Properties

Label 41.1.131...641.1
Degree $41$
Signature $[1, 20]$
Discriminant $1.319\times 10^{66}$
Root discriminant \(40.99\)
Ramified primes see page
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_{41}$ (as 41T10)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^41 - x - 1)
 
gp: K = bnfinit(y^41 - y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^41 - x - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^41 - x - 1)
 

\( x^{41} - x - 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $41$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[1, 20]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(1318788372436565706966337480963346255985889330161650994325137953641\) \(\medspace = 2251\cdot 272583857708203\cdot 761889448323748621067\cdot 2821028813073407561156582291\) 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:  \(40.99\)
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:  $2251^{1/2}272583857708203^{1/2}761889448323748621067^{1/2}2821028813073407561156582291^{1/2}\approx 1.1483851150361388e+33$
Ramified primes:   \(2251\), \(272583857708203\), \(761889448323748621067\), \(2821028813073407561156582291\) 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:  $\Q(\sqrt{13187\!\cdots\!53641}$)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$ 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:  $20$
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:   $a^{40}-1$, $a^{20}+1$, $a^{10}+1$, $a^{40}-a^{39}+a^{38}-a^{37}+a^{36}$, $a^{12}+a^{10}$, $a^{28}-a^{27}$, $a^{8}+a^{4}$, $a^{40}-a^{39}+a^{38}-a^{37}+a^{36}-a^{35}-a^{7}-1$, $a^{40}-a^{39}+a^{38}-a^{37}+a^{36}-a^{35}-a^{8}-1$, $a^{40}-a^{39}+a^{38}+a^{34}-a^{33}+a^{32}-1$, $a^{40}-a^{39}+a^{38}-a^{37}+a^{36}-a^{29}+a^{28}-a^{27}+a^{26}-a^{25}+a^{24}+a^{19}-a^{13}+a^{7}-a-1$, $a^{40}-a^{39}+a^{38}-a^{37}-a^{32}+a^{31}-a^{30}+a^{29}-a^{28}-a^{23}-a^{19}-a^{14}-a^{10}-a^{5}-a-1$, $2a^{40}-2a^{39}+2a^{38}-a^{37}+a^{36}-a^{35}+a^{34}-a^{33}+a^{32}-a^{31}+a^{30}-a^{29}-a^{21}+a^{12}+a^{5}+a^{4}-2$, $a^{40}-a^{38}+a^{37}-a^{35}+a^{34}+a^{30}+a^{26}-a^{25}+a^{23}-a^{22}+a^{20}-a^{19}-a^{14}+a^{13}-a^{11}+a^{10}-a^{8}+a^{7}-a^{5}-a-1$, $2a^{40}-2a^{39}+2a^{38}-a^{37}+2a^{36}-a^{35}+a^{34}-a^{33}-a^{31}+a^{30}+a^{28}-a^{27}-a^{25}-a^{20}-a^{19}-a^{18}-a^{12}+a^{9}-a^{6}+a^{3}+a^{2}+a-2$, $a^{36}-a^{35}+a^{34}-a^{33}+a^{31}-2a^{28}+a^{27}-a^{26}+2a^{25}-a^{22}+a^{20}-a^{19}+a^{18}-a^{17}+a^{16}-a^{15}-a^{13}+a^{11}+a^{10}+a^{9}-2a^{8}+a^{5}-a^{2}-a+1$, $a^{38}+2a^{34}-a^{33}+2a^{31}-a^{29}+a^{28}+a^{27}-a^{26}+a^{24}-a^{22}+a^{21}-a^{20}-a^{19}-a^{16}-a^{15}+a^{14}-a^{13}-2a^{12}+a^{11}-2a^{9}+a^{7}-a^{6}-a^{5}+a^{4}-a^{2}+a+1$, $2a^{39}+2a^{37}-a^{36}+a^{35}-a^{31}-a^{30}-a^{29}-a^{25}-a^{23}+a^{22}+a^{20}+a^{17}+a^{16}+a^{15}+a^{14}-2a^{8}-2a^{7}-2a^{6}-a^{5}+1$, $12a^{40}-11a^{39}+10a^{38}-9a^{37}+8a^{36}-7a^{35}+5a^{34}-3a^{33}+2a^{31}-4a^{30}+5a^{29}-5a^{28}+5a^{27}-6a^{26}+7a^{25}-7a^{24}+8a^{23}-7a^{22}+5a^{21}-3a^{20}+3a^{19}-2a^{18}+a^{17}-a^{16}+a^{14}-3a^{13}+3a^{12}-4a^{11}+4a^{10}-4a^{9}+3a^{8}-2a^{7}+3a^{6}-3a^{5}+3a^{4}-2a^{3}+2a^{2}-13$, $a^{39}-a^{37}+a^{36}-2a^{33}+2a^{32}-a^{31}-a^{29}+2a^{28}-a^{27}-a^{25}+2a^{24}-2a^{23}+a^{21}-2a^{18}+3a^{17}-2a^{16}+a^{15}-a^{14}+2a^{13}-2a^{12}-a^{11}+a^{10}-a^{7}+2a^{6}-a^{5}-a^{4}+a^{2}-a-1$ 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:  \( 28450692757863310 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{20}\cdot 28450692757863310 \cdot 1}{2\cdot\sqrt{1318788372436565706966337480963346255985889330161650994325137953641}}\cr\approx \mathstrut & 0.227825671055451 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^41 - 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^41 - 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^41 - 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^41 - 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

$S_{41}$ (as 41T10):

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 33452526613163807108170062053440751665152000000000
The 44583 conjugacy class representatives for $S_{41}$ are not computed
Character table for $S_{41}$

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 $32{,}\,{\href{/padicField/2.7.0.1}{7} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ $41$ $31{,}\,{\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ $26{,}\,{\href{/padicField/7.11.0.1}{11} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ $39{,}\,{\href{/padicField/11.2.0.1}{2} }$ $24{,}\,{\href{/padicField/13.13.0.1}{13} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ $29{,}\,{\href{/padicField/17.12.0.1}{12} }$ $19{,}\,{\href{/padicField/19.13.0.1}{13} }{,}\,{\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ $20{,}\,{\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ $23{,}\,{\href{/padicField/29.9.0.1}{9} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ $23{,}\,{\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ $32{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ $41$ $19{,}\,{\href{/padicField/43.11.0.1}{11} }{,}\,{\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ $39{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ $23{,}\,15{,}\,{\href{/padicField/53.3.0.1}{3} }$ $15{,}\,{\href{/padicField/59.14.0.1}{14} }{,}\,{\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
\(2251\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $6$$1$$6$$0$$C_6$$[\ ]^{6}$
Deg $28$$1$$28$$0$$C_{28}$$[\ ]^{28}$
\(272583857708203\) Copy content Toggle raw display $\Q_{272583857708203}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $7$$1$$7$$0$$C_7$$[\ ]^{7}$
Deg $14$$1$$14$$0$$C_{14}$$[\ ]^{14}$
Deg $15$$1$$15$$0$$C_{15}$$[\ ]^{15}$
\(761\!\cdots\!067\) Copy content Toggle raw display $\Q_{76\!\cdots\!67}$$x$$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$$[\ ]^{5}$
Deg $9$$1$$9$$0$$C_9$$[\ ]^{9}$
Deg $20$$1$$20$$0$20T1$[\ ]^{20}$
\(282\!\cdots\!291\) Copy content Toggle raw display $\Q_{28\!\cdots\!91}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $26$$1$$26$$0$$C_{26}$$[\ ]^{26}$