\frac{1}{3}a^{5}-\frac{2}{7}a$, $\frac{1}{21}a^{14}-\frac{3}{7}a^{8}-\frac{1}{3}a^{6}-\frac{2}{7}a^{2}$, $\frac{1}{21}a^{15}-\frac{2}{21}a^{9}-\frac{1}{3}a^{7}-\frac{2}{7}a^{3}-\frac{1}{3}a$, $\frac{1}{315}a^{16}-\frac{2}{105}a^{15}-\frac{1}{45}a^{14}+\frac{2}{105}a^{13}-\frac{2}{315}a^{12}+\frac{47}{315}a^{10}-\frac{1}{35}a^{9}+\frac{11}{45}a^{8}-\frac{6}{35}a^{7}+\frac{109}{315}a^{6}-\frac{1}{5}a^{5}-\frac{11}{63}a^{4}+\frac{8}{21}a^{3}-\frac{1}{9}a^{2}+\frac{44}{105}a-\frac{3}{7}$, $\frac{1}{315}a^{17}+\frac{2}{315}a^{15}-\frac{2}{105}a^{14}+\frac{4}{315}a^{13}+\frac{1}{105}a^{12}+\frac{47}{315}a^{11}-\frac{2}{15}a^{10}+\frac{38}{315}a^{9}+\frac{46}{105}a^{8}+\frac{11}{63}a^{7}-\frac{23}{105}a^{6}+\frac{92}{315}a^{5}+\frac{20}{63}a^{3}+\frac{19}{105}a^{2}+\frac{34}{105}a+\frac{1}{7}$, $\frac{1}{315}a^{18}+\frac{2}{105}a^{15}+\frac{1}{105}a^{14}+\frac{2}{105}a^{13}+\frac{2}{105}a^{12}-\frac{2}{15}a^{11}+\frac{7}{45}a^{10}+\frac{17}{105}a^{9}+\frac{4}{35}a^{8}-\frac{32}{105}a^{7}+\frac{23}{105}a^{6}+\frac{1}{15}a^{5}-\frac{1}{3}a^{4}+\frac{44}{105}a^{3}+\frac{157}{315}a^{2}+\frac{37}{105}a-\frac{2}{7}$, $\frac{1}{2205}a^{19}+\frac{1}{735}a^{18}+\frac{2}{2205}a^{17}+\frac{2}{2205}a^{16}-\frac{26}{2205}a^{15}+\frac{16}{2205}a^{14}-\frac{22}{2205}a^{13}+\frac{1}{63}a^{12}+\frac{227}{2205}a^{11}-\frac{179}{2205}a^{10}-\frac{179}{2205}a^{9}-\frac{166}{441}a^{8}+\frac{272}{2205}a^{7}+\frac{614}{2205}a^{6}+\frac{289}{2205}a^{5}+\frac{352}{2205}a^{4}-\frac{83}{245}a^{3}+\frac{401}{2205}a^{2}-\frac{107}{735}a+\frac{9}{49}$, $\frac{1}{6615}a^{20}-\frac{4}{6615}a^{17}+\frac{1}{2205}a^{16}-\frac{74}{6615}a^{15}-\frac{4}{315}a^{14}-\frac{67}{6615}a^{13}+\frac{94}{6615}a^{12}-\frac{419}{6615}a^{11}-\frac{22}{245}a^{10}+\frac{484}{6615}a^{9}-\frac{197}{2205}a^{8}+\frac{3221}{6615}a^{7}-\frac{209}{441}a^{6}+\frac{2572}{6615}a^{5}-\frac{53}{6615}a^{4}+\frac{416}{6615}a^{3}+\frac{149}{441}a^{2}-\frac{55}{147}a-\frac{16}{49}$, $\frac{1}{19845}a^{21}+\frac{1}{19845}a^{20}+\frac{17}{19845}a^{18}-\frac{22}{19845}a^{17}-\frac{8}{19845}a^{16}+\frac{178}{19845}a^{15}-\frac{88}{19845}a^{14}+\frac{149}{6615}a^{13}-\frac{73}{19845}a^{12}-\frac{677}{19845}a^{11}+\frac{352}{19845}a^{10}+\frac{2749}{19845}a^{9}+\frac{9119}{19845}a^{8}+\frac{8948}{19845}a^{7}-\frac{2453}{19845}a^{6}-\frac{2941}{19845}a^{5}+\frac{1906}{6615}a^{4}+\frac{4898}{19845}a^{3}+\frac{586}{1323}a^{2}-\frac{110}{441}a-\frac{65}{147}$, $\frac{1}{19845}a^{22}-\frac{1}{19845}a^{20}-\frac{1}{19845}a^{19}-\frac{2}{1323}a^{18}-\frac{22}{19845}a^{17}+\frac{8}{6615}a^{16}+\frac{391}{19845}a^{15}+\frac{373}{19845}a^{14}+\frac{443}{19845}a^{13}+\frac{341}{19845}a^{12}+\frac{304}{6615}a^{11}+\frac{928}{6615}a^{10}+\frac{2599}{19845}a^{9}-\frac{89}{441}a^{8}+\frac{2414}{19845}a^{7}-\frac{2972}{19845}a^{6}+\frac{6103}{19845}a^{5}+\frac{6389}{19845}a^{4