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Properties

Label 31.1.351...719.1
Degree $31$
Signature $[1, 15]$
Discriminant $-3.514\times 10^{60}$
Root discriminant \(89.76\)
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 - 2*x - 3)
 
gp: K = bnfinit(y^31 - 2*y - 3, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^31 - 2*x - 3);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^31 - 2*x - 3)
 

\( x^{31} - 2x - 3 \) 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:   \(-3514391143547463851053558201195469962323789296982843961159719\) \(\medspace = -\,3^{30}\cdot 1069\cdot 745011551\cdot 21\!\cdots\!49\) 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:  \(89.76\)
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:   \(3\), \(1069\), \(745011551\), \(21432\!\cdots\!88549\) 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:  $\Q(\sqrt{-17069\!\cdots\!34431}$)
$\card{ \Aut(K/\Q) }$:  $1$
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:   $a^{30}+a^{28}+a^{26}+a^{24}+a^{22}+a^{20}+a^{19}+a^{18}+a^{17}+a^{16}+a^{15}+a^{14}+a^{13}+a^{12}+a^{11}+a^{10}+2a^{9}+a^{8}+2a^{7}+a^{6}+2a^{5}+a^{4}+2a^{3}+a^{2}+2a-1$, $a^{30}+a^{29}+a^{28}+a^{27}+a^{26}+a^{25}+a^{24}+a^{23}+a^{22}+a^{21}+a^{20}+a^{19}+a^{18}+a^{17}+a^{16}+a^{15}+a^{14}+a^{13}+a^{12}+a^{11}+a^{10}+a^{9}+a^{8}+a^{7}+a^{6}+a^{5}+a^{4}+a^{3}+a^{2}+2a+2$, $a^{28}-a^{27}-a^{26}+a^{25}+2a^{24}-2a^{23}-2a^{22}+2a^{21}+2a^{20}-2a^{19}-a^{18}+a^{17}+a^{14}-a^{13}-2a^{12}+2a^{11}+2a^{10}-2a^{9}-2a^{8}+2a^{7}+a^{6}-a^{5}-a^{4}+a^{3}-1$, $a^{29}+a^{23}+a^{22}+a^{20}-a^{19}+a^{16}+2a^{14}-a^{13}+a^{12}-2a^{11}+2a^{8}+3a^{6}-a^{5}-a^{3}+4$, $3a^{30}-4a^{29}+5a^{28}-2a^{27}+3a^{26}-4a^{25}+a^{24}-3a^{23}+3a^{22}+a^{21}+a^{19}-4a^{18}+2a^{17}-a^{16}+4a^{15}-a^{14}-a^{13}-3a^{12}-2a^{11}+4a^{10}+a^{9}+5a^{8}-4a^{7}-a^{6}-5a^{5}+4a^{4}+5a^{2}-3a-11$, $19a^{30}+16a^{29}+9a^{28}-a^{27}-9a^{26}-20a^{25}-27a^{24}-29a^{23}-28a^{22}-20a^{21}-9a^{20}+5a^{19}+21a^{18}+34a^{17}+40a^{16}+44a^{15}+39a^{14}+23a^{13}+7a^{12}-16a^{11}-38a^{10}-54a^{9}-66a^{8}-63a^{7}-50a^{6}-29a^{5}+2a^{4}+37a^{3}+66a^{2}+87a+59$, $a^{30}-a^{29}-a^{28}+3a^{27}-a^{25}-a^{23}+a^{21}+2a^{20}-2a^{19}-2a^{18}+a^{17}-2a^{16}+3a^{15}+4a^{14}-4a^{13}-a^{12}-a^{11}-2a^{10}+5a^{9}+4a^{8}-2a^{7}-4a^{6}-2a^{5}-2a^{4}+2a^{3}+8a^{2}-2a-7$, $a^{30}+a^{29}-2a^{28}-5a^{27}-4a^{26}-8a^{25}-7a^{24}-5a^{23}-4a^{22}+2a^{21}+6a^{20}+8a^{19}+10a^{18}+10a^{17}+7a^{16}+7a^{15}+2a^{14}-5a^{13}-7a^{12}-15a^{11}-15a^{10}-12a^{9}-12a^{8}-4a^{7}+4a^{5}+13a^{4}+19a^{3}+22a^{2}+24a+14$, $4a^{30}-6a^{29}+6a^{28}-5a^{27}+a^{26}+3a^{25}-8a^{24}+10a^{23}-7a^{22}+4a^{21}+2a^{20}-6a^{19}+5a^{18}-4a^{17}+4a^{16}-5a^{15}+4a^{14}-3a^{13}-a^{12}+8a^{11}-10a^{10}+11a^{9}-8a^{8}+a^{7}+7a^{6}-13a^{5}+8a^{4}-6a^{3}+3a^{2}+3a-5$, $3a^{30}-5a^{29}+6a^{27}-5a^{26}-6a^{25}+14a^{24}-13a^{23}+3a^{22}+5a^{21}-3a^{20}-7a^{19}+13a^{18}-7a^{17}-8a^{16}+16a^{15}-9a^{14}-3a^{13}+8a^{12}-17a^{10}+19a^{9}-7a^{8}-11a^{7}+15a^{6}-2a^{5}-13a^{4}+11a^{3}+7a^{2}-26a+17$, $6a^{30}+6a^{29}+8a^{28}+9a^{27}+6a^{26}+13a^{25}+4a^{24}+11a^{23}+4a^{22}+6a^{21}+2a^{20}+a^{19}-3a^{18}-5a^{17}-7a^{16}-14a^{15}-8a^{14}-21a^{13}-14a^{12}-20a^{11}-18a^{10}-19a^{9}-15a^{8}-16a^{7}-11a^{6}-4a^{5}-5a^{4}+10a^{3}+5a^{2}+21a+11$, $6a^{30}+3a^{29}+3a^{28}+4a^{27}+3a^{25}+a^{24}-4a^{23}-7a^{21}-5a^{20}-5a^{19}-11a^{18}-8a^{17}-9a^{16}-11a^{15}-5a^{14}-10a^{13}-8a^{12}-4a^{11}-5a^{10}+a^{9}+2a^{8}+3a^{7}+9a^{6}+12a^{5}+17a^{4}+17a^{3}+19a^{2}+19a+8$, $14a^{30}+2a^{29}-16a^{28}+11a^{27}+7a^{26}-13a^{25}+9a^{24}+4a^{23}-15a^{22}+11a^{21}+5a^{20}-17a^{19}+17a^{18}+11a^{17}-28a^{16}+9a^{15}+20a^{14}-25a^{13}+4a^{12}+26a^{11}-20a^{10}-3a^{9}+19a^{8}-19a^{7}+4a^{6}+23a^{5}-27a^{4}+4a^{3}+36a^{2}-32a-38$, $4a^{30}-5a^{28}-3a^{27}-6a^{26}-a^{25}+5a^{24}+14a^{23}+15a^{22}+13a^{21}+11a^{20}+3a^{19}-5a^{18}-12a^{17}-5a^{16}-a^{15}+7a^{14}+15a^{13}+27a^{12}+26a^{11}+15a^{10}+6a^{9}-4a^{8}-13a^{7}-20a^{6}-4a^{5}+14a^{4}+30a^{3}+37a^{2}+45a+29$, $11a^{30}+2a^{29}-14a^{28}+5a^{27}+12a^{26}-13a^{25}-7a^{24}+17a^{23}-2a^{22}-19a^{21}+11a^{20}+14a^{19}-21a^{18}-4a^{17}+22a^{16}-8a^{15}-20a^{14}+17a^{13}+10a^{12}-25a^{11}+2a^{10}+23a^{9}-18a^{8}-15a^{7}+26a^{6}-3a^{5}-28a^{4}+17a^{3}+18a^{2}-33a-26$ 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:  \( 2083870736834916600 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{15}\cdot 2083870736834916600 \cdot 1}{2\cdot\sqrt{3514391143547463851053558201195469962323789296982843961159719}}\cr\approx \mathstrut & 1.04386359927944 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^31 - 2*x - 3)
 
