Properties

Label 29.29.383...121.1
Degree $29$
Signature $[29, 0]$
Discriminant $3.836\times 10^{49}$
Root discriminant \(51.26\)
Ramified prime $59$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{29}$ (as 29T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^29 - x^28 - 28*x^27 + 27*x^26 + 351*x^25 - 325*x^24 - 2600*x^23 + 2300*x^22 + 12650*x^21 - 10626*x^20 - 42504*x^19 + 33649*x^18 + 100947*x^17 - 74613*x^16 - 170544*x^15 + 116280*x^14 + 203490*x^13 - 125970*x^12 - 167960*x^11 + 92378*x^10 + 92378*x^9 - 43758*x^8 - 31824*x^7 + 12376*x^6 + 6188*x^5 - 1820*x^4 - 560*x^3 + 105*x^2 + 15*x - 1)
 
gp: K = bnfinit(y^29 - y^28 - 28*y^27 + 27*y^26 + 351*y^25 - 325*y^24 - 2600*y^23 + 2300*y^22 + 12650*y^21 - 10626*y^20 - 42504*y^19 + 33649*y^18 + 100947*y^17 - 74613*y^16 - 170544*y^15 + 116280*y^14 + 203490*y^13 - 125970*y^12 - 167960*y^11 + 92378*y^10 + 92378*y^9 - 43758*y^8 - 31824*y^7 + 12376*y^6 + 6188*y^5 - 1820*y^4 - 560*y^3 + 105*y^2 + 15*y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^29 - x^28 - 28*x^27 + 27*x^26 + 351*x^25 - 325*x^24 - 2600*x^23 + 2300*x^22 + 12650*x^21 - 10626*x^20 - 42504*x^19 + 33649*x^18 + 100947*x^17 - 74613*x^16 - 170544*x^15 + 116280*x^14 + 203490*x^13 - 125970*x^12 - 167960*x^11 + 92378*x^10 + 92378*x^9 - 43758*x^8 - 31824*x^7 + 12376*x^6 + 6188*x^5 - 1820*x^4 - 560*x^3 + 105*x^2 + 15*x - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^29 - x^28 - 28*x^27 + 27*x^26 + 351*x^25 - 325*x^24 - 2600*x^23 + 2300*x^22 + 12650*x^21 - 10626*x^20 - 42504*x^19 + 33649*x^18 + 100947*x^17 - 74613*x^16 - 170544*x^15 + 116280*x^14 + 203490*x^13 - 125970*x^12 - 167960*x^11 + 92378*x^10 + 92378*x^9 - 43758*x^8 - 31824*x^7 + 12376*x^6 + 6188*x^5 - 1820*x^4 - 560*x^3 + 105*x^2 + 15*x - 1)
 

\( x^{29} - x^{28} - 28 x^{27} + 27 x^{26} + 351 x^{25} - 325 x^{24} - 2600 x^{23} + 2300 x^{22} + 12650 x^{21} - 10626 x^{20} - 42504 x^{19} + 33649 x^{18} + 100947 x^{17} - 74613 x^{16} + \cdots - 1 \) 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:  $[29, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(38358032782038398419973086399760468678777161743121\) \(\medspace = 59^{28}\) 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:  \(51.26\)
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:  $59^{28/29}\approx 51.261126887866126$
Ramified primes:   \(59\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $29$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(59\)
Dirichlet character group:    $\lbrace$$\chi_{59}(1,·)$, $\chi_{59}(3,·)$, $\chi_{59}(4,·)$, $\chi_{59}(5,·)$, $\chi_{59}(7,·)$, $\chi_{59}(9,·)$, $\chi_{59}(12,·)$, $\chi_{59}(15,·)$, $\chi_{59}(16,·)$, $\chi_{59}(17,·)$, $\chi_{59}(19,·)$, $\chi_{59}(20,·)$, $\chi_{59}(21,·)$, $\chi_{59}(22,·)$, $\chi_{59}(25,·)$, $\chi_{59}(26,·)$, $\chi_{59}(27,·)$, $\chi_{59}(28,·)$, $\chi_{59}(29,·)$, $\chi_{59}(35,·)$, $\chi_{59}(36,·)$, $\chi_{59}(41,·)$, $\chi_{59}(45,·)$, $\chi_{59}(46,·)$, $\chi_{59}(48,·)$, $\chi_{59}(49,·)$, $\chi_{59}(51,·)$, $\chi_{59}(53,·)$, $\chi_{59}(57,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$ Copy content Toggle raw display