}+\frac{9589}{19845}a^{3}-\frac{47}{945}a^{2}+\frac{103}{315}a+\frac{53}{147}$, $\frac{1}{19845}a^{23}-\frac{1}{6615}a^{19}+\frac{13}{19845}a^{18}-\frac{1}{2835}a^{17}-\frac{4}{19845}a^{16}+\frac{101}{19845}a^{15}+\frac{283}{19845}a^{14}-\frac{247}{19845}a^{13}+\frac{209}{19845}a^{12}+\frac{1306}{19845}a^{11}-\frac{641}{3969}a^{10}+\frac{2668}{19845}a^{9}-\frac{1931}{19845}a^{8}+\frac{281}{735}a^{7}-\frac{4216}{19845}a^{6}-\frac{1555}{3969}a^{5}+\frac{9376}{19845}a^{4}-\frac{6304}{19845}a^{3}-\frac{2932}{6615}a^{2}-\frac{1012}{2205}a-\frac{26}{147}$, $\frac{1}{138915}a^{24}-\frac{2}{138915}a^{23}-\frac{1}{138915}a^{22}+\frac{1}{138915}a^{20}+\frac{2}{138915}a^{19}-\frac{61}{46305}a^{18}-\frac{16}{138915}a^{17}-\frac{1}{27783}a^{16}+\frac{503}{138915}a^{15}-\frac{2608}{138915}a^{14}-\frac{232}{19845}a^{13}-\frac{431}{138915}a^{12}+\frac{430}{9261}a^{11}+\frac{3943}{46305}a^{10}+\frac{13519}{138915}a^{9}-\frac{51641}{138915}a^{8}+\frac{2218}{5145}a^{7}-\frac{526}{2835}a^{6}-\frac{6052}{27783}a^{5}-\frac{1174}{3969}a^{4}-\frac{6835}{27783}a^{3}-\frac{5393}{46305}a^{2}+\frac{4027}{15435}a+\frac{485}{1029}$, $\frac{1}{27088425}a^{25}+\frac{1}{27088425}a^{24}-\frac{59}{3869775}a^{23}+\frac{74}{27088425}a^{22}+\frac{463}{27088425}a^{21}+\frac{53}{9029475}a^{20}+\frac{4429}{27088425}a^{19}+\frac{10978}{27088425}a^{18}-\frac{478}{3009825}a^{17}-\frac{116}{694575}a^{16}-\frac{3506}{773955}a^{15}+\frac{146807}{9029475}a^{14}-\frac{12718}{601965}a^{13}-\frac{240739}{27088425}a^{12}+\frac{2228549}{27088425}a^{11}+\frac{1223039}{27088425}a^{10}+\frac{188387}{2083725}a^{9}+\frac{1841324}{27088425}a^{8}+\frac{110636}{9029475}a^{7}+\frac{1467749}{5417685}a^{6}+\frac{1316821}{5417685}a^{5}-\frac{936944}{27088425}a^{4}+\frac{2389754}{9029475}a^{3}+\frac{807827}{3009825}a^{2}-\frac{239206}{1003275}a-\frac{29839}{66885}$, $\frac{1}{52903694025}a^{26}-\frac{2}{10580738805}a^{25}-\frac{183919}{52903694025}a^{24}-\frac{252416}{17634564675}a^{23}-\frac{3214}{1356504975}a^{22}-\frac{100484}{52903694025}a^{21}+\frac{727964}{10580738805}a^{20}-\frac{2151931}{52903694025}a^{19}+\frac{249056}{813902985}a^{18}-\frac{6205649}{17634564675}a^{17}-\frac{131272}{154238175}a^{16}+\frac{1004614001}{52903694025}a^{15}+\frac{88177298}{17634564675}a^{14}-\frac{3148927}{7557670575}a^{13}+\frac{527596213}{52903694025}a^{12}-\frac{566309914}{10580738805}a^{11}+\frac{2928621967}{52903694025}a^{10}+\frac{4538467093}{52903694025}a^{9}+\frac{1117370258}{4069514925}a^{8}-\frac{13840600103}{52903694025}a^{7}+\frac{1396623542}{3526912935}a^{6}-\frac{623798102}{1959396075}a^{5}-\frac{8426713289}{52903694025}a^{4}+\frac{1732080572}{17634564675}a^{3}-\frac{100212092}{1175637645}a^{2}+\frac{602257766}{1959396075}a+\frac{854894}{4213755}$, $\frac{1}{37\!\cdots\!75}a^{27}+\frac{821}{415205825272875}a^{26}+\frac{1745869}{37\!