Properties

Label 29.1.190...729.1
Degree $29$
Signature $[1, 14]$
Discriminant $1.904\times 10^{47}$
Root discriminant \(42.69\)
Ramified prime $2383$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_{29}$ (as 29T2)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^29 - 8*x^28 + 42*x^27 - 91*x^26 + 71*x^25 - 172*x^24 + 980*x^23 - 2477*x^22 + 4032*x^21 - 5998*x^20 + 8995*x^19 - 12396*x^18 + 15464*x^17 - 17477*x^16 + 17661*x^15 - 16777*x^14 + 14899*x^13 - 10749*x^12 + 6197*x^11 - 2594*x^10 - 1233*x^9 + 2318*x^8 - 113*x^7 - 855*x^6 + 1459*x^5 - 2625*x^4 + 2079*x^3 - 978*x^2 + 419*x + 1)
 
gp: K = bnfinit(y^29 - 8*y^28 + 42*y^27 - 91*y^26 + 71*y^25 - 172*y^24 + 980*y^23 - 2477*y^22 + 4032*y^21 - 5998*y^20 + 8995*y^19 - 12396*y^18 + 15464*y^17 - 17477*y^16 + 17661*y^15 - 16777*y^14 + 14899*y^13 - 10749*y^12 + 6197*y^11 - 2594*y^10 - 1233*y^9 + 2318*y^8 - 113*y^7 - 855*y^6 + 1459*y^5 - 2625*y^4 + 2079*y^3 - 978*y^2 + 419*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^29 - 8*x^28 + 42*x^27 - 91*x^26 + 71*x^25 - 172*x^24 + 980*x^23 - 2477*x^22 + 4032*x^21 - 5998*x^20 + 8995*x^19 - 12396*x^18 + 15464*x^17 - 17477*x^16 + 17661*x^15 - 16777*x^14 + 14899*x^13 - 10749*x^12 + 6197*x^11 - 2594*x^10 - 1233*x^9 + 2318*x^8 - 113*x^7 - 855*x^6 + 1459*x^5 - 2625*x^4 + 2079*x^3 - 978*x^2 + 419*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^29 - 8*x^28 + 42*x^27 - 91*x^26 + 71*x^25 - 172*x^24 + 980*x^23 - 2477*x^22 + 4032*x^21 - 5998*x^20 + 8995*x^19 - 12396*x^18 + 15464*x^17 - 17477*x^16 + 17661*x^15 - 16777*x^14 + 14899*x^13 - 10749*x^12 + 6197*x^11 - 2594*x^10 - 1233*x^9 + 2318*x^8 - 113*x^7 - 855*x^6 + 1459*x^5 - 2625*x^4 + 2079*x^3 - 978*x^2 + 419*x + 1)
 

\( x^{29} - 8 x^{28} + 42 x^{27} - 91 x^{26} + 71 x^{25} - 172 x^{24} + 980 x^{23} - 2477 x^{22} + 4032 x^{21} - 5998 x^{20} + 8995 x^{19} - 12396 x^{18} + 15464 x^{17} - 17477 x^{16} + \cdots + 1 \) 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:   \(190430537333205962921320156798047381287357677729\) \(\medspace = 2383^{14}\) 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:  \(42.69\)
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:  $2383^{1/2}\approx 48.815980989835694$
Ramified primes:   \(2383\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{4}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{5}$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{6}$, $\frac{1}{9}a^{15}+\frac{1}{9}a^{14}+\frac{1}{9}a^{13}+\frac{1}{9}a^{12}+\frac{1}{9}a^{11}+\frac{1}{9}a^{10}+\frac{1}{9}a^{9}+\frac{1}{9}a^{8}-\frac{4}{9}a^{7}-\frac{4}{9}a^{6}-\frac{4}{9}a^{5}-\frac{4}{9}a^{4}-\frac{4}{9}a^{3}-\frac{4}{9}a^{2}-\frac{4}{9}a-\frac{4}{9}$, $\frac{1}{9}a^{16}+\frac{1}{9}a^{8}-\frac{2}{9}$, $\frac{1}{9}a^{17}+\frac{1}{9}a^{9}-\frac{2}{9}a$, $\frac{1}{9}a^{18}+\frac{1}{9}a^{10}-\frac{2}{9}a^{2}$, $\frac{1}{9}a^{19}+\frac{1}{9}a^{11}-\frac{2}{9}a^{3}$, $\frac{1}{45}a^{20}-\frac{1}{45}a^{19}+\frac{2}{45}a^{18}+\frac{2}{45}a^{17}-\frac{1}{45}a^{16}-\frac{1}{45}a^{15}-\frac{7}{45}a^{14}-\frac{1}{45}a^{13}-\frac{1}{9}a^{11}+\frac{7}{45}a^{10}+\frac{4}{45}a^{9}-\frac{2}{45}a^{8}-\frac{14}{45}a^{7}+\frac{2}{9}a^{6}+\frac{22}{45}a^{5}+\frac{4}{9}a^{4}-\frac{1}{5}a^{3}-\frac{2}{15}a^{2}+\frac{1}{3}a-\frac{7}{15}$, $\frac{1}{45}a^{21}+\frac{1}{45}a^{19}-\frac{1}{45}a^{18}+\frac{1}{45}a^{17}-\frac{2}{45}a^{16}+\frac{2}{45}a^{15}+\frac{2}{45}a^{14}-\frac{2}{15}a^{13}+\frac{1}{9}a^{12}-\frac{1}{15}a^{11}+\frac{1}{45}a^{10}-\frac{1}{15}a^{9}-\frac{2}{15}a^{8}+\frac{1}{45}a^{7}-\frac{8}{45}a^{6}+\frac{17}{45}a^{5}+\frac{16}{45}a^{4}+\frac{1}{9}a^{3}-\frac{2}{15}a^{2}+\frac{14}{45}a