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Properties

Label 29.1.185...720.1
Degree $29$
Signature $[1, 14]$
Discriminant $1.850\times 10^{59}$
Root discriminant \(110.59\)
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 + x - 4)
 
gp: K = bnfinit(y^29 + y - 4, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^29 + x - 4);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^29 + x - 4)
 

\( x^{29} + x - 4 \) 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:   \(185021286441299404600865722102234162683483422027183523102720\) \(\medspace = 2^{57}\cdot 3\cdot 5\cdot 5205763\cdot 56893477\cdot 288984011824906401353807959\) 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:  \(110.59\)
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:  not computed
Ramified primes:   \(2\), \(3\), \(5\), \(5205763\), \(56893477\), \(288984011824906401353807959\) 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:  $\Q(\sqrt{25676\!\cdots\!60270}$)
$\card{ \Aut(K/\Q) }$:  $1$
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:  $14$
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:   $3a^{28}+2a^{27}+a^{26}-a^{25}+4a^{22}+7a^{21}+10a^{20}+11a^{19}+10a^{18}+7a^{17}+a^{16}-a^{15}-5a^{14}-4a^{13}-3a^{12}+a^{11}+a^{10}-3a^{8}-12a^{7}-17a^{6}-23a^{5}-22a^{4}-21a^{3}-10a^{2}-2a+7$, $40a^{28}+32a^{27}-13a^{26}-47a^{25}-30a^{24}+22a^{23}+52a^{22}+26a^{21}-29a^{20}-55a^{19}-24a^{18}+33a^{17}+57a^{16}+23a^{15}-36a^{14}-62a^{13}-27a^{12}+40a^{11}+74a^{10}+33a^{9}-50a^{8}-90a^{7}-36a^{6}+67a^{5}+111a^{4}+33a^{3}-96a^{2}-130a+23$, $9a^{28}+13a^{27}+8a^{26}-2a^{25}-7a^{24}-a^{23}+12a^{22}+18a^{21}+8a^{20}-5a^{19}-9a^{18}+2a^{17}+17a^{16}+22a^{15}+8a^{14}-9a^{13}-10a^{12}+5a^{11}+25a^{10}+26a^{9}+7a^{8}-15a^{7}-11a^{6}+12a^{5}+32a^{4}+31a^{3}+5a^{2}-21a-7$, $6a^{28}-4a^{27}+2a^{26}-2a^{24}+4a^{23}-6a^{22}+8a^{21}-10a^{20}+12a^{19}-14a^{18}+16a^{17}-18a^{16}+20a^{15}-22a^{14}+24a^{13}-26a^{12}+28a^{11}-30a^{10}+32a^{9}-34a^{8}+36a^{7}-38a^{6}+40a^{5}-42a^{4}+44a^{3}-45a^{2}+43a-31$, $37a^{28}+33a^{27}+16a^{26}-10a^{25}-34a^{24}-45a^{23}-39a^{22}-16a^{21}+17a^{20}+44a^{19}+54a^{18}+45a^{17}+16a^{16}-25a^{15}-57a^{14}-66a^{13}-51a^{12}-13a^{11}+36a^{10}+72a^{9}+81a^{8}+60a^{7}+9a^{6}-52a^{5}-91a^{4}-97a^{3}-69a^{2}-5a+107$, $2a^{28}-3a^{27}-9a^{25}+19a^{24}-10a^{23}+9a^{22}-19a^{21}+14a^{20}+a^{19}+3a^{18}+a^{17}-16a^{16}+20a^{15}-13a^{14}+29a^{13}-32a^{12}+12a^{11}-9a^{10}+22a^{9}+3a^{8}-27a^{7}+20a^{6}-21a^{5}+50a^{4}-45a^{3}+28a^{2}-39a+33$, $81a^{28}+20a^{27}+47a^{26}+73a^{25}-11a^{24}+38a^{23}+33a^{22}-49a^{21}+a^{20}-11a^{19}-103a^{18}-27a^{17}-54a^{16}-146a^{15}-31a^{14}-89a^{13}-141a^{12}-17a^{11}-67a^{10}-118a^{9}+52a^{8}-26a^{7}-57a^{6}+160a^{5}+11a^{4}+42a^{3}+233a^{2}+60a+175$, $23a^{28}+2a^{27}-32a^{26}+13a^{25}+20a^{24}-30a^{23}+a^{22}+28a^{21}-29a^{20}-8a^{19}+52a^{18}-10a^{17}-38a^{16}+33a^{15}-6a^{14}-53a^{13}+42a^{12}+37a^{11}-63a^{10}+10a^{9}+58a^{8}-53a^{7}-11a^{6}+81a^{5}-55a^{4}-75a^{3}+87a^{2}+10a-53$, $6a^{28}-17a^{27}-18a^{26}+23a^{25}+18a^{24}-15a^{23}-12a^{22}-10a^{21}+11a^{20}+35a^{19}-19a^{18}-37a^{17}+14a^{16}+16a^{15}+16a^{14}-51a^{12}-2a^{11}+59a^{10}+a^{9}-31a^{8}-18a^{7}-18a^{6}+57a^{5}+41a^{4}-81a^{3}-27a^{2}+39a+27$, $18a^{28}-32a^{27}+22a^{26}-21a^{25}+a^{24}+16a^{23}-28a^{22}+42a^{21}-32a^{20}+25a^{19}+5a^{18}-24a^{17}+51a^{16}-55a^{15}+41a^{14}-26a^{13}-21a^{12}+37a^{11}-72a^{10}+71a^{9}-53a^{8}+25a^{7}+33a^{6}-64a^{5}+104a^{4}-93a^{3}+75a^{2}-7a-41$, $a^{28}+2a^{27}-a^{25}-6a^{24}+2a^{23}+4a^{22}+3a^{21}+3a^{20}-6a^{19}+2a^{16}+3a^{15}-3a^{14}+a^{13}-7a^{12}+4a^{11}+10a^{10}+2a^{9}+a^{8}-12a^{7}-2a^{6}+a^{5}+9a^{4}+8a^{3}-14a^{2}+5a+1$, $18a^{28}+12a^{27}+3a^{26}-2a^{25}-10a^{24}-21a^{23}-28a^{22}-36a^{21}-44a^{20}-48a^{19}-48a^{18}-50a^{17}-49a^{16}-43a^{15}-38a^{14}-31a^{13}-17a^{12}-2a^{11}+9a^{10}+26a^{9}+44a^{8}+50a^{7}+66a^{6}+78a^{5}+81a^{4}+88a^{3}+90a^{2}+86a+89$, $32a^{28}-114a^{27}+137a^{26}-73a^{25}-42a^{24}+132a^{23}-159a^{22}+106a^{21}+18a^{20}-161a^{19}+216a^{18}-139a^{17}-22a^{16}+172a^{15}-253a^{14}+205a^{13}-18a^{12}-217a^{11}+329a^{10}-252a^{9}+37a^{8}+212a^{7}-393a^{6}+367a^{5}-96a^{4}-270a^{3}+483a^{2}-439a+195$, $20a^{28}-90a^{27}+31a^{26}-93a^{25}+41a^{24}-85a^{23}+61a^{22}-70a^{21}+74a^{20}-48a^{19}+92a^{18}-19a^{17}+92a^{16}+9a^{15}+88a^{14}+41a^{13}+59a^{12}+62a^{11}+22a^{10}+85a^{9}-40a^{8}+94a^{7}-108a^{6}+113a^{5}-193a^{4}+122a^{3}-271a^{2}+151a-325$ 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:  \( 7158511448931481000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{14}\cdot 7158511448931481000 \cdot 1}{2\cdot\sqrt{185021286441299404600865722102234162683483422027183523102720}}\cr\approx \mathstrut & 2.48731066929119 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^29 + x - 4)
 