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

Monogenic:  Yes
Index:  $1$
Inessential primes:  None

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:  $28$
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:   $a^{25}-25a^{23}+275a^{21}-1750a^{19}+7125a^{17}-19380a^{15}+35700a^{13}-44200a^{11}+35750a^{9}-17875a^{7}+5005a^{5}-650a^{3}+25a$, $a^{27}-27a^{25}+324a^{23}-2277a^{21}+10395a^{19}-32319a^{17}+69768a^{15}-104652a^{13}+107406a^{11}-72930a^{9}+30888a^{7}-7371a^{5}+819a^{3}-27a$, $a^{5}-5a^{3}+5a$, $a^{18}-18a^{16}+135a^{14}-546a^{12}+1287a^{10}-1782a^{8}+1386a^{6}-540a^{4}+81a^{2}-1$, $a^{17}-17a^{15}+119a^{13}-442a^{11}+935a^{9}-1122a^{7}+714a^{5}-204a^{3}+17a$, $a^{10}-10a^{8}+35a^{6}-50a^{4}+25a^{2}-2$, $a^{20}-20a^{18}+170a^{16}-800a^{14}+2275a^{12}-4004a^{10}+4290a^{8}-2640a^{6}+825a^{4}-100a^{2}+2$, $a^{21}-21a^{19}+189a^{17}-952a^{15}+2940a^{13}-5733a^{11}+7007a^{9}-5148a^{7}+2079a^{5}-385a^{3}+21a$, $a^{9}-9a^{7}+27a^{5}-30a^{3}+9a$, $a^{25}-25a^{23}+275a^{21}-a^{20}-1750a^{19}+20a^{18}+7125a^{17}-170a^{16}-19379a^{15}+800a^{14}+35685a^{13}-2275a^{12}-44110a^{11}+4003a^{10}+35475a^{9}-4280a^{8}-17425a^{7}+2605a^{6}+4628a^{5}-775a^{4}-515a^{3}+75a^{2}+15a-1$, $a^{28}-28a^{26}+350a^{24}-2576a^{22}+12398a^{20}-40984a^{18}+95132a^{16}-155840a^{14}+178634a^{12}-140152a^{10}+72412a^{8}-23136a^{6}+4116a^{4}-336a^{2}+8$, $a^{24}-24a^{22}+252a^{20}-1520a^{18}+5814a^{16}-14688a^{14}+24752a^{12}-27456a^{10}+19306a^{8}-8016a^{6}+1736a^{4}-160a^{2}+4$, $a^{16}-16a^{14}+104a^{12}-352a^{10}+660a^{8}-672a^{6}+336a^{4}-64a^{2}+2$, $a$, $a^{28}-27a^{26}+325a^{24}-2300a^{22}+10626a^{20}-33649a^{18}+74613a^{16}-116280a^{14}+125970a^{12}-92378a^{10}+43758a^{8}-12376a^{6}+1820a^{4}-105a^{2}+1$, $a^{10}-10a^{8}+35a^{6}-50a^{4}+25a^{2}-1$, $a^{18}-18a^{16}+135a^{14}-546a^{12}+1287a^{10}-1782a^{8}+1386a^{6}-540a^{4}+81a^{2}-2$, $a^{5}-5a^{3}+5a-1$, $a^{3}-3a-1$, $a^{28}-28a^{26}-a^{25}+350a^{24}+25a^{23}-2575a^{22}-275a^{21}+12375a^{20}+1750a^{19}-40755a^{18}-7125a^{17}+93840a^{16}+19380a^{15}-151300a^{14}-35700a^{13}+168350a^{12}+44200a^{11}-125125a^{10}-35751a^{9}+58630a^{8}+17884a^{7}-15664a^{6}-5032a^{5}+1969a^{4}+679a^{3}-66a^{2}-31a-1$, $a^{7}-7a^{5}+14a^{3}-7a$, $a^{26}-26a^{24}+299a^{22}-2002a^{20}-a^{19}+8645a^{18}+19a^{17}-25194a^{16}-152a^{15}+50388a^{14}+665a^{13}-68952a^{12}-1729a^{11}+63206a^{10}+2717a^{9}-37180a^{8}-2509a^{7}+13013a^{6}+1261a^{5}-2366a^{4}-299a^{3}+169a^{2}+26a-1$, $a^{21}-21a^{19}+189a^{17}-952a^{15}-a^{14}+2940a^{13}+14a^{12}-5733a^{11}-77a^{10}+7007a^{9}+210a^{8}-5147a^{7}-294a^{6}+2072a^{5}+196a^{4}-371a^{3}-49a^{2}+14a+1$, $a^{28}-a^{27}-27a^{26}+27a^{25}+324a^{24}-325a^{23}-2276a^{22}+2299a^{21}+10373a^{20}-10604a^{19}-32109a^{18}+33440a^{17}+68628a^{16}-73491a^{15}-100777a^{14}+112540a^{13}+98853a^{12}-117962a^{11}-60644a^{10}+81368a^{9}+19723a^{8}-34330a^{7}-1386a^{6}+7692a^{5}-806a^{4}-660a^{3}+124a^{2}+9a-2$, $a^{24}-24a^{22}+252a^{20}-1520a^{18}+5814a^{16}-14688a^{14}+24752a^{12}-27456a^{10}+19305a^{8}-8008a^{6}+1716a^{4}-144a^{2}+2$, $a^{12}-12a^{10}+54a^{8}-112a^{6}+105a^{4}-36a^{2}+2$, $a^{6}-6a^{4}+9a^{2}-2$, $a^{6}-6a^{4}+9a^{2}-1$ 