-\frac{16}{45}$, $\frac{1}{135}a^{22}-\frac{1}{135}a^{21}+\frac{4}{135}a^{19}+\frac{1}{27}a^{18}-\frac{1}{27}a^{16}-\frac{4}{135}a^{15}+\frac{1}{15}a^{14}-\frac{8}{135}a^{13}-\frac{13}{135}a^{12}-\frac{2}{45}a^{11}-\frac{11}{135}a^{10}-\frac{22}{135}a^{9}-\frac{2}{45}a^{8}-\frac{13}{27}a^{7}-\frac{5}{27}a^{6}-\frac{11}{45}a^{5}-\frac{56}{135}a^{4}-\frac{67}{135}a^{3}-\frac{1}{15}a^{2}-\frac{4}{27}a-\frac{58}{135}$, $\frac{1}{135}a^{23}-\frac{1}{135}a^{21}+\frac{1}{135}a^{20}-\frac{1}{45}a^{19}-\frac{1}{135}a^{18}+\frac{4}{135}a^{17}-\frac{2}{45}a^{16}-\frac{7}{135}a^{15}+\frac{7}{135}a^{14}+\frac{4}{45}a^{13}+\frac{11}{135}a^{12}+\frac{13}{135}a^{11}+\frac{7}{45}a^{10}+\frac{1}{27}a^{9}+\frac{2}{27}a^{8}+\frac{4}{45}a^{7}-\frac{28}{135}a^{6}-\frac{1}{27}a^{5}-\frac{11}{45}a^{4}-\frac{4}{135}a^{3}-\frac{41}{135}a^{2}-\frac{1}{45}a-\frac{5}{27}$, $\frac{1}{135}a^{24}+\frac{1}{9}a^{8}-\frac{16}{135}$, $\frac{1}{2295}a^{25}+\frac{2}{2295}a^{24}-\frac{7}{2295}a^{23}+\frac{2}{2295}a^{22}+\frac{8}{2295}a^{21}+\frac{2}{2295}a^{20}-\frac{22}{2295}a^{19}-\frac{43}{2295}a^{18}-\frac{52}{2295}a^{17}+\frac{122}{2295}a^{16}+\frac{53}{2295}a^{15}-\frac{298}{2295}a^{14}-\frac{67}{2295}a^{13}-\frac{28}{2295}a^{12}-\frac{7}{2295}a^{11}-\frac{73}{2295}a^{10}-\frac{112}{2295}a^{9}-\frac{103}{2295}a^{8}-\frac{1072}{2295}a^{7}-\frac{28}{2295}a^{6}+\frac{653}{2295}a^{5}+\frac{512}{2295}a^{4}-\frac{322}{2295}a^{3}-\frac{478}{2295}a^{2}-\frac{458}{2295}a-\frac{5}{153}$, $\frac{1}{20655}a^{26}-\frac{1}{6885}a^{25}-\frac{2}{1215}a^{24}+\frac{4}{4131}a^{23}-\frac{2}{20655}a^{22}+\frac{1}{255}a^{21}+\frac{2}{20655}a^{20}-\frac{341}{20655}a^{19}+\frac{7}{153}a^{18}+\frac{518}{20655}a^{17}+\frac{164}{4131}a^{16}-\frac{182}{6885}a^{15}-\frac{2164}{20655}a^{14}-\frac{622}{4131}a^{13}+\frac{1087}{6885}a^{12}-\frac{470}{4131}a^{11}-\frac{286}{4131}a^{10}+\frac{205}{1377}a^{9}+\frac{2639}{20655}a^{8}+\frac{10126}{20655}a^{7}+\frac{916}{6885}a^{6}+\frac{8093}{20655}a^{5}+\frac{5176}{20655}a^{4}+\frac{587}{6885}a^{3}+\frac{3496}{20655}a^{2}-\frac{2171}{20655}a+\frac{7039}{20655}$, $\frac{1}{119117385}a^{27}-\frac{2564}{119117385}a^{26}-\frac{22546}{119117385}a^{25}+\frac{163486}{119117385}a^{24}+\frac{53738}{39705795}a^{23}+\frac{400906}{119117385}a^{22}+\frac{1144649}{119117385}a^{21}-\frac{100477}{13235265}a^{20}+\frac{62698}{119117385}a^{19}+\frac{3106148}{119117385}a^{18}+\frac{1372672}{39705795}a^{17}+\frac{1246973}{23823477}a^{16}+\frac{444833}{119117385}a^{15}-\frac{2936167}{39705795}a^{14}-\frac{4276}{7006905}a^{13}-\frac{3153319}{119117385}a^{12}-\frac{1041382}{39705795}a^{11}-\frac{15334577}{119117385}a^{10}+\frac{1573253}{119117385}a^{9}-\frac{9031}{60435}a^{8}-\frac{4949653}{23823477}a^{7}-\frac{59524924}{119117385}a^{6}+\frac{6300764}{13235265}a^{5}+\frac{16270786}{119117385}a^{4}+\frac{2873897}{7006905}a^{3}-\frac{16947886}{119117385}a^{2}+\frac{2148520}{23823477}a+\frac{58725184}{119117385}$, $\frac{1}{24\!\cdots\!85}a^{28}-\frac{12\!\cdots\!84}{27\!\cdots\!65}a^{27}+\frac{19\!\cdots\!69}{83\!\cdots\!95}a^{26}+\frac{10\!\cdots\!55}{49\!\cdots\!17}a^{25}+\frac{22\!\cdots\!37}{83\!\cdots\!95}a^{24}+\frac{92\!\cdots\!01}{24\!\cdots\!85}a^{23}-\frac{17\!\cdots\!70}{55\!\cdots\!13}a^{22}-\frac{12\!\cdots\!54}{24\!\cdots\!85}a^{21}+\frac{14\!\cdots\!81}{24\!\cdots\!85}a^{20}-\frac{37\!\cdots\!67}{83\!\cdots\!95}a^{19}+\frac{26\!\cdots\!98}{13\!\cdots\!15}a^{18}+\frac{27\!\cdots\!66}{24\!\cdots\!85}a^{17}-\frac{51\!\cdots\!58}{16\!\cdots\!39}a^{16}+\frac{10\!\cdots\!17}{24\!\cdots\!85}a^{15}-\frac{38\!\cdots\!19}{24\!\cdots\!85}a^{14}-\frac{50\!\cdots\!81}{11\!