Properties

Label 29.1.587...445.1
Degree $29$
Signature $[1, 14]$
Discriminant $5.874\times 10^{55}$
Root discriminant \(83.77\)
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 - 3)
 
gp: K = bnfinit(y^29 + y - 3, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^29 + x - 3);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^29 + x - 3)
 

\( x^{29} + x - 3 \) 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:   \(58740423215346262667355847452891580849174093037099352445\) \(\medspace = 5\cdot 11\cdot 16381\cdot 127865750920013\cdot 1994744457920647\cdot 255618611114956521989\) 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:  \(83.77\)
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:  $5^{1/2}11^{1/2}16381^{1/2}127865750920013^{1/2}1994744457920647^{1/2}255618611114956521989^{1/2}\approx 7.664230112369165e+27$
Ramified primes:   \(5\), \(11\), \(16381\), \(127865750920013\), \(1994744457920647\), \(255618611114956521989\) 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:  $\Q(\sqrt{58740\!\cdots\!52445}$)
$\card{ \Aut(K/\Q) }$:  $1$
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:   $a-1$, $a^{28}+a^{27}-a^{25}-a^{24}-a^{23}-a^{22}+a^{20}+a^{19}+a^{18}+a^{17}-a^{15}-a^{14}-a^{13}-a^{12}+a^{10}+a^{9}+a^{8}+a^{7}-a^{5}-a^{4}-a^{3}-a^{2}+2$, $a^{26}-a^{24}+a^{23}-2a^{21}+a^{20}+a^{19}-2a^{18}+a^{17}+2a^{16}-2a^{15}+2a^{13}-2a^{12}-a^{11}+2a^{10}-a^{9}-a^{8}+2a^{7}-a^{5}+a^{4}-a^{2}+1$, $a^{28}+2a^{27}+2a^{26}+2a^{25}+a^{24}-a^{22}-2a^{21}-2a^{20}-2a^{19}-a^{18}-a^{17}+a^{16}+2a^{14}+2a^{12}+2a^{10}+2a^{8}-a^{7}+a^{6}-3a^{5}-a^{4}-5a^{3}-a^{2}-4a+2$, $6a^{28}+4a^{27}-a^{26}-5a^{25}-8a^{24}-9a^{23}-6a^{22}-a^{21}+4a^{20}+7a^{19}+7a^{18}+4a^{17}-a^{16}-8a^{15}-12a^{14}-10a^{13}-7a^{12}-a^{11}+7a^{10}+10a^{9}+9a^{8}+4a^{7}-4a^{6}-11a^{5}-15a^{4}-14a^{3}-7a^{2}+2a+14$, $a^{28}-4a^{27}+4a^{26}-10a^{24}+4a^{23}+9a^{22}-5a^{21}-a^{20}+6a^{19}-9a^{18}-5a^{17}+14a^{16}-9a^{14}+7a^{13}+2a^{12}-14a^{11}+10a^{10}+10a^{9}-14a^{8}-3a^{7}+16a^{6}-11a^{5}-3a^{4}+17a^{3}-8a^{2}-22a+22$, $6a^{28}-3a^{27}-7a^{26}-11a^{25}+16a^{24}+10a^{23}-10a^{22}-7a^{21}-9a^{20}+15a^{19}+17a^{18}-18a^{17}-9a^{16}-2a^{15}+11a^{14}+23a^{13}-24a^{12}-15a^{11}+9a^{10}+9a^{9}+23a^{8}-24a^{7}-27a^{6}+22a^{5}+12a^{4}+15a^{3}-17a^{2}-42a+40$, $a^{28}+3a^{27}-2a^{26}-5a^{25}+3a^{24}+3a^{23}-2a^{22}-2a^{21}+a^{20}-3a^{19}+3a^{18}+4a^{17}-5a^{16}-6a^{15}+7a^{14}+2a^{13}-4a^{12}-a^{11}-3a^{9}+6a^{8}+3a^{7}-10a^{6}-3a^{5}+10a^{4}-6a^{2}+2a-4$, $5a^{28}+2a^{27}-4a^{26}-4a^{25}+a^{24}+5a^{23}+3a^{22}-4a^{21}-6a^{20}+a^{19}+8a^{18}+3a^{17}-7a^{16}-7a^{15}+3a^{14}+10a^{13}+2a^{12}-11a^{11}-8a^{10}+8a^{9}+13a^{8}-3a^{7}-14a^{6}-4a^{5}+11a^{4}+11a^{3}-5a^{2}-14a+2$, $4a^{28}-2a^{27}+3a^{26}+4a^{25}-a^{24}-2a^{23}-7a^{22}+2a^{21}+4a^{20}-4a^{19}+8a^{18}+3a^{17}-8a^{16}-2a^{15}-4a^{14}+4a^{13}+3a^{12}-2a^{11}+12a^{10}-5a^{9}-13a^{8}+4a^{7}-3a^{6}+4a^{5}+5a^{4}+a^{3}+11a^{2}-17a-7$, $3a^{28}+15a^{27}+26a^{26}+26a^{25}+34a^{24}+18a^{23}+20a^{22}-5a^{21}-9a^{20}-28a^{19}-34a^{18}-34a^{17}-36a^{16}-16a^{15}-13a^{14}+18a^{13}+18a^{12}+47a^{11}+36a^{10}+47a^{9}+30a^{8}+11a^{7}+4a^{6}-38a^{5}-26a^{4}-66a^{3}-38a^{2}-54a-16$, $6a^{28}-4a^{27}+2a^{26}+2a^{25}-4a^{24}+2a^{23}-5a^{22}+9a^{21}-10a^{20}+9a^{19}-6a^{18}+a^{17}-a^{16}+2a^{15}+5a^{14}-11a^{13}+14a^{12}-15a^{11}+11a^{10}-9a^{9}+12a^{8}-10a^{7}+6a^{5}-13a^{4}+18a^{3}-15a^{2}+17a-17$, $3a^{28}+4a^{27}-10a^{26}+14a^{25}-15a^{24}+13a^{23}-8a^{22}-a^{21}+12a^{20}-22a^{19}+27a^{18}-24a^{17}+12a^{16}+8a^{15}-28a^{14}+41a^{13}-41a^{12}+26a^{11}-29a^{9}+50a^{8}-54a^{7}+39a^{6}-12a^{5}-20a^{4}+46a^{3}-56a^{2}+50a-28$, $3a^{28}-a^{27}-3a^{26}+2a^{25}-2a^{24}+5a^{22}-3a^{21}+2a^{19}-4a^{18}-2a^{17}+5a^{16}-3a^{15}+3a^{14}+3a^{13}-7a^{12}+a^{11}+2a^{10}-6a^{9}+4a^{8}+6a^{7}-6a^{6}+4a^{5}-a^{4}-10a^{3}+9a^{2}-2$ 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:  \( 42821205896121990 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{14}\cdot 42821205896121990 \cdot 1}{2\cdot\sqrt{58740423215346262667355847452891580849174093037099352445}}\cr\approx \mathstrut & 0.835042375123828 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^29 + x - 3)
 
