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Properties

Label 31.1.158...375.1
Degree $31$
Signature $[1, 15]$
Discriminant $-1.590\times 10^{67}$
Root discriminant \(147.16\)
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 + x - 5)
 
gp: K = bnfinit(y^31 + y - 5, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^31 + x - 5);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^31 + x - 5)
 

\( x^{31} + x - 5 \) 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:   \(-15896907197985086551505014396253074975127944915555417537689208984375\) \(\medspace = -\,5^{30}\cdot 19\cdot 173\cdot 2149711\cdot 24\!\cdots\!51\) 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:  \(147.16\)
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:   \(5\), \(19\), \(173\), \(2149711\), \(24156\!\cdots\!40951\) 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:  $\Q(\sqrt{-17069\!\cdots\!34207}$)
$\card{ \Aut(K/\Q) }$:  $1$
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:   $a^{25}+2a^{19}+2a^{13}+a^{7}+1$, $a^{30}+2a^{29}+a^{26}-2a^{25}+3a^{24}-a^{23}+2a^{22}-a^{20}-2a^{19}-a^{18}-7a^{17}-9a^{15}+a^{14}-3a^{13}-3a^{9}+4a^{8}-8a^{7}+4a^{6}-4a^{5}+2a^{4}+8a^{3}+3a^{2}+11a+9$, $a^{29}+2a^{27}+2a^{25}+a^{23}-a^{19}-2a^{17}-2a^{15}-a^{13}+a^{9}+2a^{7}+2a^{5}+a^{3}+1$, $14a^{29}+14a^{28}+19a^{27}+15a^{26}+6a^{25}+3a^{24}-13a^{23}-13a^{22}-10a^{21}-3a^{20}+13a^{18}+27a^{17}+5a^{15}-2a^{14}-36a^{13}-38a^{12}-27a^{11}-20a^{10}-26a^{9}+25a^{8}+33a^{7}+6a^{6}+33a^{5}+5a^{4}-24a^{3}-40a^{2}-25a-21$, $58a^{30}-152a^{29}-241a^{28}-120a^{27}+133a^{26}+296a^{25}+186a^{24}-126a^{23}-345a^{22}-247a^{21}+94a^{20}+380a^{19}+343a^{18}-35a^{17}-448a^{16}-472a^{15}+538a^{13}+570a^{12}+22a^{11}-580a^{10}-652a^{9}-106a^{8}+590a^{7}+789a^{6}+226a^{5}-634a^{4}-931a^{3}-307a^{2}+644a+1056$, $33a^{30}+41a^{29}+13a^{28}-25a^{27}-60a^{26}-31a^{25}+33a^{24}+53a^{23}+38a^{22}-22a^{21}-80a^{20}-42a^{19}+27a^{18}+71a^{17}+79a^{16}-7a^{15}-85a^{14}-59a^{13}+4a^{12}+98a^{11}+133a^{10}+13a^{9}-91a^{8}-104a^{7}-42a^{6}+122a^{5}+169a^{4}+30a^{3}-91a^{2}-174a-76$, $83a^{30}+33a^{29}+79a^{28}+196a^{27}+286a^{26}+260a^{25}+136a^{24}+37a^{23}+72a^{22}+224a^{21}+364a^{20}+369a^{19}+229a^{18}+69a^{17}+57a^{16}+237a^{15}+458a^{14}+510a^{13}+342a^{12}+121a^{11}+64a^{10}+244a^{9}+535a^{8}+693a^{7}+553a^{6}+221a^{5}+31a^{4}+221a^{3}+648a^{2}+916a+866$, $425a^{30}+1534a^{29}+81a^{28}-1737a^{27}-573a^{26}+1647a^{25}+1274a^{24}-1503a^{23}-1853a^{22}+1009a^{21}+2415a^{20}-270a^{19}-2842a^{18}-550a^{17}+2807a^{16}+1713a^{15}-