\cdots\!89}a^{13}-\frac{20\!\cdots\!44}{24\!\cdots\!85}a^{12}+\frac{21\!\cdots\!82}{11\!\cdots\!35}a^{11}+\frac{13\!\cdots\!11}{83\!\cdots\!95}a^{10}+\frac{19\!\cdots\!89}{24\!\cdots\!85}a^{9}-\frac{39\!\cdots\!12}{24\!\cdots\!85}a^{8}+\frac{38\!\cdots\!28}{83\!\cdots\!95}a^{7}+\frac{24\!\cdots\!73}{67\!\cdots\!05}a^{6}-\frac{11\!\cdots\!76}{24\!\cdots\!85}a^{5}-\frac{90\!\cdots\!92}{24\!\cdots\!85}a^{4}-\frac{19\!\cdots\!94}{14\!\cdots\!05}a^{3}+\frac{17\!\cdots\!08}{49\!\cdots\!17}a^{2}-\frac{21\!\cdots\!35}{49\!\cdots\!17}a+\frac{16\!\cdots\!52}{24\!\cdots\!85}$ 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:  $3$, $5$

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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:   $\frac{33\!\cdots\!44}{24\!\cdots\!85}a^{28}-\frac{89\!\cdots\!31}{83\!\cdots\!95}a^{27}+\frac{53\!\cdots\!96}{92\!\cdots\!55}a^{26}-\frac{33\!\cdots\!93}{24\!\cdots\!85}a^{25}+\frac{10\!\cdots\!61}{83\!\cdots\!95}a^{24}-\frac{59\!\cdots\!62}{24\!\cdots\!85}a^{23}+\frac{10\!\cdots\!33}{83\!\cdots\!95}a^{22}-\frac{89\!\cdots\!84}{24\!\cdots\!85}a^{21}+\frac{15\!\cdots\!67}{24\!\cdots\!85}a^{20}-\frac{74\!\cdots\!98}{83\!\cdots\!95}a^{19}+\frac{22\!\cdots\!17}{17\!\cdots\!85}a^{18}-\frac{45\!\cdots\!92}{24\!\cdots\!85}a^{17}+\frac{18\!\cdots\!59}{83\!\cdots\!95}a^{16}-\frac{12\!\cdots\!39}{49\!\cdots\!17}a^{15}+\frac{12\!\cdots\!23}{49\!\cdots\!17}a^{14}-\frac{42\!\cdots\!37}{18\!\cdots\!15}a^{13}+\frac{51\!\cdots\!56}{24\!\cdots\!85}a^{12}-\frac{17\!\cdots\!34}{11\!\cdots\!35}a^{11}+\frac{37\!\cdots\!16}{55\!\cdots\!13}a^{10}-\frac{31\!\cdots\!56}{24\!\cdots\!85}a^{9}-\frac{20\!\cdots\!02}{49\!\cdots\!17}a^{8}+\frac{16\!\cdots\!62}{27\!\cdots\!65}a^{7}-\frac{13\!\cdots\!29}{67\!\cdots\!05}a^{6}-\frac{21\!\cdots\!88}{24\!\cdots\!85}a^{5}+\frac{41\!\cdots\!91}{24\!\cdots\!85}a^{4}-\frac{87\!\cdots\!08}{24\!\cdots\!85}a^{3}+\frac{66\!\cdots\!82}{24\!\cdots\!85}a^{2}-\frac{28\!\cdots\!44}{24\!\cdots\!85}a+\frac{14\!\cdots\!63}{24\!\cdots\!85}$, $\frac{50\!\cdots\!74}{24\!\cdots\!85}a^{28}-\frac{11\!\cdots\!31}{83\!\cdots\!95}a^{27}+\frac{12\!\cdots\!55}{16\!\cdots\!39}a^{26}-\frac{33\!\cdots\!04}{24\!\cdots\!85}a^{25}+\frac{50\!\cdots\!78}{37\!\cdots\!05}a^{24}-\frac{10\!\cdots\!81}{24\!\cdots\!85}a^{23}+\frac{94\!\cdots\!42}{55\!\cdots\!13}a^{22}-\frac{98\!\cdots\!78}{24\!\cdots\!85}a^{21}+\frac{17\!\cdots\!96}{24\!\cdots\!85}a^{20}-\frac{93\!\cdots\!61}{83\!\cdots\!95}a^{19}+\frac{23\!\cdots\!11}{13\!\cdots\!15}a^{18}-\frac{59\!\cdots\!74}{24\!\cdots\!85}a^{17}+\frac{24\!\cdots\!89}{83\!\cdots\!95}a^{16}-\frac{84\!\cdots\!88}{24\!\cdots\!85}a^{15}+\frac{11\!\cdots\!34}{31\!\cdots\!15}a^{14}-\frac{38\!\cdots\!26}{11\!\cdots\!89}a^{13}+\frac{76\!\cdots\!19}{24\!\cdots\!85}a^{12}-\frac{27\!\cdots\!56}{11\!\cdots\!35}a^{11}+\frac{11\!\cdots\!48}{83\!\cdots\!95}a^{10}-\frac{24\!\cdots\!83}{49\!\cdots\!17}a^{9}-\frac{10\!\cdots\!99}{49\!\cdots\!17}a^{8}+\frac{43\!\cdots\!22}{83\!\cdots\!95}a^{7}-\frac{19\!\cdots\!84}{13\!\cdots\!41}a^{6}-\frac{48\!\cdots\!29}{24\!\cdots\!85}a^{5}+\frac{58\!\cdots\!76}{29\!\cdots\!01}a^{4}-\frac{13\!\cdots\!06}{24\!\cdots\!85}a^{3}+\frac{15\!\cdots\!63}{24\!\cdots\!85}a^{2}-\frac{33\!\cdots\!49}{24\!\cdots\!85}a+\frac{19\!\cdots\!66}{24\!\cdots\!85}$, $\frac{10\!\cdots\!56}{24\!\cdots\!85}a^{28}-\frac{23\!\cdots\!77}{83\!\cdots\!95}a^{27}+\frac{11\!\cdots\!76}{83\!\cdots\!95}a^{26}-\frac{54\!\cdots\!13}{24\!\cdots\!85}a^{25}+\frac{15\!\cdots\!12}{27\!\cdots\!65}a^{24}-\frac{16\!\cdots\!26}{24\!\cdots\!85}a^{23}+\frac{30\!\cdots\!22}{92\!\cdots\!55}a^{22}-\frac{16\!\cdots\!07}{24\!