Properties

Label 29.3.955...720.1
Degree $29$
Signature $[3, 13]$
Discriminant $-9.554\times 10^{57}$
Root discriminant \(99.84\)
Ramified primes see page
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_{29}$ (as 29T8)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^29 - 4*x - 2)
 
gp: K = bnfinit(y^29 - 4*y - 2, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^29 - 4*x - 2);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^29 - 4*x - 2)
 

\( x^{29} - 4x - 2 \) 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:  $[3, 13]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-9553545905420272548496712387718055513666469828515038494720\) \(\medspace = -\,2^{28}\cdot 5\cdot 71\!\cdots\!99\) 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:  \(99.84\)
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:  $2^{28/29}5^{1/2}7117946375474536827576691201119166250278405717399^{1/2}\approx 1.1649626699833909e+25$
Ramified primes:   \(2\), \(5\), \(71179\!\cdots\!17399\) 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:  $\Q(\sqrt{-35589\!\cdots\!86995}$)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$ 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:  $15$
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:   $a+1$, $2a^{15}-4a-1$, $a^{14}-a^{13}+a^{12}-a^{11}+a^{10}-a^{9}+a^{8}-a^{7}+a^{6}-a^{5}+a^{4}-a^{3}+a^{2}-a-1$, $16a^{28}-8a^{27}+5a^{26}+a^{25}+4a^{24}+2a^{23}-a^{21}-2a^{20}-a^{19}-2a^{18}-3a^{17}-5a^{16}-4a^{15}-a^{14}+2a^{13}+4a^{12}+3a^{11}+3a^{10}+2a^{9}+4a^{8}+2a^{7}-a^{6}-7a^{5}-8a^{4}-7a^{3}-3a^{2}-a-65$, $3a^{28}-6a^{26}+2a^{25}+4a^{24}-2a^{23}+2a^{22}+2a^{21}-7a^{20}+7a^{18}-3a^{17}-4a^{16}+2a^{15}-7a^{14}-3a^{13}+12a^{12}+2a^{11}-7a^{10}+6a^{9}+a^{8}-3a^{7}+15a^{6}+5a^{5}-17a^{4}+4a^{2}-10a-5$, $4a^{28}+2a^{27}+2a^{26}-3a^{25}-5a^{24}+2a^{22}+2a^{21}-2a^{19}-2a^{18}+4a^{17}+6a^{16}-6a^{14}-8a^{13}-3a^{12}+4a^{11}+12a^{10}+3a^{9}-6a^{8}-3a^{7}-3a^{6}+6a^{5}+4a^{4}-7a^{3}-13a^{2}-5a-1$, $3a^{28}-4a^{27}-2a^{26}+5a^{25}+2a^{24}-8a^{23}-4a^{22}+8a^{21}+6a^{20}-6a^{19}-6a^{18}+6a^{17}+4a^{16}-10a^{15}-5a^{14}+12a^{13}+10a^{12}-11a^{11}-13a^{10}+8a^{9}+9a^{8}-10a^{7}-8a^{6}+16a^{5}+14a^{4}-19a^{3}-22a^{2}+13a+9$, $5a^{28}+7a^{27}+8a^{26}+7a^{25}+14a^{24}+19a^{23}+8a^{22}+18a^{21}+18a^{20}+14a^{19}+6a^{18}+14a^{17}+5a^{16}-2a^{15}-11a^{14}-3a^{13}-20a^{12}-31a^{11}-23a^{10}-32a^{9}-38a^{8}-46a^{7}-26a^{6}-35a^{5}-38a^{4}-21a^{3}+3a^{2}-14a-9$, $4a^{28}-4a^{27}+6a^{26}-6a^{25}+4a^{23}-3a^{22}+2a^{21}-9a^{20}+13a^{19}-13a^{18}+7a^{17}-6a^{16}+11a^{15}-18a^{14}+10a^{13}-a^{11}-3a^{10}+8a^{8}-15a^{7}+12a^{6}-4a^{5}+13a^{4}-28a^{3}+25a^{2}-8a-15$, $a^{28}-3a^{27}-a^{26}+6a^{25}-7a^{24}+2a^{23}-a^{22}+4a^{21}-7a^{20}+2a^{19}+2a^{18}-a^{17}-5a^{16}+2a^{15}+6a^{14}-12a^{13}+5a^{12}+a^{11}-10a^{9}+10a^{8}-4a^{7}-2a^{6}-6a^{5}+9a^{4}-2a^{3}-14a^{2}+10a-1$, $3a^{28}+2a^{27}+4a^{26}+4a^{25}+4a^{24}+2a^{23}+4a^{22}+4a^{21}+4a^{20}+3a^{19}+6a^{18}+7a^{17}+7a^{16}+5a^{15}+7a^{14}+7a^{13}+7a^{12}+4a^{11}+8a^{10}+10a^{9}+11a^{8}+9a^{7}+13a^{6}+14a^{5}+12a^{4}+8a^{3}+12a^{2}+15a+5$, $9a^{28}+9a^{27}+4a^{26}-7a^{25}-8a^{24}+13a^{22}+12a^{21}+a^{20}-13a^{19}-8a^{18}+6a^{17}+20a^{16}+13a^{15}-6a^{14}-18a^{13}-6a^{12}+18a^{11}+26a^{10}+11a^{9}-18a^{8}-21a^{7}+2a^{6}+35a^{5}+31a^{4}+a^{3}-32a^{2}-22a-15$, $5a^{28}+a^{27}-5a^{26}+8a^{25}-4a^{24}-2a^{23}+8a^{22}-5a^{21}-a^{20}+11a^{19}-15a^{18}+11a^{17}+a^{16}-8a^{15}+12a^{14}-4a^{13}-5a^{12}+12a^{11}-4a^{10}-2a^{9}+13a^{8}-12a^{7}+6a^{6}+5a^{5}-3a^{4}+13a^{2}-16a-11$, $48a^{28}-34a^{27}+6a^{26}-14a^{25}-6a^{24}-13a^{23}-11a^{22}-9a^{21}-6a^{20}-7a^{19}-6a^{18}-2a^{17}+4a^{16}+9a^{15}+9a^{14}+11a^{13}+18a^{12}+24a^{11}+25a^{10}+23a^{9}+21a^{8}+25a^{7}+26a^{6}+17a^{5}+7a^{4}+a^{3}-2a^{2}-8a-217$, $5a^{28}+7a^{27}-5a^{26}-7a^{25}+4a^{24}+7a^{23}-4a^{22}-6a^{21}+4a^{20}+6a^{19}-5a^{18}-5a^{17}+7a^{16}+7a^{15}-9a^{14}-8a^{13}+12a^{12}+13a^{11}-14a^{10}-17a^{9}+15a^{8}+23a^{7}-16a^{6}-29a^{5}+13a^{4}+33a^{3}-13a^{2}-38a-13$ 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:  \( 819283530647138000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{13}\cdot 819283530647138000 \cdot 1}{2\cdot\sqrt{9553545905420272548496712387718055513666469828515038494720}}\cr\approx \mathstrut & 0.797535837477643 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^29 - 4*x - 2)
 