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 - 4, 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 - 4);
 
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 - 4);
 
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 $S_{29}$
Character table for $S_{29}$

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 R R $18{,}\,{\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ $15{,}\,{\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ $26{,}\,{\href{/padicField/13.3.0.1}{3} }$ ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.8.0.1}{8} }^{2}{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ $22{,}\,{\href{/padicField/23.7.0.1}{7} }$ ${\href{/padicField/29.2.0.1}{2} }^{14}{,}\,{\href{/padicField/29.1.0.1}{1} }$ $27{,}\,{\href{/padicField/31.2.0.1}{2} }$ $26{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ $15{,}\,{\href{/padicField/41.14.0.1}{14} }$ $26{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ $23{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ $23{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ $20{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$

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$Trivial$[\ ]$
2.4.9.7$x^{4} + 2 x^{2} + 6$$4$$1$$9$$D_{4}$$[2, 3, 7/2]$
2.12.24.266$x^{12} + 14 x^{10} + 8 x^{9} - 2 x^{8} + 80 x^{7} - 224 x^{6} + 160 x^{5} + 1316 x^{4} + 2592 x^{3} + 5240 x^{2} + 3424 x + 6392$$4$$3$$24$12T87$[2, 2, 2, 3, 3, 3]^{3}$
2.12.24.163$x^{12} + 12 x^{11} + 46 x^{10} + 52 x^{9} + 58 x^{8} + 320 x^{7} + 288 x^{6} + 944 x^{5} + 724 x^{4} + 1168 x^{3} + 536 x^{2} + 432 x + 72$$4$$3$$24$12T134$[2, 2, 2, 2, 3, 3]^{6}$
\(3\) Copy content Toggle raw display 3.2.1.1$x^{2} + 6$$2$$1$$1$$C_2$$[\ ]_{2}$
3.7.0.1$x^{7} + 2 x^{2} + 1$$1$$7$$0$$C_7$$[\ ]^{7}$
3.8.0.1$x^{8} + 2 x^{5} + x^{4} + 2 x^{2} + 2 x + 2$$1$$8$$0$$C_8$$[\ ]^{8}$
3.12.0.1$x^{12} + x^{6} + x^{5} + x^{4} + x^{2} + 2$$1$$12$$0$$C_{12}$$[\ ]^{12}$
\(5\) Copy content Toggle raw display 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}$
5.2.0.1$x^{2} + 4 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.23.0.1$x^{23} + 2 x^{2} + 3$$1$$23$$0$$C_{23}$$[\ ]^{23}$
\(5205763\) Copy content Toggle raw display $\Q_{5205763}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{5205763}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $10$$1$$10$$0$$C_{10}$$[\ ]^{10}$
Deg $13$$1$$13$$0$$C_{13}$$[\ ]^{13}$
\(56893477\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $17$$1$$17$$0$$C_{17}$$[\ ]^{17}$
\(288\!\cdots\!959\) Copy content Toggle raw display $\Q_{28\!\cdots\!59}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{28\!\cdots\!59}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $5$$1$$5$$0$$C_5$$[\ ]^{5}$
Deg $14$$1$$14$$0$$C_{14}$$[\ ]^{14}$