\cdots\!85}a^{21}+\frac{23\!\cdots\!08}{24\!\cdots\!85}a^{20}-\frac{12\!\cdots\!73}{83\!\cdots\!95}a^{19}+\frac{29\!\cdots\!82}{13\!\cdots\!15}a^{18}-\frac{13\!\cdots\!80}{49\!\cdots\!17}a^{17}+\frac{55\!\cdots\!37}{16\!\cdots\!39}a^{16}-\frac{89\!\cdots\!36}{24\!\cdots\!85}a^{15}+\frac{86\!\cdots\!48}{24\!\cdots\!85}a^{14}-\frac{18\!\cdots\!43}{55\!\cdots\!45}a^{13}+\frac{67\!\cdots\!32}{24\!\cdots\!85}a^{12}-\frac{38\!\cdots\!89}{23\!\cdots\!47}a^{11}+\frac{78\!\cdots\!18}{83\!\cdots\!95}a^{10}-\frac{77\!\cdots\!57}{49\!\cdots\!17}a^{9}-\frac{15\!\cdots\!63}{24\!\cdots\!85}a^{8}+\frac{18\!\cdots\!51}{83\!\cdots\!95}a^{7}+\frac{24\!\cdots\!09}{13\!\cdots\!41}a^{6}-\frac{30\!\cdots\!27}{24\!\cdots\!85}a^{5}+\frac{12\!\cdots\!79}{24\!\cdots\!85}a^{4}-\frac{13\!\cdots\!14}{24\!\cdots\!85}a^{3}+\frac{63\!\cdots\!02}{24\!\cdots\!85}a^{2}-\frac{82\!\cdots\!68}{49\!\cdots\!17}a+\frac{29\!\cdots\!59}{49\!\cdots\!17}$, $\frac{11\!\cdots\!06}{24\!\cdots\!85}a^{28}-\frac{18\!\cdots\!66}{54\!\cdots\!15}a^{27}+\frac{14\!\cdots\!36}{83\!\cdots\!95}a^{26}-\frac{89\!\cdots\!53}{24\!\cdots\!85}a^{25}+\frac{16\!\cdots\!82}{83\!\cdots\!95}a^{24}-\frac{16\!\cdots\!32}{24\!\cdots\!85}a^{23}+\frac{34\!\cdots\!54}{83\!\cdots\!95}a^{22}-\frac{24\!\cdots\!06}{24\!\cdots\!85}a^{21}+\frac{73\!\cdots\!76}{49\!\cdots\!17}a^{20}-\frac{65\!\cdots\!24}{30\!\cdots\!85}a^{19}+\frac{45\!\cdots\!28}{13\!\cdots\!15}a^{18}-\frac{21\!\cdots\!81}{49\!\cdots\!17}a^{17}+\frac{47\!\cdots\!31}{92\!\cdots\!55}a^{16}-\frac{13\!\cdots\!93}{24\!\cdots\!85}a^{15}+\frac{13\!\cdots\!18}{24\!\cdots\!85}a^{14}-\frac{16\!\cdots\!96}{32\!\cdots\!85}a^{13}+\frac{10\!\cdots\!22}{24\!\cdots\!85}a^{12}-\frac{35\!\cdots\!91}{13\!\cdots\!91}a^{11}+\frac{83\!\cdots\!58}{83\!\cdots\!95}a^{10}-\frac{38\!\cdots\!08}{24\!\cdots\!85}a^{9}-\frac{44\!\cdots\!04}{49\!\cdots\!17}a^{8}+\frac{63\!\cdots\!29}{83\!\cdots\!95}a^{7}+\frac{39\!\cdots\!47}{67\!\cdots\!05}a^{6}-\frac{16\!\cdots\!59}{24\!\cdots\!85}a^{5}+\frac{15\!\cdots\!47}{24\!\cdots\!85}a^{4}-\frac{54\!\cdots\!75}{49\!\cdots\!17}a^{3}+\frac{36\!\cdots\!37}{49\!\cdots\!17}a^{2}-\frac{10\!\cdots\!56}{49\!\cdots\!17}a+\frac{24\!\cdots\!97}{24\!\cdots\!85}$, $\frac{17\!\cdots\!12}{24\!\cdots\!85}a^{28}-\frac{43\!\cdots\!94}{83\!\cdots\!95}a^{27}+\frac{14\!\cdots\!83}{55\!\cdots\!13}a^{26}-\frac{13\!\cdots\!71}{24\!\cdots\!85}a^{25}+\frac{31\!\cdots\!88}{83\!\cdots\!95}a^{24}-\frac{62\!\cdots\!55}{49\!\cdots\!17}a^{23}+\frac{52\!\cdots\!66}{83\!\cdots\!95}a^{22}-\frac{36\!\cdots\!88}{24\!\cdots\!85}a^{21}+\frac{60\!\cdots\!89}{24\!\cdots\!85}a^{20}-\frac{30\!\cdots\!09}{83\!\cdots\!95}a^{19}+\frac{77\!\cdots\!82}{13\!\cdots\!15}a^{18}-\frac{19\!\cdots\!52}{24\!\cdots\!85}a^{17}+\frac{81\!\cdots\!12}{83\!\cdots\!95}a^{16}-\frac{27\!\cdots\!36}{24\!\cdots\!85}a^{15}+\frac{29\!\cdots\!74}{24\!\cdots\!85}a^{14}-\frac{21\!\cdots\!87}{18\!\cdots\!15}a^{13}+\frac{25\!\cdots\!71}{24\!\cdots\!85}a^{12}-\frac{93\!\cdots\!03}{11\!\cdots\!35}a^{11}+\frac{29\!\cdots\!24}{55\!\cdots\!13}a^{10}-\frac{13\!\cdots\!58}{49\!\cdots\!17}a^{9}-\frac{74\!\cdots\!71}{24\!\cdots\!85}a^{8}+\frac{95\!\cdots\!97}{92\!\cdots\!55}a^{7}-\frac{77\!\cdots\!79}{13\!\cdots\!41}a^{6}+\frac{83\!\cdots\!09}{24\!\cdots\!85}a^{5}+\frac{14\!\cdots\!31}{49\!\cdots\!17}a^{4}-\frac{31\!\cdots\!47}{49\!\cdots\!17}a^{3}+\frac{17\!\cdots\!99}{24\!\cdots\!85}a^{2}-\frac{84\!\cdots\!04}{24\!\cdots\!85}a+\frac{31\!\cdots\!53}{24\!\cdots\!85}$, $\frac{13\!\cdots\!88}{35\!\cdots\!35}a^{28}-\frac{29\!\cdots\!07}{11\!\cdots\!45}a^{27}+\frac{46\!\cdots\!41}{39\!\cdots\!15}a^{26}-\frac{49\!\cdots\!56}{35\!