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 - 4*x - 2, 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 - 4*x - 2);
 
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 - 4*x - 2);
 
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

$S_{29}$ (as 29T8):

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 non-solvable group of order 8841761993739701954543616000000
The 4565 conjugacy class representatives for $S_{29}$ are not computed
Character table for $S_{29}$ 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 R $22{,}\,{\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ R $21{,}\,{\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ $16{,}\,{\href{/padicField/11.7.0.1}{7} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ $17{,}\,{\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ $15{,}\,{\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ $24{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.14.0.1}{14} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.13.0.1}{13} }{,}\,{\href{/padicField/31.9.0.1}{9} }{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ $20{,}\,{\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.9.0.1}{9} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.13.0.1}{13} }{,}\,{\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.6.0.1}{6} }$ ${\href{/padicField/47.11.0.1}{11} }{,}\,{\href{/padicField/47.9.0.1}{9} }{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ $16{,}\,{\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ $21{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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 Deg $29$$29$$1$$28$
\(5\) Copy content Toggle raw display 5.2.0.1$x^{2} + 4 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} + 4 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.1.1$x^{2} + 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.23.0.1$x^{23} + 2 x^{2} + 3$$1$$23$$0$$C_{23}$$[\ ]^{23}$
\(711\!\cdots\!399\) Copy content Toggle raw display Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $9$$1$$9$$0$$C_9$$[\ ]^{9}$
Deg $14$$1$$14$$0$$C_{14}$$[\ ]^{14}$