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Properties

Label 31.1.127...144.1
Degree $31$
Signature $[1, 15]$
Discriminant $-1.272\times 10^{59}$
Root discriminant \(80.65\)
Ramified primes see page
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_{31}$ (as 31T12)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^31 + 3*x - 2)
 
gp: K = bnfinit(y^31 + 3*y - 2, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^31 + 3*x - 2);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^31 + 3*x - 2)
 

\( x^{31} + 3x - 2 \) Copy content Toggle raw display

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

Invariants

Degree:  $31$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[1, 15]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-127191802711813906924700680583351835033630345588173537542144\) \(\medspace = -\,2^{31}\cdot 59\cdot 316799009\cdot 31\!\cdots\!63\) 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:  \(80.65\)
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:  not computed
Ramified primes:   \(2\), \(59\), \(316799009\), \(31687\!\cdots\!53263\) 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:  $\Q(\sqrt{-11845\!\cdots\!31306}$)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$ 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:  $15$
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:   $a^{27}+a^{24}-a^{18}-2a^{15}-a^{12}+a^{9}+3a^{6}-a^{5}+2a^{3}-a^{2}-1$, $a^{28}-a^{26}+a^{25}-a^{24}-2a^{23}+2a^{22}-3a^{21}+a^{20}-a^{18}-a^{17}+3a^{16}-4a^{15}+4a^{14}-a^{13}+a^{11}+3a^{10}-5a^{9}+7a^{8}-6a^{7}+2a^{6}-a^{5}+a^{4}-6a^{3}+8a^{2}-10a+5$, $2a^{29}+a^{28}+a^{27}+a^{26}-a^{24}+a^{22}+a^{21}+2a^{20}-2a^{19}-4a^{18}+a^{16}-3a^{12}-a^{11}+2a^{10}-a^{9}+3a^{8}+5a^{7}-a^{6}-4a^{5}+3a^{3}+a^{2}+3a-3$, $3a^{30}+3a^{29}+2a^{28}+a^{27}+a^{26}+2a^{25}-a^{20}-a^{18}+a^{16}+a^{15}+a^{13}-a^{9}-2a^{8}+a^{7}-2a^{5}+a^{4}+2a^{3}+11$, $3a^{30}+a^{29}+2a^{28}+a^{25}-a^{23}+2a^{22}-2a^{21}-a^{20}+a^{19}-2a^{18}-a^{17}+3a^{16}-2a^{15}+a^{14}+3a^{13}-2a^{12}+a^{11}+2a^{10}-2a^{9}+a^{8}+2a^{7}-2a^{6}+2a^{5}-a^{4}-a^{3}-a^{2}-a+7$, $2a^{30}+2a^{29}+a^{28}+2a^{26}-2a^{25}+2a^{24}-a^{22}+2a^{21}-2a^{20}+3a^{19}-3a^{18}+3a^{17}-a^{16}-a^{15}+4a^{14}-4a^{13}+4a^{12}-2a^{11}+2a^{10}-a^{8}+4a^{7}-5a^{6}+5a^{5}-a^{4}-2a^{3}+4a^{2}-3a+9$, $2a^{29}+3a^{28}+4a^{27}+4a^{26}+4a^{25}+3a^{24}+a^{23}-a^{21}-2a^{20}-2a^{19}-2a^{18}-2a^{17}-3a^{16}-5a^{15}-4a^{14}-3a^{13}-a^{12}+2a^{11}+4a^{10}+6a^{9}+4a^{8}+2a^{7}+2a^{6}+2a^{5}+2a^{4}+2a^{3}+a^{2}+a-3$, $a^{30}-a^{28}+a^{26}-a^{24}+a^{22}-a^{20}+a^{18}-a^{16}+a^{14}-a^{12}+a^{10}-a^{8}+a^{6}-a^{4}+a^{2}+a-1$, $12a^{30}+2a^{29}-12a^{28}-7a^{27}+10a^{26}+11a^{25}-6a^{24}-15a^{23}+a^{22}+16a^{21}+6a^{20}-16a^{19}-12a^{18}+12a^{17}+18a^{16}-7a^{15}-21a^{14}-a^{13}+23a^{12}+9a^{11}-20a^{10}-19a^{9}+15a^{8}+26a^{7}-5a^{6}-31a^{5}-6a^{4}+32a^{3}+19a^{2}-28a+5$, $a^{30}-a^{29}-5a^{28}+2a^{27}-7a^{26}+3a^{25}-4a^{24}-a^{23}+4a^{22}-5a^{21}+8a^{20}-5a^{19}+6a^{18}+2a^{17}-3a^{16}+9a^{15}-8a^{14}+10a^{13}-6a^{12}+5a^{10}-12a^{9}+10a^{8}-15a^{7}+6a^{6}-3a^{5}-9a^{4}+13a^{3}-16a^{2}+19a-7$, $a^{28}+a^{27}-a^{26}-2a^{25}-2a^{24}-2a^{23}-2a^{22}-a^{21}+a^{20}+a^{19}+2a^{18}+3a^{17}+2a^{16}-a^{13}-3a^{12}-4a^{11}-a^{10}-a^{9}+3a^{7}+4a^{6}+2a^{5}+2a^{4}+a^{3}-a^{2}-6a-3$, $19a^{30}+13a^{29}+7a^{28}-2a^{27}-10a^{26}-17a^{25}-23a^{24}-26a^{23}-26a^{22}-23a^{21}-19a^{20}-11a^{19}-a^{18}+6a^{17}+15a^{16}+23a^{15}+25a^{14}+29a^{13}+28a^{12}+22a^{11}+16a^{10}+8a^{9}-3a^{8}-13a^{7}-21a^{6}-26a^{5}-32a^{4}-29a^{3}-27a^{2}-24a+47$, $20a^{30}+15a^{29}+28a^{28}+31a^{27}+18a^{26}+25a^{25}+37a^{24}+24a^{23}+19a^{22}+34a^{21}+31a^{20}+15a^{19}+21a^{18}+33a^{17}+17a^{16}+4a^{15}+23a^{14}+22a^{13}-7a^{12}+3a^{11}+21a^{10}-8a^{9}-18a^{8}+8a^{7}-2a^{6}-30a^{5}-17a^{4}+a^{3}-24a^{2}-44a+49$, $a^{30}-3a^{29}-5a^{28}-5a^{27}+a^{26}+10a^{25}+8a^{24}-4a^{23}-9a^{22}-5a^{21}+4a^{19}+10a^{18}+6a^{17}-7a^{16}-13a^{15}-5a^{14}+6a^{13}+10a^{12}+9a^{11}-2a^{10}-11a^{9}-12a^{8}+8a^{6}+15a^{5}+8a^{4}-10a^{3}-21a^{2}-6a+15$, $154a^{30}+105a^{29}+69a^{28}+50a^{27}+29a^{26}+18a^{25}+13a^{24}+11a^{23}+8a^{22}+4a^{21}+2a^{20}-3a^{19}+4a^{18}+3a^{17}+3a^{16}-4a^{15}-2a^{13}+2a^{12}+7a^{11}-5a^{10}-2a^{9}-3a^{8}+5a^{7}+5a^{5}-6a^{4}-5a^{3}+3a^{2}+5a+465$ 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:  \( 1049301058301323900 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{15}\cdot 1049301058301323900 \cdot 1}{2\cdot\sqrt{127191802711813906924700680583351835033630345588173537542144}}\cr\approx \mathstrut & 2.76292168429242 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^31 + 3*x - 2)
 