\cdots\!35}a^{25}-\frac{11\!\cdots\!98}{11\!\cdots\!45}a^{24}-\frac{19\!\cdots\!78}{35\!\cdots\!35}a^{23}+\frac{72\!\cdots\!35}{23\!\cdots\!09}a^{22}-\frac{16\!\cdots\!97}{35\!\cdots\!35}a^{21}+\frac{16\!\cdots\!94}{35\!\cdots\!35}a^{20}-\frac{10\!\cdots\!51}{11\!\cdots\!45}a^{19}+\frac{30\!\cdots\!68}{19\!\cdots\!65}a^{18}-\frac{64\!\cdots\!38}{35\!\cdots\!35}a^{17}+\frac{22\!\cdots\!22}{11\!\cdots\!45}a^{16}-\frac{72\!\cdots\!11}{35\!\cdots\!35}a^{15}+\frac{16\!\cdots\!06}{70\!\cdots\!27}a^{14}-\frac{21\!\cdots\!04}{86\!\cdots\!55}a^{13}+\frac{38\!\cdots\!56}{20\!\cdots\!55}a^{12}-\frac{76\!\cdots\!37}{97\!\cdots\!05}a^{11}+\frac{11\!\cdots\!78}{13\!\cdots\!05}a^{10}-\frac{35\!\cdots\!52}{35\!\cdots\!35}a^{9}-\frac{56\!\cdots\!49}{35\!\cdots\!35}a^{8}+\frac{63\!\cdots\!89}{39\!\cdots\!15}a^{7}+\frac{41\!\cdots\!52}{94\!\cdots\!55}a^{6}-\frac{16\!\cdots\!11}{35\!\cdots\!35}a^{5}+\frac{33\!\cdots\!66}{35\!\cdots\!35}a^{4}-\frac{11\!\cdots\!69}{41\!\cdots\!31}a^{3}+\frac{81\!\cdots\!82}{35\!\cdots\!35}a^{2}-\frac{10\!\cdots\!28}{70\!\cdots\!27}a+\frac{33\!\cdots\!57}{35\!\cdots\!35}$, $\frac{72\!\cdots\!88}{83\!\cdots\!95}a^{28}-\frac{57\!\cdots\!31}{83\!\cdots\!95}a^{27}+\frac{41\!\cdots\!31}{11\!\cdots\!15}a^{26}-\frac{63\!\cdots\!28}{83\!\cdots\!95}a^{25}+\frac{36\!\cdots\!68}{83\!\cdots\!95}a^{24}-\frac{95\!\cdots\!17}{83\!\cdots\!95}a^{23}+\frac{69\!\cdots\!18}{83\!\cdots\!95}a^{22}-\frac{58\!\cdots\!89}{27\!\cdots\!65}a^{21}+\frac{25\!\cdots\!14}{83\!\cdots\!95}a^{20}-\frac{35\!\cdots\!64}{83\!\cdots\!95}a^{19}+\frac{22\!\cdots\!27}{34\!\cdots\!83}a^{18}-\frac{73\!\cdots\!17}{83\!\cdots\!95}a^{17}+\frac{87\!\cdots\!08}{83\!\cdots\!95}a^{16}-\frac{20\!\cdots\!74}{18\!\cdots\!05}a^{15}+\frac{89\!\cdots\!64}{83\!\cdots\!95}a^{14}-\frac{52\!\cdots\!48}{55\!\cdots\!45}a^{13}+\frac{25\!\cdots\!32}{30\!\cdots\!85}a^{12}-\frac{20\!\cdots\!61}{39\!\cdots\!45}a^{11}+\frac{14\!\cdots\!91}{83\!\cdots\!95}a^{10}-\frac{16\!\cdots\!54}{10\!\cdots\!95}a^{9}-\frac{15\!\cdots\!49}{83\!\cdots\!95}a^{8}+\frac{18\!\cdots\!61}{83\!\cdots\!95}a^{7}+\frac{65\!\cdots\!29}{74\!\cdots\!45}a^{6}-\frac{12\!\cdots\!39}{83\!\cdots\!95}a^{5}+\frac{19\!\cdots\!78}{27\!\cdots\!65}a^{4}-\frac{15\!\cdots\!46}{83\!\cdots\!95}a^{3}+\frac{11\!\cdots\!72}{83\!\cdots\!95}a^{2}-\frac{10\!\cdots\!74}{55\!\cdots\!13}a-\frac{92\!\cdots\!02}{83\!\cdots\!95}$, $\frac{10\!\cdots\!26}{24\!\cdots\!85}a^{28}-\frac{26\!\cdots\!29}{83\!\cdots\!95}a^{27}+\frac{33\!\cdots\!09}{21\!\cdots\!41}a^{26}-\frac{73\!\cdots\!23}{24\!\cdots\!85}a^{25}+\frac{55\!\cdots\!35}{55\!\cdots\!13}a^{24}-\frac{13\!\cdots\!71}{24\!\cdots\!85}a^{23}+\frac{10\!\cdots\!91}{27\!\cdots\!65}a^{22}-\frac{20\!\cdots\!84}{24\!\cdots\!85}a^{21}+\frac{28\!\cdots\!26}{24\!\cdots\!85}a^{20}-\frac{13\!\cdots\!83}{83\!\cdots\!95}a^{19}+\frac{34\!\cdots\!39}{13\!\cdots\!15}a^{18}-\frac{82\!\cdots\!01}{24\!\cdots\!85}a^{17}+\frac{31\!\cdots\!57}{83\!\cdots\!95}a^{16}-\frac{10\!\cdots\!68}{24\!\cdots\!85}a^{15}+\frac{93\!\cdots\!48}{24\!\cdots\!85}a^{14}-\frac{37\!\cdots\!08}{11\!\cdots\!89}a^{13}+\frac{70\!\cdots\!87}{24\!\cdots\!85}a^{12}-\frac{17\!\cdots\!19}{11\!\cdots\!35}a^{11}+\frac{21\!\cdots\!98}{48\!\cdots\!35}a^{10}+\frac{65\!\cdots\!38}{24\!\cdots\!85}a^{9}-\frac{21\!\cdots\!87}{24\!\cdots\!85}a^{8}+\frac{54\!\cdots\!73}{83\!\cdots\!95}a^{7}+\frac{36\!\cdots\!23}{67\!\cdots\!05}a^{6}-\frac{22\!\cdots\!14}{49\!\cdots\!17}a^{5}+\frac{66\!\cdots\!57}{24\!\cdots\!85}a^{4}-\frac{37\!\cdots\!53}{49\!\cdots\!17}a^{3}+\frac{97\!\cdots\!44}{24\!\cdots\!85}a^{2}+\frac{24\!\cdots\!97}{24\!\cdots\!85}a-\frac{43\!