\cdots\!75}a^{25}+\frac{11145908744}{37\!\cdots\!75}a^{24}+\frac{4494257857}{415205825272875}a^{23}+\frac{13093801414}{37\!\cdots\!75}a^{22}-\frac{79672810747}{37\!\cdots\!75}a^{21}-\frac{4587726397}{138401941757625}a^{20}+\frac{242452810126}{12\!\cdots\!25}a^{19}-\frac{4552757117873}{37\!\cdots\!75}a^{18}-\frac{3061180624}{6280424247825}a^{17}+\frac{590017502918}{747370485491175}a^{16}-\frac{83468905952197}{37\!\cdots\!75}a^{15}+\frac{998235674908}{106767212213025}a^{14}-\frac{21368293834}{4464578766375}a^{13}-\frac{6439205916593}{747370485491175}a^{12}+\frac{456545248975394}{37\!\cdots\!75}a^{11}+\frac{65337171020813}{37\!\cdots\!75}a^{10}-\frac{845768502286}{46133980585875}a^{9}+\frac{26138513117611}{249123495163725}a^{8}-\frac{356122894320383}{37\!\cdots\!75}a^{7}+\frac{499560336094147}{12\!\cdots\!25}a^{6}+\frac{319940647733249}{747370485491175}a^{5}+\frac{133398428952886}{287450186727375}a^{4}-\frac{18600159882922}{49824699032745}a^{3}-\frac{96116996514706}{415205825272875}a^{2}+\frac{20659279039946}{138401941757625}a-\frac{18139786768}{42519797775}$, $\frac{1}{27\!\cdots\!75}a^{28}-\frac{67\!\cdots\!14}{79\!\cdots\!25}a^{27}-\frac{13\!\cdots\!27}{27\!\cdots\!75}a^{26}-\frac{10\!\cdots\!69}{92\!\cdots\!25}a^{25}+\frac{86\!\cdots\!71}{39\!\cdots\!25}a^{24}+\frac{68\!\cdots\!07}{27\!\cdots\!75}a^{23}-\frac{10\!\cdots\!71}{92\!\cdots\!25}a^{22}-\frac{56\!\cdots\!96}{27\!\cdots\!75}a^{21}-\frac{46\!\cdots\!07}{92\!\cdots\!25}a^{20}-\frac{38\!\cdots\!56}{29\!\cdots\!25}a^{19}-\frac{13\!\cdots\!44}{10\!\cdots\!25}a^{18}-\frac{95\!\cdots\!44}{18\!\cdots\!25}a^{17}-\frac{32\!\cdots\!13}{30\!\cdots\!75}a^{16}-\frac{33\!\cdots\!44}{21\!\cdots\!75}a^{15}-\frac{63\!\cdots\!88}{27\!\cdots\!75}a^{14}-\frac{93\!\cdots\!73}{27\!\cdots\!75}a^{13}+\frac{18\!\cdots\!14}{89\!\cdots\!25}a^{12}-\frac{34\!\cdots\!63}{27\!\cdots\!75}a^{11}-\frac{42\!\cdots\!23}{27\!\cdots\!75}a^{10}-\frac{52\!\cdots\!22}{92\!\cdots\!25}a^{9}-\frac{43\!\cdots\!28}{27\!\cdots\!75}a^{8}+\frac{44\!\cdots\!43}{27\!\cdots\!75}a^{7}-\frac{11\!\cdots\!37}{43\!\cdots\!75}a^{6}-\frac{61\!\cdots\!11}{39\!\cdots\!25}a^{5}-\frac{27\!\cdots\!37}{27\!\cdots\!75}a^{4}-\frac{23\!\cdots\!48}{92\!\cdots\!25}a^{3}-\frac{32\!\cdots\!36}{23\!\cdots\!75}a^{2}-\frac{13\!\cdots\!39}{10\!\cdots\!25}a-\frac{10\!\cdots\!56}{22\!\cdots\!25}$ 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:  $3$, $5$, $7$

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 - 13*x^28 - 55*x^27 + 712*x^26 + 11351*x^25 - 127986*x^24 + 66127*x^23 + 4301081*x^22 - 11864947*x^21 - 126534305*x^20 + 781815928*x^19 + 394279608*x^18 - 18262666572*x^17 + 39703600309*x^16 + 333330481344*x^15 - 1935249895004*x^14 + 1288520615067*x^13 + 24287341394015*x^12 - 64823993632866*x^11 - 202566749188801*x^10 + 1440509151901048*x^9 - 2780340418135933*x^8 - 4068418180429144*x^7 + 3309165651131592*x^6 + 104817181048914519*x^5 - 418797024905501732*x^4 + 871709514773509500*x^3 - 878388606425292930*x^2 + 5204661351059478078*x - 13446098969951310345)
 
