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Properties

Label 39.1.51845326008...3647.1
Degree $39$
Signature $[1, 19]$
Discriminant $-\,3^{39}\cdot 14058971690735203\cdot 72182134610038326703\cdot 1260658330770821965649$
Root discriminant $87.38$
Ramified primes $3, 14058971690735203, 72182134610038326703, 1260658330770821965649$
Class number Not computed
Class group Not computed
Galois group $S_{39}$ (as 39T306)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 3, 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, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^39 + 3*x - 1)
 
gp: K = bnfinit(x^39 + 3*x - 1, 1)
 

Normalized defining polynomial

\( x^{39} + 3 x - 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $39$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 19]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-5184532600841803944263915674895587210566658055846276632709425783895556943647=-\,3^{39}\cdot 14058971690735203\cdot 72182134610038326703\cdot 1260658330770821965649\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $87.38$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $3, 14058971690735203, 72182134610038326703, 1260658330770821965649$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $\frac{1}{29} a^{38} - \frac{5}{29} a^{37} - \frac{4}{29} a^{36} - \frac{9}{29} a^{35} - \frac{13}{29} a^{34} + \frac{7}{29} a^{33} - \frac{6}{29} a^{32} + \frac{1}{29} a^{31} - \frac{5}{29} a^{30} - \frac{4}{29} a^{29} - \frac{9}{29} a^{28} - \frac{13}{29} a^{27} + \frac{7}{29} a^{26} - \frac{6}{29} a^{25} + \frac{1}{29} a^{24} - \frac{5}{29} a^{23} - \frac{4}{29} a^{22} - \frac{9}{29} a^{21} - \frac{13}{29} a^{20} + \frac{7}{29} a^{19} - \frac{6}{29} a^{18} + \frac{1}{29} a^{17} - \frac{5}{29} a^{16} - \frac{4}{29} a^{15} - \frac{9}{29} a^{14} - \frac{13}{29} a^{13} + \frac{7}{29} a^{12} - \frac{6}{29} a^{11} + \frac{1}{29} a^{10} - \frac{5}{29} a^{9} - \frac{4}{29} a^{8} - \frac{9}{29} a^{7} - \frac{13}{29} a^{6} + \frac{7}{29} a^{5} - \frac{6}{29} a^{4} + \frac{1}{29} a^{3} - \frac{5}{29} a^{2} - \frac{4}{29} a - \frac{6}{29}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $19$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_{39}$ (as 39T306):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 20397882081197443358640281739902897356800000000
The 31185 conjugacy class representatives for $S_{39}$ are not computed
Character table for $S_{39}$ 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 $26{,}\,{\href{/LocalNumberField/2.7.0.1}{7} }{,}\,{\href{/LocalNumberField/2.6.0.1}{6} }$ R $19{,}\,{\href{/LocalNumberField/5.11.0.1}{11} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.11.0.1}{11} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ $36{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ $27{,}\,{\href{/LocalNumberField/13.11.0.1}{11} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ $20{,}\,{\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ $24{,}\,15$ $32{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ $25{,}\,{\href{/LocalNumberField/29.7.0.1}{7} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ $35{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ $21{,}\,{\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $36{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ $26{,}\,{\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ $34{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ $30{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ $17{,}\,16{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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];
 
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]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$3$3.3.3.1$x^{3} + 6 x + 3$$3$$1$$3$$S_3$$[3/2]_{2}$
3.9.9.7$x^{9} + 18 x^{3} + 54 x + 27$$3$$3$$9$$C_3^2 : S_3 $$[3/2, 3/2]_{2}^{3}$
3.9.9.2$x^{9} + 18 x^{3} + 27 x + 27$$3$$3$$9$$C_3^2 : S_3 $$[3/2, 3/2]_{2}^{3}$
3.9.9.3$x^{9} + 9 x^{4} + 18 x^{3} + 54$$3$$3$$9$$(C_3^2:C_3):C_2$$[3/2, 3/2, 3/2]_{2}^{3}$
3.9.9.7$x^{9} + 18 x^{3} + 54 x + 27$$3$$3$$9$$C_3^2 : S_3 $$[3/2, 3/2]_{2}^{3}$
14058971690735203Data not computed
72182134610038326703Data not computed
1260658330770821965649Data not computed