\cdots\!73}{24\!\cdots\!85}$, $\frac{74\!\cdots\!19}{14\!\cdots\!05}a^{28}-\frac{36\!\cdots\!64}{10\!\cdots\!95}a^{27}+\frac{29\!\cdots\!82}{16\!\cdots\!39}a^{26}-\frac{14\!\cdots\!13}{49\!\cdots\!17}a^{25}+\frac{47\!\cdots\!66}{48\!\cdots\!35}a^{24}-\frac{19\!\cdots\!48}{24\!\cdots\!85}a^{23}+\frac{35\!\cdots\!28}{83\!\cdots\!95}a^{22}-\frac{21\!\cdots\!64}{24\!\cdots\!85}a^{21}+\frac{31\!\cdots\!97}{24\!\cdots\!85}a^{20}-\frac{52\!\cdots\!82}{27\!\cdots\!65}a^{19}+\frac{39\!\cdots\!59}{13\!\cdots\!15}a^{18}-\frac{18\!\cdots\!01}{49\!\cdots\!17}a^{17}+\frac{12\!\cdots\!96}{27\!\cdots\!65}a^{16}-\frac{12\!\cdots\!41}{24\!\cdots\!85}a^{15}+\frac{11\!\cdots\!34}{24\!\cdots\!85}a^{14}-\frac{47\!\cdots\!36}{11\!\cdots\!89}a^{13}+\frac{18\!\cdots\!58}{49\!\cdots\!17}a^{12}-\frac{25\!\cdots\!29}{11\!\cdots\!35}a^{11}+\frac{19\!\cdots\!00}{16\!\cdots\!39}a^{10}-\frac{52\!\cdots\!64}{24\!\cdots\!85}a^{9}-\frac{20\!\cdots\!06}{24\!\cdots\!85}a^{8}+\frac{35\!\cdots\!37}{83\!\cdots\!95}a^{7}+\frac{24\!\cdots\!38}{92\!\cdots\!85}a^{6}-\frac{60\!\cdots\!78}{24\!\cdots\!85}a^{5}+\frac{14\!\cdots\!39}{24\!\cdots\!85}a^{4}-\frac{19\!\cdots\!94}{24\!\cdots\!85}a^{3}+\frac{19\!\cdots\!78}{49\!\cdots\!17}a^{2}-\frac{44\!\cdots\!03}{24\!\cdots\!85}a+\frac{31\!\cdots\!13}{29\!\cdots\!01}$, $\frac{13\!\cdots\!49}{24\!\cdots\!85}a^{28}-\frac{39\!\cdots\!84}{83\!\cdots\!95}a^{27}+\frac{21\!\cdots\!06}{83\!\cdots\!95}a^{26}-\frac{16\!\cdots\!59}{24\!\cdots\!85}a^{25}+\frac{54\!\cdots\!13}{83\!\cdots\!95}a^{24}-\frac{25\!\cdots\!14}{24\!\cdots\!85}a^{23}+\frac{49\!\cdots\!03}{83\!\cdots\!95}a^{22}-\frac{85\!\cdots\!50}{49\!\cdots\!17}a^{21}+\frac{73\!\cdots\!76}{24\!\cdots\!85}a^{20}-\frac{71\!\cdots\!59}{16\!\cdots\!45}a^{19}+\frac{91\!\cdots\!26}{13\!\cdots\!15}a^{18}-\frac{22\!\cdots\!91}{24\!\cdots\!85}a^{17}+\frac{32\!\cdots\!06}{27\!\cdots\!65}a^{16}-\frac{33\!\cdots\!28}{24\!\cdots\!85}a^{15}+\frac{34\!\cdots\!48}{24\!\cdots\!85}a^{14}-\frac{14\!\cdots\!29}{11\!\cdots\!89}a^{13}+\frac{29\!\cdots\!54}{24\!\cdots\!85}a^{12}-\frac{10\!\cdots\!96}{11\!\cdots\!35}a^{11}+\frac{42\!\cdots\!84}{83\!\cdots\!95}a^{10}-\frac{55\!\cdots\!18}{24\!\cdots\!85}a^{9}-\frac{14\!\cdots\!14}{24\!\cdots\!85}a^{8}+\frac{22\!\cdots\!41}{10\!\cdots\!05}a^{7}-\frac{46\!\cdots\!56}{85\!\cdots\!95}a^{6}-\frac{18\!\cdots\!82}{24\!\cdots\!85}a^{5}+\frac{24\!\cdots\!19}{24\!\cdots\!85}a^{4}-\frac{96\!\cdots\!65}{49\!\cdots\!17}a^{3}+\frac{91\!\cdots\!17}{49\!\cdots\!17}a^{2}-\frac{20\!\cdots\!24}{24\!\cdots\!85}a+\frac{86\!\cdots\!39}{24\!\cdots\!85}$, $\frac{32\!\cdots\!59}{24\!\cdots\!85}a^{28}-\frac{29\!\cdots\!00}{20\!\cdots\!19}a^{27}+\frac{66\!\cdots\!59}{83\!\cdots\!95}a^{26}-\frac{59\!\cdots\!46}{24\!\cdots\!85}a^{25}+\frac{18\!\cdots\!56}{83\!\cdots\!95}a^{24}-\frac{37\!\cdots\!94}{24\!\cdots\!85}a^{23}+\frac{15\!\cdots\!19}{83\!\cdots\!95}a^{22}-\frac{15\!\cdots\!81}{24\!\cdots\!85}a^{21}+\frac{48\!\cdots\!98}{49\!\cdots\!17}a^{20}-\frac{36\!\cdots\!83}{27\!\cdots\!65}a^{19}+\frac{56\!\cdots\!83}{27\!\cdots\!23}a^{18}-\frac{72\!\cdots\!16}{24\!\cdots\!85}a^{17}+\frac{99\!\cdots\!89}{27\!\cdots\!65}a^{16}-\frac{19\!\cdots\!55}{49\!\cdots\!17}a^{15}+\frac{98\!\cdots\!61}{24\!\cdots\!85}a^{14}-\frac{40\!\cdots\!56}{11\!\cdots\!89}a^{13}+\frac{99\!\cdots\!43}{29\!\cdots\!01}a^{12}-\frac{55\!\cdots\!67}{23\!\cdots\!47}a^{11}+\frac{87\!\cdots\!93}{83\!\cdots\!95}a^{10}-\frac{14\!\cdots\!69}{24\!\cdots\!85}a^{9}-\frac{90\!\cdots\!77}{24\!\cdots\!85}a^{8}+\frac{78\!\cdots\!96}{83\!\cdots\!95}a^{7}+\frac{15\!\cdots\!03}{67\!\cdots\!05}a^{6}-\frac{99\!\cdots\!34}{24\!\cdots\!85}a^{5}-\frac{47\!