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 - 13*x^28 - 55*x^27 + 712*x^26 + 11351*x^25 - 127986*x^24 + 66127*x^23 + 4301081*x^22 - 11864947*x^21 - 126534305*x^20 + 781815928*x^19 + 394279608*x^18 - 18262666572*x^17 + 39703600309*x^16 + 333330481344*x^15 - 1935249895004*x^14 + 1288520615067*x^13 + 24287341394015*x^12 - 64823993632866*x^11 - 202566749188801*x^10 + 1440509151901048*x^9 - 2780340418135933*x^8 - 4068418180429144*x^7 + 3309165651131592*x^6 + 104817181048914519*x^5 - 418797024905501732*x^4 + 871709514773509500*x^3 - 878388606425292930*x^2 + 5204661351059478078*x - 13446098969951310345, 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 - 13*x^28 - 55*x^27 + 712*x^26 + 11351*x^25 - 127986*x^24 + 66127*x^23 + 4301081*x^22 - 11864947*x^21 - 126534305*x^20 + 781815928*x^19 + 394279608*x^18 - 18262666572*x^17 + 39703600309*x^16 + 333330481344*x^15 - 1935249895004*x^14 + 1288520615067*x^13 + 24287341394015*x^12 - 64823993632866*x^11 - 202566749188801*x^10 + 1440509151901048*x^9 - 2780340418135933*x^8 - 4068418180429144*x^7 + 3309165651131592*x^6 + 104817181048914519*x^5 - 418797024905501732*x^4 + 871709514773509500*x^3 - 878388606425292930*x^2 + 5204661351059478078*x - 13446098969951310345);
 
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 - 13*x^28 - 55*x^27 + 712*x^26 + 11351*x^25 - 127986*x^24 + 66127*x^23 + 4301081*x^22 - 11864947*x^21 - 126534305*x^20 + 781815928*x^19 + 394279608*x^18 - 18262666572*x^17 + 39703600309*x^16 + 333330481344*x^15 - 1935249895004*x^14 + 1288520615067*x^13 + 24287341394015*x^12 - 64823993632866*x^11 - 202566749188801*x^10 + 1440509151901048*x^9 - 2780340418135933*x^8 - 4068418180429144*x^7 + 3309165651131592*x^6 + 104817181048914519*x^5 - 418797024905501732*x^4 + 871709514773509500*x^3 - 878388606425292930*x^2 + 5204661351059478078*x - 13446098969951310345);
 
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 $29$ ${\href{/padicField/3.2.0.1}{2} }^{14}{,}\,{\href{/padicField/3.1.0.1}{1} }$ ${\href{/padicField/5.2.0.1}{2} }^{14}{,}\,{\href{/padicField/5.1.0.1}{1} }$ R $29$ ${\href{/padicField/13.2.0.1}{2} }^{14}{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.2.0.1}{2} }^{14}{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.2.0.1}{2} }^{14}{,}\,{\href{/padicField/19.1.0.1}{1} }$ $29$ $29$ ${\href{/padicField/31.2.0.1}{2} }^{14}{,}\,{\href{/padicField/31.1.0.1}{1} }$ $29$ ${\href{/padicField/41.2.0.1}{2} }^{14}{,}\,{\href{/padicField/41.1.0.1}{1} }$ $29$ ${\href{/padicField/47.2.0.1}{2} }^{14}{,}\,{\href{/padicField/47.1.0.1}{1} }$ $29$ ${\href{/padicField/59.2.0.1}{2} }^{14}{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