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^31 + 3*x - 2, 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^31 + 3*x - 2);
 
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^31 + 3*x - 2);
 
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

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

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 8222838654177922817725562880000000
The 6842 conjugacy class representatives for $S_{31}$ are not computed
Character table for $S_{31}$

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 R $30{,}\,{\href{/padicField/3.1.0.1}{1} }$ $31$ $17{,}\,{\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.9.0.1}{9} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }$ $18{,}\,{\href{/padicField/13.13.0.1}{13} }$ $15{,}\,{\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ $17{,}\,{\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ $17{,}\,{\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ $15{,}\,{\href{/padicField/29.13.0.1}{13} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ $15^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ $17{,}\,{\href{/padicField/37.11.0.1}{11} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ ${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.5.0.1}{5} }$ ${\href{/padicField/43.14.0.1}{14} }{,}\,{\href{/padicField/43.13.0.1}{13} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ ${\href{/padicField/53.14.0.1}{14} }{,}\,{\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ R

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
\(2\) Copy content Toggle raw display $\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.4.4.2$x^{4} + 4 x^{3} + 4 x^{2} + 12$$2$$2$$4$$C_4$$[2]^{2}$
2.8.8.6$x^{8} + 2 x^{7} + 24 x^{6} + 84 x^{5} + 264 x^{4} + 408 x^{3} + 384 x^{2} - 208 x + 80$$2$$4$$8$$(C_8:C_2):C_2$$[2, 2, 2]^{4}$
2.8.8.2$x^{8} + 8 x^{7} + 56 x^{6} + 240 x^{5} + 816 x^{4} + 2048 x^{3} + 3776 x^{2} + 4928 x + 3760$$2$$4$$8$$C_2^2:C_4$$[2, 2]^{4}$
2.8.8.6$x^{8} + 2 x^{7} + 24 x^{6} + 84 x^{5} + 264 x^{4} + 408 x^{3} + 384 x^{2} - 208 x + 80$$2$$4$$8$$(C_8:C_2):C_2$$[2, 2, 2]^{4}$
\(59\) Copy content Toggle raw display 59.2.1.1$x^{2} + 118$$2$$1$$1$$C_2$$[\ ]_{2}$
59.4.0.1$x^{4} + 2 x^{2} + 40 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
59.8.0.1$x^{8} + 16 x^{4} + 32 x^{3} + 2 x^{2} + 50 x + 2$$1$$8$$0$$C_8$$[\ ]^{8}$
59.17.0.1$x^{17} + 9 x + 57$$1$$17$$0$$C_{17}$$[\ ]^{17}$
\(316799009\) Copy content Toggle raw display $\Q_{316799009}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $22$$1$$22$$0$22T1$[\ ]^{22}$
\(316\!\cdots\!263\) Copy content Toggle raw display $\Q_{31\!\cdots\!63}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $10$$1$$10$$0$$C_{10}$$[\ ]^{10}$
Deg $16$$1$$16$$0$$C_{16}$$[\ ]^{16}$