\cdots\!42}{24\!\cdots\!85}a^{4}-\frac{16\!\cdots\!84}{24\!\cdots\!85}a^{3}+\frac{14\!\cdots\!68}{24\!\cdots\!85}a^{2}+\frac{16\!\cdots\!72}{34\!\cdots\!45}a+\frac{42\!\cdots\!65}{49\!\cdots\!17}$, $\frac{49\!\cdots\!32}{14\!\cdots\!05}a^{28}-\frac{20\!\cdots\!32}{83\!\cdots\!95}a^{27}+\frac{37\!\cdots\!29}{30\!\cdots\!85}a^{26}-\frac{52\!\cdots\!88}{24\!\cdots\!85}a^{25}+\frac{48\!\cdots\!43}{83\!\cdots\!95}a^{24}-\frac{94\!\cdots\!26}{24\!\cdots\!85}a^{23}+\frac{21\!\cdots\!17}{83\!\cdots\!95}a^{22}-\frac{30\!\cdots\!50}{49\!\cdots\!17}a^{21}+\frac{21\!\cdots\!52}{24\!\cdots\!85}a^{20}-\frac{85\!\cdots\!42}{83\!\cdots\!95}a^{19}+\frac{18\!\cdots\!53}{13\!\cdots\!15}a^{18}-\frac{42\!\cdots\!28}{24\!\cdots\!85}a^{17}+\frac{15\!\cdots\!86}{83\!\cdots\!95}a^{16}-\frac{81\!\cdots\!22}{49\!\cdots\!17}a^{15}+\frac{22\!\cdots\!41}{24\!\cdots\!85}a^{14}-\frac{14\!\cdots\!59}{61\!\cdots\!05}a^{13}-\frac{53\!\cdots\!94}{24\!\cdots\!85}a^{12}+\frac{50\!\cdots\!93}{69\!\cdots\!55}a^{11}-\frac{30\!\cdots\!54}{27\!\cdots\!65}a^{10}+\frac{66\!\cdots\!38}{49\!\cdots\!17}a^{9}-\frac{31\!\cdots\!82}{24\!\cdots\!85}a^{8}+\frac{10\!\cdots\!34}{27\!\cdots\!65}a^{7}+\frac{28\!\cdots\!67}{39\!\cdots\!65}a^{6}-\frac{24\!\cdots\!97}{24\!\cdots\!85}a^{5}+\frac{15\!\cdots\!53}{24\!\cdots\!85}a^{4}-\frac{30\!\cdots\!49}{24\!\cdots\!85}a^{3}-\frac{21\!\cdots\!69}{49\!\cdots\!17}a^{2}+\frac{99\!\cdots\!91}{24\!\cdots\!85}a-\frac{56\!\cdots\!81}{49\!\cdots\!17}$, $\frac{39\!\cdots\!78}{49\!\cdots\!17}a^{28}-\frac{15\!\cdots\!03}{27\!\cdots\!65}a^{27}+\frac{80\!\cdots\!02}{27\!\cdots\!65}a^{26}-\frac{24\!\cdots\!02}{49\!\cdots\!17}a^{25}+\frac{10\!\cdots\!16}{55\!\cdots\!13}a^{24}-\frac{61\!\cdots\!09}{49\!\cdots\!17}a^{23}+\frac{56\!\cdots\!23}{83\!\cdots\!95}a^{22}-\frac{35\!\cdots\!37}{24\!\cdots\!85}a^{21}+\frac{10\!\cdots\!15}{49\!\cdots\!17}a^{20}-\frac{26\!\cdots\!09}{83\!\cdots\!95}a^{19}+\frac{65\!\cdots\!57}{13\!\cdots\!15}a^{18}-\frac{15\!\cdots\!68}{24\!\cdots\!85}a^{17}+\frac{61\!\cdots\!57}{83\!\cdots\!95}a^{16}-\frac{20\!\cdots\!22}{24\!\cdots\!85}a^{15}+\frac{38\!\cdots\!12}{49\!\cdots\!17}a^{14}-\frac{13\!\cdots\!56}{18\!\cdots\!15}a^{13}+\frac{14\!\cdots\!48}{24\!\cdots\!85}a^{12}-\frac{86\!\cdots\!20}{23\!\cdots\!47}a^{11}+\frac{59\!\cdots\!08}{30\!\cdots\!85}a^{10}-\frac{94\!\cdots\!39}{24\!\cdots\!85}a^{9}-\frac{19\!\cdots\!03}{14\!\cdots\!05}a^{8}+\frac{21\!\cdots\!93}{27\!\cdots\!65}a^{7}+\frac{37\!\cdots\!52}{67\!\cdots\!05}a^{6}-\frac{13\!\cdots\!98}{24\!\cdots\!85}a^{5}+\frac{25\!\cdots\!43}{24\!\cdots\!85}a^{4}-\frac{32\!\cdots\!81}{24\!\cdots\!85}a^{3}+\frac{16\!\cdots\!94}{24\!\cdots\!85}a^{2}-\frac{64\!\cdots\!22}{24\!\cdots\!85}a-\frac{16\!\cdots\!34}{24\!\cdots\!85}$, $a$ Copy content Toggle raw display (assuming GRH)
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:  \( 6346995452652.517 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{14}\cdot 6346995452652.517 \cdot 1}{2\cdot\sqrt{190430537333205962921320156798047381287357677729}}\cr\approx \mathstrut & 2.17379230843342 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^29 - 8*x^28 + 42*x^27 - 91*x^26 + 71*x^25 - 172*x^24 + 980*x^23 - 2477*x^22 + 4032*x^21 - 5998*x^20 + 8995*x^19 - 12396*x^18 + 15464*x^17 - 17477*x^16 + 17661*x^15 - 16777*x^14 + 14899*x^13 - 10749*x^12 + 6197*x^11 - 2594*x^10 - 1233*x^9 + 2318*x^8 - 113*x^7 - 855*x^6 + 1459*x^5 - 2625*x^4 + 2079*x^3 - 978*x^2 + 419*x + 1)
 