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:  \( 2275980944744796.5 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{29}\cdot(2\pi)^{0}\cdot 2275980944744796.5 \cdot 1}{2\cdot\sqrt{38358032782038398419973086399760468678777161743121}}\cr\approx \mathstrut & 0.0986461947872962 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^29 - x^28 - 28*x^27 + 27*x^26 + 351*x^25 - 325*x^24 - 2600*x^23 + 2300*x^22 + 12650*x^21 - 10626*x^20 - 42504*x^19 + 33649*x^18 + 100947*x^17 - 74613*x^16 - 170544*x^15 + 116280*x^14 + 203490*x^13 - 125970*x^12 - 167960*x^11 + 92378*x^10 + 92378*x^9 - 43758*x^8 - 31824*x^7 + 12376*x^6 + 6188*x^5 - 1820*x^4 - 560*x^3 + 105*x^2 + 15*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 - x^28 - 28*x^27 + 27*x^26 + 351*x^25 - 325*x^24 - 2600*x^23 + 2300*x^22 + 12650*x^21 - 10626*x^20 - 42504*x^19 + 33649*x^18 + 100947*x^17 - 74613*x^16 - 170544*x^15 + 116280*x^14 + 203490*x^13 - 125970*x^12 - 167960*x^11 + 92378*x^10 + 92378*x^9 - 43758*x^8 - 31824*x^7 + 12376*x^6 + 6188*x^5 - 1820*x^4 - 560*x^3 + 105*x^2 + 15*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 - x^28 - 28*x^27 + 27*x^26 + 351*x^25 - 325*x^24 - 2600*x^23 + 2300*x^22 + 12650*x^21 - 10626*x^20 - 42504*x^19 + 33649*x^18 + 100947*x^17 - 74613*x^16 - 170544*x^15 + 116280*x^14 + 203490*x^13 - 125970*x^12 - 167960*x^11 + 92378*x^10 + 92378*x^9 - 43758*x^8 - 31824*x^7 + 12376*x^6 + 6188*x^5 - 1820*x^4 - 560*x^3 + 105*x^2 + 15*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 - x^28 - 28*x^27 + 27*x^26 + 351*x^25 - 325*x^24 - 2600*x^23 + 2300*x^22 + 12650*x^21 - 10626*x^20 - 42504*x^19 + 33649*x^18 + 100947*x^17 - 74613*x^16 - 170544*x^15 + 116280*x^14 + 203490*x^13 - 125970*x^12 - 167960*x^11 + 92378*x^10 + 92378*x^9 - 43758*x^8 - 31824*x^7 + 12376*x^6 + 6188*x^5 - 1820*x^4 - 560*x^3 + 105*x^2 + 15*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

$C_{29}$ (as 29T1):

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 cyclic group of order 29
The 29 conjugacy class representatives for $C_{29}$
Character table for $C_{29}$ is not computed

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 $29$ $29$ $29$ $29$ $29$ $29$ $29$ $29$ $29$ $29$ $29$ $29$ $29$ $29$ $29$ $29$ R

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
\(59\) Copy content Toggle raw display Deg $29$$29$$1$$28$