Properties

Label 31.3.949...569.1
Degree $31$
Signature $[3, 14]$
Discriminant $9.495\times 10^{62}$
Root discriminant \(107.53\)
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 - 4*x - 1)
 
gp: K = bnfinit(y^31 - 4*y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^31 - 4*x - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^31 - 4*x - 1)
 

\( x^{31} - 4x - 1 \) 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:  $[3, 14]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(949505255199079208246627035002460041389356970940685243955265569\) \(\medspace = 24365060088671\cdot 38\!\cdots\!39\) 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:  \(107.53\)
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:  $24365060088671^{1/2}38969953357125920236474466375786993070484887309439^{1/2}\approx 3.0814043149172737e+31$
Ramified primes:   \(24365060088671\), \(38969\!\cdots\!09439\) 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:  $\Q(\sqrt{94950\!\cdots\!65569}$)
$\card{ \Aut(K/\Q) }$:  $1$
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:  $16$
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:   $a^{30}-4$, $a^{16}+2a$, $a^{29}+a^{28}+a^{27}+a^{26}+a^{25}+a^{24}+a^{23}+a^{22}+a^{21}+a^{20}+a^{19}+a^{18}+a^{17}+a^{16}+a^{15}+a^{14}+a^{13}+a^{12}+a^{11}+a^{10}+a^{9}+a^{8}+a^{7}+a^{6}+a^{5}+a^{4}+a^{3}+a^{2}+a+2$, $6a^{30}+a^{29}+2a^{28}+2a^{27}-3a^{26}+4a^{25}-7a^{24}-7a^{22}+a^{21}+5a^{20}+2a^{19}+8a^{18}-5a^{17}+6a^{16}-5a^{15}-a^{14}-7a^{13}-10a^{12}+5a^{11}-a^{10}+17a^{9}-2a^{8}+7a^{7}-a^{6}-3a^{5}+a^{4}-22a^{3}-13a-5$, $2a^{30}-10a^{29}+5a^{28}+9a^{27}-11a^{26}-a^{25}+13a^{24}-8a^{23}-8a^{22}+12a^{21}-14a^{19}+5a^{18}+11a^{17}-13a^{16}-6a^{15}+20a^{14}-6a^{13}-17a^{12}+22a^{11}+6a^{10}-26a^{9}+12a^{8}+17a^{7}-24a^{6}-7a^{5}+24a^{4}-10a^{3}-25a^{2}+23a+8$, $14a^{30}-18a^{28}-37a^{27}-49a^{26}-50a^{25}-42a^{24}-29a^{23}-13a^{22}+12a^{21}+40a^{20}+65a^{19}+76a^{18}+70a^{17}+55a^{16}+33a^{15}+4a^{14}-37a^{13}-78a^{12}-107a^{11}-112a^{10}-95a^{9}-69a^{8}-31a^{7}+20a^{6}+84a^{5}+139a^{4}+167a^{3}+159a^{2}+125a+25$, $2a^{30}+2a^{29}-a^{28}+4a^{27}+4a^{24}+a^{23}-4a^{22}+8a^{21}-6a^{20}+4a^{19}-2a^{18}-2a^{17}+a^{16}-2a^{15}-8a^{14}+5a^{13}-6a^{12}-7a^{11}+5a^{10}-11a^{9}+4a^{8}-6a^{7}-3a^{6}+3a^{5}+3a^{4}-11a^{3}+18a^{2}-6a-4$, $a^{30}-3a^{29}+2a^{28}-3a^{27}+a^{25}-a^{24}-2a^{23}+7a^{22}-8a^{21}+8a^{20}-2a^{18}+7a^{17}-2a^{16}+5a^{15}-a^{14}+5a^{13}-a^{12}+a^{11}+4a^{10}-2a^{9}-6a^{8}+13a^{7}-19a^{6}+10a^{5}-6a^{4}-10a^{3}+6a^{2}-12a-3$, $12a^{30}-a^{29}+8a^{28}-8a^{27}+2a^{26}-15a^{25}-4a^{24}-15a^{23}-5a^{22}-8a^{21}+a^{20}+6a^{19}+6a^{18}+19a^{17}+9a^{16}+23a^{15}+a^{14}+17a^{13}-15a^{12}+a^{11}-28a^{10}-11a^{9}-32a^{8}-12a^{7}-16a^{6}-2a^{5}+12a^{4}+17a^{3}+39a^{2}+21a+1$, $4a^{30}-5a^{29}+4a^{28}-3a^{27}+3a^{26}-a^{25}+a^{24}+a^{23}-3a^{22}+4a^{21}-7a^{20}+8a^{19}-9a^{18}+12a^{17}-10a^{16}+15a^{15}-13a^{14}+14a^{13}-14a^{12}+9a^{11}-12a^{10}+7a^{9}-7a^{8}+5a^{7}+3a^{6}+9a^{4}-5a^{3}+9a^{2}-13a-5$, $20a^{30}+2a^{29}-28a^{28}+37a^{27}-12a^{26}-30a^{25}+50a^{24}-32a^{23}-19a^{22}+58a^{21}-51a^{20}+3a^{19}+54a^{18}-64a^{17}+22a^{16}+36a^{15}-63a^{14}+33a^{13}+14a^{12}-42a^{11}+35a^{10}-5a^{9}-11a^{8}+16a^{7}-13a^{6}+15a^{5}-26a^{4}+8a^{3}+31a^{2}-72a-19$, $45a^{30}+118a^{29}-77a^{28}-116a^{27}+106a^{26}+106a^{25}-135a^{24}-85a^{23}+172a^{22}+65a^{21}-195a^{20}-27a^{19}+223a^{18}-15a^{17}-244a^{16}+62a^{15}+247a^{14}-126a^{13}-255a^{12}+182a^{11}+238a^{10}-249a^{9}-207a^{8}+321a^{7}+172a^{6}-380a^{5}-105a^{4}+441a^{3}+24a^{2}-494a-117$, $2a^{30}-a^{29}-9a^{26}-5a^{25}+6a^{24}-3a^{23}-6a^{22}-a^{21}-9a^{20}+2a^{19}+11a^{18}-7a^{17}-7a^{16}+5a^{15}+4a^{14}+9a^{13}+10a^{12}-9a^{11}-2a^{10}+24a^{9}+9a^{8}-2a^{7}+6a^{6}-8a^{5}+10a^{4}+25a^{3}-14a^{2}-19a-4$, $7a^{30}+44a^{29}+30a^{28}-2a^{27}-55a^{26}-31a^{25}-a^{24}+65a^{23}+36a^{22}-73a^{20}-48a^{19}+4a^{18}+76a^{17}+65a^{16}-12a^{15}-77a^{14}-85a^{13}+22a^{12}+82a^{11}+108a^{10}-27a^{9}-92a^{8}-128a^{7}+24a^{6}+110a^{5}+140a^{4}-16a^{3}-142a^{2}-151a-28$, $86a^{30}-6a^{29}-89a^{28}+63a^{27}+54a^{26}-102a^{25}+6a^{24}+108a^{23}-81a^{22}-70a^{21}+129a^{20}-a^{19}-140a^{18}+90a^{17}+94a^{16}-153a^{15}-2a^{14}+178a^{13}-112a^{12}-127a^{11}+198a^{10}+18a^{9}-229a^{8}+120a^{7}+157a^{6}-233a^{5}-22a^{4}+278a^{3}-155a^{2}-210a-40$, $59a^{30}-138a^{29}-253a^{28}-25a^{27}+281a^{26}+279a^{25}-20a^{24}-232a^{23}-120a^{22}+21a^{21}-124a^{20}-153a^{19}+279a^{18}+543a^{17}+43a^{16}-533a^{15}-393a^{14}+73a^{13}+167a^{12}+39a^{11}+155a^{10}+416a^{9}+233a^{8}-543a^{7}-933a^{6}-142a^{5}+822a^{4}+571a^{3}-135a^{2}+2a+14$ 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:  \( 38197936820101820000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{14}\cdot 38197936820101820000 \cdot 1}{2\cdot\sqrt{949505255199079208246627035002460041389356970940685243955265569}}\cr\approx \mathstrut & 0.741087295199414 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^31 - 4*x - 1)
 