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 - 8*x^28 + 42*x^27 - 91*x^26 + 71*x^25 - 172*x^24 + 980*x^23 - 2477*x^22 + 4032*x^21 - 5998*x^20 + 8995*x^19 - 12396*x^18 + 15464*x^17 - 17477*x^16 + 17661*x^15 - 16777*x^14 + 14899*x^13 - 10749*x^12 + 6197*x^11 - 2594*x^10 - 1233*x^9 + 2318*x^8 - 113*x^7 - 855*x^6 + 1459*x^5 - 2625*x^4 + 2079*x^3 - 978*x^2 + 419*x + 1, 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 - 8*x^28 + 42*x^27 - 91*x^26 + 71*x^25 - 172*x^24 + 980*x^23 - 2477*x^22 + 4032*x^21 - 5998*x^20 + 8995*x^19 - 12396*x^18 + 15464*x^17 - 17477*x^16 + 17661*x^15 - 16777*x^14 + 14899*x^13 - 10749*x^12 + 6197*x^11 - 2594*x^10 - 1233*x^9 + 2318*x^8 - 113*x^7 - 855*x^6 + 1459*x^5 - 2625*x^4 + 2079*x^3 - 978*x^2 + 419*x + 1);
 
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 - 8*x^28 + 42*x^27 - 91*x^26 + 71*x^25 - 172*x^24 + 980*x^23 - 2477*x^22 + 4032*x^21 - 5998*x^20 + 8995*x^19 - 12396*x^18 + 15464*x^17 - 17477*x^16 + 17661*x^15 - 16777*x^14 + 14899*x^13 - 10749*x^12 + 6197*x^11 - 2594*x^10 - 1233*x^9 + 2318*x^8 - 113*x^7 - 855*x^6 + 1459*x^5 - 2625*x^4 + 2079*x^3 - 978*x^2 + 419*x + 1);
 
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 $29$ ${\href{/padicField/3.2.0.1}{2} }^{14}{,}\,{\href{/padicField/3.1.0.1}{1} }$ ${\href{/padicField/5.2.0.1}{2} }^{14}{,}\,{\href{/padicField/5.1.0.1}{1} }$ $29$ $29$ $29$ ${\href{/padicField/17.2.0.1}{2} }^{14}{,}\,{\href{/padicField/17.1.0.1}{1} }$ $29$ $29$ $29$ $29$ ${\href{/padicField/37.2.0.1}{2} }^{14}{,}\,{\href{/padicField/37.1.0.1}{1} }$ $29$ $29$ $29$ ${\href{/padicField/53.2.0.1}{2} }^{14}{,}\,{\href{/padicField/53.1.0.1}{1} }$ $29$

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
\(2383\) Copy content Toggle raw display $\Q_{2383}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.2383.2t1.a.a$1$ $ 2383 $ \(\Q(\sqrt{-2383}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.2383.29t2.a.b$2$ $ 2383 $ 29.1.190430537333205962921320156798047381287357677729.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2383.29t2.a.l$2$ $ 2383 $ 29.1.190430537333205962921320156798047381287357677729.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2383.29t2.a.k$2$ $ 2383 $ 29.1.190430537333205962921320156798047381287357677729.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2383.29t2.a.e$2$ $ 2383 $ 29.1.190430537333205962921320156798047381287357677729.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2383.29t2.a.f$2$ $ 2383 $ 29.1.190430537333205962921320156798047381287357677729.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2383.29t2.a.c$2$ $ 2383 $ 29.1.190430537333205962921320156798047381287357677729.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2383.29t2.a.i$2$ $ 2383 $ 29.1.190430537333205962921320156798047381287357677729.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2383.29t2.a.a$2$ $ 2383 $ 29.1.190430537333205962921320156798047381287357677729.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2383.29t2.a.d$2$ $ 2383 $ 29.1.190430537333205962921320156798047381287357677729.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2383.29t2.a.h$2$ $ 2383 $ 29.1.190430537333205962921320156798047381287357677729.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2383.29t2.a.j$2$ $ 2383 $ 29.1.190430537333205962921320156798047381287357677729.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2383.29t2.a.n$2$ $ 2383 $ 29.1.190430537333205962921320156798047381287357677729.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2383.29t2.a.m$2$ $ 2383 $ 29.1.190430537333205962921320156798047381287357677729.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2383.29t2.a.g$2$ $ 2383 $ 29.1.190430537333205962921320156798047381287357677729.1 $D_{29}$ (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 *.