\(7\) Copy content Toggle raw display $\Q_{7}$$x + 4$$1$$1$$0$Trivial$[\ ]$
7.2.1.2$x^{2} + 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 7$$2$$1$$1$$C_2$$[\ ]_{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$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.7.2t1.a.a$1$ $ 7 $ \(\Q(\sqrt{-7}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.380023.29t2.a.a$2$ $ 7 \cdot 233^{2}$ 29.1.1310202274198394060556452494826429036980688154429108157810117051380399416309809.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.380023.29t2.a.n$2$ $ 7 \cdot 233^{2}$ 29.1.1310202274198394060556452494826429036980688154429108157810117051380399416309809.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.380023.29t2.a.d$2$ $ 7 \cdot 233^{2}$ 29.1.1310202274198394060556452494826429036980688154429108157810117051380399416309809.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.380023.29t2.a.b$2$ $ 7 \cdot 233^{2}$ 29.1.1310202274198394060556452494826429036980688154429108157810117051380399416309809.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.380023.29t2.a.l$2$ $ 7 \cdot 233^{2}$ 29.1.1310202274198394060556452494826429036980688154429108157810117051380399416309809.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.380023.29t2.a.k$2$ $ 7 \cdot 233^{2}$ 29.1.1310202274198394060556452494826429036980688154429108157810117051380399416309809.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.380023.29t2.a.m$2$ $ 7 \cdot 233^{2}$ 29.1.1310202274198394060556452494826429036980688154429108157810117051380399416309809.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.380023.29t2.a.j$2$ $ 7 \cdot 233^{2}$ 29.1.1310202274198394060556452494826429036980688154429108157810117051380399416309809.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.380023.29t2.a.i$2$ $ 7 \cdot 233^{2}$ 29.1.1310202274198394060556452494826429036980688154429108157810117051380399416309809.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.380023.29t2.a.c$2$ $ 7 \cdot 233^{2}$ 29.1.1310202274198394060556452494826429036980688154429108157810117051380399416309809.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.380023.29t2.a.g$2$ $ 7 \cdot 233^{2}$ 29.1.1310202274198394060556452494826429036980688154429108157810117051380399416309809.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.380023.29t2.a.e$2$ $ 7 \cdot 233^{2}$ 29.1.1310202274198394060556452494826429036980688154429108157810117051380399416309809.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.380023.29t2.a.f$2$ $ 7 \cdot 233^{2}$ 29.1.1310202274198394060556452494826429036980688154429108157810117051380399416309809.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.380023.29t2.a.h$2$ $ 7 \cdot 233^{2}$ 29.1.1310202274198394060556452494826429036980688154429108157810117051380399416309809.1 $D_{29}$ (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 *.