# 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)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^29 + x - 3)

gp: K = bnfinit(x^29 + x - 3, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]);

$$x^{29} + x - 3$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $29$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[1, 14]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$58740423215346262667355847452891580849174093037099352445$$$$\medspace = 5\cdot 11\cdot 16381\cdot 127865750920013\cdot 1994744457920647\cdot 255618611114956521989$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $83.77$ sage: (K.disc().abs())^(1./K.degree())  gp: abs(K.disc)^(1/poldegree(K.pol))  magma: Abs(Discriminant(Integers(K)))^(1/Degree(K)); Ramified primes: $5, 11, 16381, 127865750920013, 1994744457920647, 255618611114956521989$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Aut(K/\Q)|$: $1$ This field is not Galois over $\Q$. This is not a CM field.

## Integral basis (with respect to field generator $$a$$)

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

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

## Class group and class number

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

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

## Unit group

sage: UK = K.unit_group()

magma: UK, f := UnitGroup(K);

 Rank: $14$ sage: UK.rank()  gp: K.fu  magma: UnitRank(K); Torsion generator: $$-1$$ (order $2$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental units: Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$42821205896121990$$ (assuming GRH) sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{14}\cdot 42821205896121990 \cdot 1}{2\sqrt{58740423215346262667355847452891580849174093037099352445}}\approx 0.835042375123828$ (assuming GRH)

## Galois group

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

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 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 $\Q$.

## 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.

sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:

sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:

gp: idealfactors = idealprimedec(K, p); \\ get the data

gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:

magma: idealfactors := Factorization(p*Integers(K)); // get the data

magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];

## Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
5Data not computed
$11$$\Q_{11}$$x + 3$$1$$1$$0Trivial[\ ] 11.2.1.1x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5} 11.7.0.1x^{7} - x + 4$$1$$7$$0$$C_7$$[\ ]^{7}$
11.14.0.1$x^{14} + x^{2} - 5 x + 8$$1$$14$$0$$C_{14}$$[\ ]^{14}$
16381Data not computed
127865750920013Data not computed
1994744457920647Data not computed
255618611114956521989Data not computed