Properties

Label 31.3.958...281.1
Degree $31$
Signature $[3, 14]$
Discriminant $9.588\times 10^{65}$
Root discriminant $134.41$
Ramified primes see page
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_{31}$ (as 31T12)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^31 - 5*x - 3)
 
gp: K = bnfinit(x^31 - 5*x - 3, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, -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, 1]);
 

\( x^{31} - 5 x - 3 \)

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

Invariants

Degree:  $31$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[3, 14]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(958\!\cdots\!281\)\(\medspace = 3^{30}\cdot 13\cdot 16690100963\cdot 21461804096776561542209810433141867740951\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $134.41$
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:  $3, 13, 16690100963, 21461804096776561542209810433141867740951$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$

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:  $16$
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:  \( 1258080429033531000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{3}\cdot(2\pi)^{14}\cdot 1258080429033531000000 \cdot 1}{2\sqrt{958751781772834738039164297352952377530037676210703017156038840281}}\approx 0.768127763566694$ (assuming GRH)

Galois group

$S_{31}$ (as 31T12):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A non-solvable group of order 8222838654177922817725562880000000
The 6842 conjugacy class representatives for $S_{31}$ are not computed
Character table for $S_{31}$ 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 $21{,}\,{\href{/LocalNumberField/2.7.0.1}{7} }{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{10}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.11.0.1}{11} }{,}\,{\href{/LocalNumberField/7.9.0.1}{9} }{,}\,{\href{/LocalNumberField/7.7.0.1}{7} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ $16{,}\,{\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ R $20{,}\,{\href{/LocalNumberField/17.11.0.1}{11} }$ $16{,}\,15$ $24{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ $21{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{10}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ $16{,}\,{\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ $28{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ $17{,}\,{\href{/LocalNumberField/43.13.0.1}{13} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $19{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ $26{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ $21{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }$

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
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.6.6.1$x^{6} + 3 x^{5} - 2$$3$$2$$6$$C_3^2:C_4$$[3/2, 3/2]_{2}^{2}$
3.12.12.18$x^{12} + 42 x^{11} - 48 x^{10} - 114 x^{9} - 99 x^{8} - 54 x^{7} - 90 x^{6} - 108 x^{5} + 27 x^{4} - 27 x^{3} + 81 x^{2} + 81 x - 81$$3$$4$$12$12T46$[3/2, 3/2]_{2}^{4}$
3.12.12.10$x^{12} + 9 x^{11} + 36 x^{10} - 72 x^{9} + 18 x^{8} + 45 x^{7} + 99 x^{6} + 54 x^{5} + 81 x^{4} - 81 x^{3} + 81 x^{2} - 81 x + 81$$3$$4$$12$12T173$[3/2, 3/2, 3/2, 3/2]_{2}^{4}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.11.0.1$x^{11} - 4 x + 2$$1$$11$$0$$C_{11}$$[\ ]^{11}$
13.13.0.1$x^{13} - x + 2$$1$$13$$0$$C_{13}$$[\ ]^{13}$
16690100963Data not computed
21461804096776561542209810433141867740951Data not computed