2696a^{14}-2614a^{13}+1982a^{12}+3632a^{11}-1048a^{10}-4416a^{9}-180a^{8}+4678a^{7}+2016a^{6}-4828a^{5}-3560a^{4}+3847a^{3}+5516a^{2}-2584a-6429$, $83a^{30}-78a^{29}+113a^{28}-43a^{27}-4a^{26}+48a^{25}-81a^{24}+123a^{23}-112a^{22}+129a^{21}-8a^{20}-63a^{19}+129a^{18}-165a^{17}+166a^{16}-134a^{15}+110a^{14}+91a^{13}-167a^{12}+269a^{11}-239a^{10}+178a^{9}-92a^{8}+24a^{7}+254a^{6}-302a^{5}+446a^{4}-279a^{3}+137a^{2}+35a-99$, $71a^{30}-45a^{29}+26a^{28}+12a^{27}-40a^{26}+127a^{25}-125a^{24}+129a^{23}-107a^{22}+93a^{21}-6a^{20}+13a^{19}+17a^{18}-16a^{17}+28a^{16}-4a^{15}+145a^{14}-212a^{13}+255a^{12}-236a^{11}+253a^{10}-94a^{9}+22a^{8}+111a^{7}-230a^{6}+345a^{5}-194a^{4}+198a^{3}-202a^{2}+257a-176$, $97a^{30}-314a^{29}+114a^{28}+292a^{27}-266a^{26}-214a^{25}+367a^{24}+142a^{23}-505a^{22}+18a^{21}+648a^{20}-431a^{19}-338a^{18}+391a^{17}+380a^{16}-686a^{15}-50a^{14}+730a^{13}-351a^{12}-445a^{11}+473a^{10}+191a^{9}-482a^{8}-43a^{7}+711a^{6}-686a^{5}-82a^{4}+755a^{3}-444a^{2}-513a+669$, $270a^{30}-263a^{29}-54a^{28}+296a^{27}-231a^{26}-186a^{25}+389a^{24}-233a^{23}-276a^{22}+426a^{21}-149a^{20}-417a^{19}+493a^{18}-45a^{17}-555a^{16}+548a^{15}+99a^{14}-668a^{13}+516a^{12}+368a^{11}-869a^{10}+540a^{9}+543a^{8}-904a^{7}+300a^{6}+966a^{5}-1161a^{4}+216a^{3}+1120a^{2}-1063a-41$, $305a^{30}+274a^{29}+158a^{28}+201a^{27}+36a^{26}-339a^{25}-373a^{24}-196a^{23}-243a^{22}-121a^{21}+368a^{20}+521a^{19}+281a^{18}+304a^{17}+259a^{16}-339a^{15}-677a^{14}-383a^{13}-340a^{12}-399a^{11}+282a^{10}+883a^{9}+577a^{8}+409a^{7}+593a^{6}-110a^{5}-1057a^{4}-817a^{3}-456a^{2}-758a+186$, $297a^{30}+98a^{29}-404a^{28}+367a^{27}+6a^{26}-426a^{25}+529a^{24}-169a^{23}-432a^{22}+820a^{21}-658a^{20}+24a^{19}+607a^{18}-720a^{17}+184a^{16}+600a^{15}-971a^{14}+525a^{13}+513a^{12}-1389a^{11}+1376a^{10}-388a^{9}-869a^{8}+1400a^{7}-717a^{6}-707a^{5}+1721a^{4}-1365a^{3}-309a^{2}+2145a-2398$, $294a^{30}+435a^{29}+314a^{28}+50a^{27}-255a^{26}-379a^{25}-317a^{24}-154a^{23}-18a^{22}+20a^{21}+118a^{20}+254a^{19}+474a^{18}+470a^{17}+226a^{16}-311a^{15}-772a^{14}-867a^{13}-480a^{12}+202a^{11}+658a^{10}+791a^{9}+481a^{8}+218a^{7}-27a^{6}-83a^{5}-312a^{4}-698a^{3}-1029a^{2}-929a+262$ 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:  \( 4634927539611405600000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{15}\cdot 4634927539611405600000 \cdot 1}{2\cdot\sqrt{15896907197985086551505014396253074975127944915555417537689208984375}}\cr\approx \mathstrut & 1.09165310623249 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^31 + x - 5)
 