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 - 2*x - 3, 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 - 2*x - 3);
 
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 - 2*x - 3);
 
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 ${\href{/padicField/2.5.0.1}{5} }^{6}{,}\,{\href{/padicField/2.1.0.1}{1} }$ R $31$ $30{,}\,{\href{/padicField/7.1.0.1}{1} }$ $20{,}\,{\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ $16{,}\,{\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ $30{,}\,{\href{/padicField/17.1.0.1}{1} }$ $20{,}\,{\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ $17{,}\,{\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ $26{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.5.0.1}{5} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }$ $16{,}\,{\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.7.0.1}{7} }$ $17{,}\,{\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ $30{,}\,{\href{/padicField/43.1.0.1}{1} }$ $24{,}\,{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ $21{,}\,{\href{/padicField/53.5.0.1}{5} }^{2}$ $18{,}\,{\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
\(3\) Copy content Toggle raw display $\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.6.6.2$x^{6} - 6 x^{5} + 39 x^{4} + 60 x^{3} - 18 x + 9$$3$$2$$6$$C_3^2:C_4$$[3/2, 3/2]_{2}^{2}$
3.12.12.18$x^{12} - 36 x^{11} + 438 x^{10} + 120 x^{9} - 4563 x^{8} + 1188 x^{7} + 22410 x^{6} + 10692 x^{5} - 17658 x^{4} - 3780 x^{3} + 6804 x^{2} - 1296 x + 81$$3$$4$$12$12T46$[3/2, 3/2]_{2}^{4}$
3.12.12.20$x^{12} + 30 x^{10} + 228 x^{9} + 1872 x^{8} + 5778 x^{7} + 15336 x^{6} + 18036 x^{5} + 12879 x^{4} + 7074 x^{3} + 3240 x^{2} + 810 x + 81$$3$$4$$12$12T173$[3/2, 3/2, 3/2, 3/2]_{2}^{4}$
\(1069\) Copy content Toggle raw display $\Q_{1069}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{1069}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $8$$1$$8$$0$$C_8$$[\ ]^{8}$
Deg $9$$1$$9$$0$$C_9$$[\ ]^{9}$
Deg $10$$1$$10$$0$$C_{10}$$[\ ]^{10}$
\(745011551\) Copy content Toggle raw display $\Q_{745011551}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{745011551}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{745011551}$$x$$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$$[\ ]^{5}$
Deg $6$$1$$6$$0$$C_6$$[\ ]^{6}$
Deg $11$$1$$11$$0$$C_{11}$$[\ ]^{11}$
\(214\!\cdots\!549\) Copy content Toggle raw display $\Q_{21\!\cdots\!49}$$x$$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}$