# Properties

 Label 38.2.912...125.1 Degree $38$ Signature $[2, 18]$ Discriminant $9.125\times 10^{84}$ Root discriminant $172.11$ Ramified primes see page Class number not computed Class group not computed Galois group $S_{38}$ (as 38T76)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^38 - 5*x - 5)

gp: K = bnfinit(x^38 - 5*x - 5, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5, -5, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]);

$$x^{38} - 5 x - 5$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $38$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[2, 18]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$912\!\cdots\!125$$$$\medspace = 5^{37}\cdot 131\cdot 181\cdot 2011\cdot 898776624172831\cdot 2926290836145496407027696425922931571$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $172.11$ 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: $5, 131, 181, 2011, 898776624172831, 2926290836145496407027696425922931571$ 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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $\frac{1}{3} a^{37} + \frac{1}{3} a^{36} + \frac{1}{3} a^{35} + \frac{1}{3} a^{34} + \frac{1}{3} a^{33} + \frac{1}{3} a^{32} + \frac{1}{3} a^{31} + \frac{1}{3} a^{30} + \frac{1}{3} a^{29} + \frac{1}{3} a^{28} + \frac{1}{3} a^{27} + \frac{1}{3} a^{26} + \frac{1}{3} a^{25} + \frac{1}{3} a^{24} + \frac{1}{3} a^{23} + \frac{1}{3} a^{22} + \frac{1}{3} a^{21} + \frac{1}{3} a^{20} + \frac{1}{3} a^{19} + \frac{1}{3} a^{18} + \frac{1}{3} a^{17} + \frac{1}{3} a^{16} + \frac{1}{3} a^{15} + \frac{1}{3} a^{14} + \frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

## Class group and class number

not computed

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

## Unit group

sage: UK = K.unit_group()

magma: UK, f := UnitGroup(K);

 Rank: $19$ 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: not computed sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: not computed sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s)$ not computed

## Galois group

$S_{38}$ (as 38T76):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A non-solvable group of order 523022617466601111760007224100074291200000000 The 26015 conjugacy class representatives for $S_{38}$ are not computed Character table for $S_{38}$ 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 $33{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ $22{,}\,{\href{/LocalNumberField/3.7.0.1}{7} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R $28{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ $25{,}\,{\href{/LocalNumberField/11.7.0.1}{7} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ $38$ $21{,}\,{\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ $31{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/23.11.0.1}{11} }{,}\,{\href{/LocalNumberField/23.9.0.1}{9} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ $20{,}\,{\href{/LocalNumberField/29.13.0.1}{13} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ $20{,}\,{\href{/LocalNumberField/31.11.0.1}{11} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ $38$ $35{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ $17{,}\,{\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $21{,}\,15{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ $20{,}\,15{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ $20{,}\,{\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
131Data not computed
$181$$\Q_{181}$$x + 2$$1$$1$$0Trivial[\ ] 181.2.1.1x^{2} - 181$$2$$1$$1$$C_2$$[\ ]_{2}$
181.6.0.1$x^{6} - x + 23$$1$$6$$0$$C_6$$[\ ]^{6} 181.6.0.1x^{6} - x + 23$$1$$6$$0$$C_6$$[\ ]^{6}$
181.9.0.1$x^{9} - x + 58$$1$$9$$0$$C_9$$[\ ]^{9} 181.14.0.1x^{14} - x + 2$$1$$14$$0$$C_{14}$$[\ ]^{14}$
2011Data not computed
898776624172831Data not computed
2926290836145496407027696425922931571Data not computed