Properties

Label 37.1.106...653.1
Degree $37$
Signature $[1, 18]$
Discriminant $1.066\times 10^{58}$
Root discriminant \(37.01\)
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 / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^37 + x - 1)
 
gp: K = bnfinit(y^37 + y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^37 + x - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^37 + x - 1)
 

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

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

Invariants

Degree:  $37$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[1, 18]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(10661522314701499938886043561748289339867563939000058733653\) \(\medspace = 19\cdot 11323755688122007063\cdot 49\!\cdots\!49\) 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:  \(37.01\)
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:  $19^{1/2}11323755688122007063^{1/2}49553590598385949034316572009816214049^{1/2}\approx 1.0325464790846705e+29$
Ramified primes:   \(19\), \(11323755688122007063\), \(49553\!\cdots\!14049\) 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:  $\Q(\sqrt{10661\!\cdots\!33653}$)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$ 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:  $18$
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:   $a^{36}+1$, $a^{14}+a^{2}$, $a^{9}+a^{5}$, $a^{36}+a^{35}+a^{34}+a^{33}+a^{32}+a^{31}+a^{30}-a^{12}+1$, $a^{36}+a^{35}+a^{34}+a^{33}+a^{32}+a^{31}+a^{30}+a^{29}+a^{28}+a^{27}+a^{26}+a^{25}+a^{23}+a^{21}+1$, $a^{36}+a^{35}+a^{34}+a^{33}+a^{32}+a^{31}+a^{30}+a^{29}+a^{28}+a^{27}+a^{26}+a^{24}+a^{22}+a^{20}+1$, $a^{36}+a^{35}+a^{29}+a^{28}+a^{27}+a^{26}+a^{25}+a^{18}-a^{13}+a^{8}-a^{3}+1$, $a^{36}+a^{35}-a^{33}-a^{32}-a^{31}-a^{30}+a^{28}+a^{27}-a^{25}-a^{24}-a^{23}-a^{22}+a^{20}+a^{19}-a^{17}-a^{16}+a^{12}+a^{11}-a^{9}-a^{8}+a^{6}+a^{3}-a$, $a^{36}+a^{35}+a^{34}+a^{33}+a^{26}+a^{11}+a^{4}+1$, $a^{36}+a^{35}+a^{34}+a^{33}+a^{32}+a^{31}+a^{30}+a^{29}+a^{28}+a^{27}+a^{26}+a^{25}+a^{24}+a^{22}+a^{21}+a^{19}+a^{16}+1$, $a^{36}+a^{34}+a^{33}-a^{27}-a^{24}-a^{19}+a^{18}+a^{17}+a^{13}+a^{12}-a^{11}+a^{9}-a^{6}+a^{4}-a^{3}+1$, $a^{36}+a^{33}+a^{32}+a^{29}+a^{28}+a^{25}+a^{24}-a^{18}-a^{17}-2a^{14}-a^{13}+a^{12}-a^{10}+a^{8}+a^{3}-a+1$, $a^{36}+2a^{35}+2a^{34}+a^{33}+2a^{32}+a^{31}+a^{29}+a^{28}+a^{27}+a^{26}+2a^{25}+a^{24}-a^{23}+a^{22}+a^{19}+a^{18}+a^{17}-a^{16}-a^{13}+a^{12}+a^{11}-2a^{6}+a^{5}+a^{4}-a^{3}+a^{2}+1$, $a^{35}+a^{34}+2a^{33}+a^{32}+2a^{31}+2a^{30}+2a^{29}+2a^{28}+3a^{27}+2a^{26}+2a^{25}+2a^{24}+2a^{23}+a^{22}+2a^{21}+2a^{20}+a^{19}+a^{18}+a^{17}+a^{14}-a^{9}-a^{7}-a^{5}-a^{3}-a$, $a^{36}+a^{30}+a^{29}+a^{28}+a^{27}+a^{26}+a^{25}+a^{24}+a^{23}+a^{22}+a^{21}+a^{20}+a^{19}-a^{12}+a^{7}+a^{5}+a+1$, $2a^{36}+a^{35}+a^{34}+a^{33}+a^{32}+a^{29}+a^{28}+a^{25}+a^{24}+a^{23}+a^{21}+a^{10}-a+2$, $a^{36}+2a^{35}+a^{34}+a^{33}+a^{31}+a^{24}+a^{20}+a^{19}+a^{12}-a^{11}+a^{10}+a^{8}-a^{7}+1$, $a^{36}-a^{33}+a^{31}+a^{29}+a^{24}-a^{22}-a^{21}+2a^{19}-a^{17}+a^{15}+2a^{14}-a^{12}+a^{9}-a^{8}-a^{5}+a^{4}+a^{2}$ 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:  \( 232694775426930.7 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{18}\cdot 232694775426930.7 \cdot 1}{2\cdot\sqrt{10661522314701499938886043561748289339867563939000058733653}}\cr\approx \mathstrut & 0.524946026498318 \end{aligned}\] (assuming GRH)

# 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))))
 
# 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)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); 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)));
 
# 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):

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 13763753091226345046315979581580902400000000
The 21637 conjugacy class representatives for $S_{37}$ are not computed
Character table for $S_{37}$ 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 $19{,}\,18$ $17{,}\,{\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ $30{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }$ $36{,}\,{\href{/padicField/7.1.0.1}{1} }$ $36{,}\,{\href{/padicField/11.1.0.1}{1} }$ $31{,}\,{\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ $24{,}\,{\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ R $27{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ $16{,}\,{\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.9.0.1}{9} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ $21{,}\,15{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.2.0.1}{2} }^{18}{,}\,{\href{/padicField/37.1.0.1}{1} }$ $17{,}\,{\href{/padicField/41.9.0.1}{9} }{,}\,{\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ $20{,}\,{\href{/padicField/43.11.0.1}{11} }{,}\,{\href{/padicField/43.6.0.1}{6} }$ $21{,}\,{\href{/padicField/47.14.0.1}{14} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ ${\href{/padicField/53.13.0.1}{13} }{,}\,{\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.5.0.1}{5} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ $15{,}\,{\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
\(19\) Copy content Toggle raw display 19.2.1.1$x^{2} + 38$$2$$1$$1$$C_2$$[\ ]_{2}$
19.9.0.1$x^{9} + 11 x^{3} + 14 x^{2} + 16 x + 17$$1$$9$$0$$C_9$$[\ ]^{9}$
19.11.0.1$x^{11} + 8 x + 17$$1$$11$$0$$C_{11}$$[\ ]^{11}$
19.15.0.1$x^{15} + x^{7} + 10 x^{6} + 11 x^{5} + 13 x^{4} + 15 x^{3} + 14 x^{2} + 17$$1$$15$$0$$C_{15}$$[\ ]^{15}$
\(11323755688122007063\) Copy content Toggle raw display $\Q_{11323755688122007063}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $34$$1$$34$$0$$C_{34}$$[\ ]^{34}$
\(495\!\cdots\!049\) Copy content Toggle raw display Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $28$$1$$28$$0$$C_{28}$$[\ ]^{28}$