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Properties

Label 37.1.104...381.1
Degree $37$
Signature $[1, 18]$
Discriminant $1.045\times 10^{58}$
Root discriminant \(36.99\)
Ramified primes see page
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_{37}$ (as 37T11)

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Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^37 - x - 1)
 
Copy content gp:K = bnfinit(y^37 - y - 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^37 - x - 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^37 - x - 1)
 

\( x^{37} - x - 1 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $37$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $[1, 18]$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(10448747596854066889270616610243376552387228227740340151381\) \(\medspace = 983\cdot 10\!\cdots\!07\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(36.99\)
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $983^{1/2}10629448216535164688983333275934258954615695043479491507^{1/2}\approx 1.0221911561373473e+29$
Ramified primes:   \(983\), \(10629\!\cdots\!91507\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant((OK)))
 
Discriminant root field:  $\Q(\sqrt{10448\!\cdots\!51381}$)
$\Aut(K/\Q)$:   $C_1$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  Yes
Index:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $18$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $a$, $a^{18}+1$, $a^{13}-a$, $a^{8}+a^{2}$, $a^{36}-a^{35}-a^{7}-1$, $a^{30}-a^{29}+a^{28}$, $a^{26}-a^{25}$, $a^{11}+a^{9}$, $a^{35}-a^{34}+a^{33}-a^{32}+a^{31}$, $a^{36}-a^{35}+a^{34}-a^{33}+a^{32}-a^{31}-a-1$, $a^{34}-a^{32}+a^{31}-a^{30}+a^{29}-a^{28}+a^{24}+a^{20}-a^{19}-a^{17}+a^{16}-a^{14}+a^{13}+a^{11}-a^{4}+a^{2}$, $5a^{36}-4a^{35}+4a^{34}-4a^{33}+3a^{32}-4a^{31}+4a^{30}-3a^{29}+3a^{28}-3a^{27}+a^{26}-a^{25}+2a^{24}-a^{23}+a^{22}-2a^{21}+2a^{20}+a^{18}-a^{17}-a^{16}+a^{14}-a^{11}+a^{9}-a^{6}+a^{4}+a^{3}-6$, $a^{35}-a^{29}-a^{27}-a^{25}+a^{24}+a^{22}+a^{20}-a^{19}-a^{17}-a^{15}+a^{12}+a^{10}+a^{9}+a^{8}-a^{5}-a^{4}-a^{3}+1$, $9a^{36}-8a^{35}+7a^{34}-5a^{33}+4a^{32}-3a^{31}+3a^{30}-3a^{29}+3a^{28}-3a^{27}+3a^{26}-3a^{25}+2a^{24}-a^{23}-a^{20}+a^{19}-a^{18}+a^{11}-2a^{10}+2a^{9}-2a^{8}+a^{7}-a^{6}+a^{5}+a-10$, $2a^{36}-a^{35}+a^{34}+a^{31}-a^{30}+a^{29}-a^{28}-a^{25}-a^{23}-a^{21}-a^{19}-a^{17}-a^{16}-a^{14}+a^{13}-a^{12}+a^{11}+a^{8}+a^{6}+a^{4}+a^{3}+a^{2}+a-1$, $a^{34}-a^{33}+a^{32}-a^{31}+a^{30}-a^{29}+a^{28}-2a^{27}+a^{26}+2a^{24}-2a^{23}+a^{22}-a^{21}+a^{20}-2a^{19}+a^{18}-a^{17}+a^{16}-a^{15}+2a^{14}-2a^{11}+a^{10}-a^{7}+a^{6}+a^{4}-a$, $2a^{36}-a^{35}-a^{32}+a^{31}+a^{28}-a^{27}-a^{26}+2a^{25}-a^{24}+a^{23}-a^{22}-a^{21}+a^{20}+a^{17}-a^{16}-a^{15}+2a^{12}-a^{10}-a^{9}+a^{6}+a^{5}-a^{3}-a^{2}$, $a^{36}-a^{35}+a^{34}-2a^{33}+2a^{32}-2a^{31}+2a^{30}-2a^{29}+a^{24}-a^{23}+2a^{22}-a^{21}+a^{17}-a^{15}-a^{13}-a^{10}+a^{9}-a^{5}+a^{4}+a^{3}+a-2$ Copy content Toggle raw display (assuming GRH)
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 94459733489984.19 \) (assuming GRH)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{18}\cdot 94459733489984.19 \cdot 1}{2\cdot\sqrt{10448747596854066889270616610243376552387228227740340151381}}\cr\approx \mathstrut & 0.215254493967513 \end{aligned}\] (assuming GRH)

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^37 - 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))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^37 - 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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^37 - 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)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = PolynomialRing(QQ); K, a = NumberField(x^37 - 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

$S_{37}$ (as 37T11):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A non-solvable group of order 13763753091226345046315979581580902400000000
The 21637 conjugacy class representatives for $S_{37}$ are not computed
Character table for $S_{37}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content 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 $19{,}\,18$ $29{,}\,{\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ $28{,}\,{\href{/padicField/5.5.0.1}{5} }{,}\,{\href{/padicField/5.4.0.1}{4} }$ $37$ ${\href{/padicField/11.14.0.1}{14} }^{2}{,}\,{\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ $37$ $28{,}\,{\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ $20{,}\,{\href{/padicField/19.11.0.1}{11} }{,}\,{\href{/padicField/19.6.0.1}{6} }$ $18{,}\,{\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ $18{,}\,{\href{/padicField/29.13.0.1}{13} }{,}\,{\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ $15{,}\,{\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ $37$ $19{,}\,{\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }$ $26{,}\,{\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ $19{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ $37$ $27{,}\,{\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.3.0.1}{3} }$

Cycle lengths which are repeated in a cycle type are indicated by exponents.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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
\(983\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $8$$1$$8$$0$$C_8$$$[\ ]^{8}$$
Deg $11$$1$$11$$0$$C_{11}$$$[\ ]^{11}$$
Deg $14$$1$$14$$0$$C_{14}$$$[\ ]^{14}$$
\(106\!\cdots\!507\) Copy content Toggle raw display $\Q_{10\!\cdots\!07}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $3$$1$$3$$0$$C_3$$$[\ ]^{3}$$
Deg $4$$1$$4$$0$$C_4$$$[\ ]^{4}$$
Deg $7$$1$$7$$0$$C_7$$$[\ ]^{7}$$
Deg $20$$1$$20$$0$20T1$$[\ ]^{20}$$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)