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 - 3, 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 - 3);
 
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 - 3);
 
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 $27{,}\,{\href{/padicField/2.2.0.1}{2} }$ ${\href{/padicField/3.6.0.1}{6} }^{4}{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }$ R $18{,}\,{\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ R ${\href{/padicField/13.11.0.1}{11} }{,}\,{\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.14.0.1}{14} }{,}\,{\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ $20{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ ${\href{/padicField/23.14.0.1}{14} }{,}\,{\href{/padicField/23.11.0.1}{11} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.2.0.1}{2} }^{14}{,}\,{\href{/padicField/29.1.0.1}{1} }$ $27{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ $17{,}\,{\href{/padicField/37.11.0.1}{11} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ $17{,}\,{\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.4.0.1}{4} }$ $24{,}\,{\href{/padicField/43.5.0.1}{5} }$ $25{,}\,{\href{/padicField/47.4.0.1}{4} }$ $17{,}\,{\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ $15{,}\,{\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$

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
\(5\) Copy content Toggle raw display 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}$
5.2.0.1$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}$
\(11\) Copy content Toggle raw display $\Q_{11}$$x + 9$$1$$1$$0$Trivial$[\ ]$
11.2.1.1$x^{2} + 22$$2$$1$$1$$C_2$$[\ ]_{2}$
11.5.0.1$x^{5} + 10 x^{2} + 9$$1$$5$$0$$C_5$$[\ ]^{5}$
11.7.0.1$x^{7} + 4 x + 9$$1$$7$$0$$C_7$$[\ ]^{7}$
11.14.0.1$x^{14} + 2 x^{7} + 9 x^{6} + 6 x^{5} + 4 x^{4} + 8 x^{3} + 6 x^{2} + 10 x + 2$$1$$14$$0$$C_{14}$$[\ ]^{14}$
\(16381\) Copy content Toggle raw display $\Q_{16381}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $5$$1$$5$$0$$C_5$$[\ ]^{5}$
Deg $8$$1$$8$$0$$C_8$$[\ ]^{8}$
Deg $13$$1$$13$$0$$C_{13}$$[\ ]^{13}$
\(127865750920013\) Copy content Toggle raw display $\Q_{127865750920013}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $6$$1$$6$$0$$C_6$$[\ ]^{6}$
Deg $6$$1$$6$$0$$C_6$$[\ ]^{6}$
Deg $14$$1$$14$$0$$C_{14}$$[\ ]^{14}$
\(1994744457920647\) Copy content Toggle raw display $\Q_{1994744457920647}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{1994744457920647}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $21$$1$$21$$0$$C_{21}$$[\ ]^{21}$
\(255\!\cdots\!989\) Copy content Toggle raw display $\Q_{25\!\cdots\!89}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{25\!\cdots\!89}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $23$$1$$23$$0$$C_{23}$$[\ ]^{23}$