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 - 4*x - 1, 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 - 4*x - 1);
 
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 - 4*x - 1);
 
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 $S_{31}$ is not computed

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 ${\href{/padicField/2.5.0.1}{5} }^{6}{,}\,{\href{/padicField/2.1.0.1}{1} }$ $29{,}\,{\href{/padicField/3.2.0.1}{2} }$ $31$ $17{,}\,{\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ $25{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ $16{,}\,{\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ $30{,}\,{\href{/padicField/17.1.0.1}{1} }$ $26{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ $17{,}\,{\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.6.0.1}{6} }$ $30{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.5.0.1}{5} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }$ $29{,}\,{\href{/padicField/37.2.0.1}{2} }$ $25{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ $19{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ $19{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.5.0.1}{5} }$ $19{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.14.0.1}{14} }{,}\,{\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
\(24365060088671\) Copy content Toggle raw display $\Q_{24365060088671}$$x$$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$$[\ ]^{4}$
Deg $8$$1$$8$$0$$C_8$$[\ ]^{8}$
Deg $14$$1$$14$$0$$C_{14}$$[\ ]^{14}$
\(389\!\cdots\!439\) Copy content Toggle raw display $\Q_{38\!\cdots\!39}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{38\!\cdots\!39}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $5$$1$$5$$0$$C_5$$[\ ]^{5}$
Deg $8$$1$$8$$0$$C_8$$[\ ]^{8}$
Deg $10$$1$$10$$0$$C_{10}$$[\ ]^{10}$