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 + x - 5, 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 + x - 5);
 
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 + x - 5);
 
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 $21{,}\,{\href{/padicField/2.7.0.1}{7} }{,}\,{\href{/padicField/2.3.0.1}{3} }$ $16{,}\,{\href{/padicField/3.11.0.1}{11} }{,}\,{\href{/padicField/3.3.0.1}{3} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ R $25{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ $17{,}\,{\href{/padicField/11.13.0.1}{13} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ $30{,}\,{\href{/padicField/13.1.0.1}{1} }$ $24{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ R $21{,}\,{\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ $16{,}\,{\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.2.0.1}{2} }^{15}{,}\,{\href{/padicField/31.1.0.1}{1} }$ $30{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.9.0.1}{9} }{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ $31$ $30{,}\,{\href{/padicField/47.1.0.1}{1} }$ $21{,}\,{\href{/padicField/53.9.0.1}{9} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ $15{,}\,{\href{/padicField/59.11.0.1}{11} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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

# 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
\(5\) Copy content Toggle raw display $\Q_{5}$$x + 3$$1$$1$$0$Trivial$[\ ]$
5.5.5.1$x^{5} + 20 x + 5$$5$$1$$5$$F_5$$[5/4]_{4}$
5.5.5.3$x^{5} + 15 x + 5$$5$$1$$5$$F_5$$[5/4]_{4}$
5.10.10.6$x^{10} - 40 x^{6} + 10 x^{5} + 200 x^{2} - 200 x + 25$$5$$2$$10$$C_5^2 : C_8$$[5/4, 5/4]_{4}^{2}$
5.10.10.2$x^{10} - 20 x^{6} + 10 x^{5} + 50 x^{2} - 100 x + 25$$5$$2$$10$$C_5^2 : C_8$$[5/4, 5/4]_{4}^{2}$
\(19\) Copy content Toggle raw display $\Q_{19}$$x + 17$$1$$1$$0$Trivial$[\ ]$
19.2.1.2$x^{2} + 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} + 18 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.8.0.1$x^{8} + x^{4} + 12 x^{3} + 10 x^{2} + 3 x + 2$$1$$8$$0$$C_8$$[\ ]^{8}$
19.18.0.1$x^{18} + 10 x^{11} + 9 x^{10} + 7 x^{9} + 17 x^{8} + 5 x^{7} + 16 x^{5} + 5 x^{4} + 7 x^{3} + 3 x^{2} + 14 x + 2$$1$$18$$0$$C_{18}$$[\ ]^{18}$
\(173\) Copy content Toggle raw display $\Q_{173}$$x + 171$$1$$1$$0$Trivial$[\ ]$
$\Q_{173}$$x + 171$$1$$1$$0$Trivial$[\ ]$
$\Q_{173}$$x + 171$$1$$1$$0$Trivial$[\ ]$
173.2.1.1$x^{2} + 173$$2$$1$$1$$C_2$$[\ ]_{2}$
173.3.0.1$x^{3} + 2 x + 171$$1$$3$$0$$C_3$$[\ ]^{3}$
173.23.0.1$x^{23} + 5 x + 171$$1$$23$$0$$C_{23}$$[\ ]^{23}$
\(2149711\) Copy content Toggle raw display $\Q_{2149711}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $6$$1$$6$$0$$C_6$$[\ ]^{6}$
Deg $10$$1$$10$$0$$C_{10}$$[\ ]^{10}$
Deg $12$$1$$12$$0$$C_{12}$$[\ ]^{12}$
\(241\!\cdots\!951\) Copy content Toggle raw display $\Q_{24\!\cdots\!51}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $6$$1$$6$$0$$C_6$$[\ ]^{6}$
Deg $22$$1$$22$$0$22T1$[\ ]^{22}$