Properties

Label 41.1.549...641.1
Degree $41$
Signature $[1, 20]$
Discriminant $5.498\times 10^{92}$
Root discriminant \(182.79\)
Ramified primes see page
Class number not computed
Class group not computed
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 + 5*x - 1)
 
gp: K = bnfinit(y^41 + 5*y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^41 + 5*x - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^41 + 5*x - 1)
 

\( x^{41} + 5x - 1 \) 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:   \(549\!\cdots\!641\) \(\medspace = 43\cdot 191\cdot 9901\cdot 55082201453159\cdot 467781923539898069\cdot 251911473477379412779567\cdot 10\!\cdots\!01\) 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:  \(182.79\)
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:  $43^{1/2}191^{1/2}9901^{1/2}55082201453159^{1/2}467781923539898069^{1/2}251911473477379412779567^{1/2}1041564752951083406029502313101^{1/2}\approx 2.3446872155748193e+46$
Ramified primes:   \(43\), \(191\), \(9901\), \(55082201453159\), \(467781923539898069\), \(251911473477379412779567\), \(10415\!\cdots\!13101\) 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:  $\Q(\sqrt{54975\!\cdots\!53641}$)
$\card{ \Aut(K/\Q) }$:  $1$
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

not computed

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:  not computed
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:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^41 + 5*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 + 5*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 + 5*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 + 5*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 $32{,}\,{\href{/padicField/2.7.0.1}{7} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ $41$ $20^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }$ $24{,}\,15{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ $40{,}\,{\href{/padicField/11.1.0.1}{1} }$ $22{,}\,19$ $25{,}\,{\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ $16{,}\,{\href{/padicField/19.13.0.1}{13} }{,}\,{\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.14.0.1}{14} }^{2}{,}\,{\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ $19{,}\,17{,}\,{\href{/padicField/29.5.0.1}{5} }$ $27{,}\,{\href{/padicField/31.7.0.1}{7} }^{2}$ $22{,}\,15{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ $20^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ R $32{,}\,{\href{/padicField/47.9.0.1}{9} }$ $34{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }$ $25{,}\,{\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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
\(43\) Copy content Toggle raw display $\Q_{43}$$x + 40$$1$$1$$0$Trivial$[\ ]$
43.2.0.1$x^{2} + 42 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.1.2$x^{2} + 43$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $36$$1$$36$$0$$C_{36}$$[\ ]^{36}$
\(191\) Copy content Toggle raw display 191.2.1.2$x^{2} + 191$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.0.1$x^{2} + 190 x + 19$$1$$2$$0$$C_2$$[\ ]^{2}$
191.4.0.1$x^{4} + 7 x^{2} + 100 x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
191.8.0.1$x^{8} + 2 x^{4} + 164 x^{3} + 139 x^{2} + 171 x + 19$$1$$8$$0$$C_8$$[\ ]^{8}$
Deg $25$$1$$25$$0$$C_{25}$$[\ ]^{25}$
\(9901\) Copy content Toggle raw display Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $5$$1$$5$$0$$C_5$$[\ ]^{5}$
Deg $12$$1$$12$$0$$C_{12}$$[\ ]^{12}$
Deg $22$$1$$22$$0$22T1$[\ ]^{22}$
\(55082201453159\) Copy content Toggle raw display $\Q_{55082201453159}$$x$$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}$$[\ ]^{10}$
Deg $22$$1$$22$$0$22T1$[\ ]^{22}$
\(467781923539898069\) Copy content Toggle raw display $\Q_{467781923539898069}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{467781923539898069}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{467781923539898069}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $32$$1$$32$$0$32T33$[\ ]^{32}$
\(251\!\cdots\!567\) Copy content Toggle raw display Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $7$$1$$7$$0$$C_7$$[\ ]^{7}$
Deg $8$$1$$8$$0$$C_8$$[\ ]^{8}$
Deg $24$$1$$24$$0$$C_{24}$$[\ ]^{24}$
\(104\!\cdots\!101\) Copy content Toggle raw display $\Q_{10\!\cdots\!01}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{10\!\cdots\!01}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $5$$1$$5$$0$$C_5$$[\ ]^{5}$
Deg $7$$1$$7$$0$$C_7$$[\ ]^{7}$
Deg $17$$1$$17$$0$$C_{17}$$[\ ]^{17}$