Properties

Label 41.41.374...801.1
Degree $41$
Signature $[41, 0]$
Discriminant $3.747\times 10^{116}$
Root discriminant \(697.05\)
Ramified prime $821$
Class number not computed
Class group not computed
Galois group $C_{41}$ (as 41T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^41 - x^40 - 400*x^39 + 267*x^38 + 70434*x^37 - 25456*x^36 - 7231147*x^35 + 643597*x^34 + 483795648*x^33 + 68617520*x^32 - 22359647160*x^31 - 7251158053*x^30 + 739255603225*x^29 + 335708547372*x^28 - 17885796411269*x^27 - 9479269329758*x^26 + 321562652641396*x^25 + 177909044828159*x^24 - 4339729918343900*x^23 - 2288515231871121*x^22 + 44211233079107427*x^21 + 20209478217861146*x^20 - 340291820674520880*x^19 - 118610646663806737*x^18 + 1969458668208622674*x^17 + 410494333593592611*x^16 - 8467703539109471908*x^15 - 372233629251247247*x^14 + 26446416435739805624*x^13 - 3641024063367368116*x^12 - 57732002041736983193*x^11 + 19074905803558500162*x^10 + 82370621416601788582*x^9 - 43840068640516285889*x^8 - 67556477435299552660*x^7 + 51180521959924527335*x^6 + 23228035973272766493*x^5 - 26391119726982867791*x^4 + 202102108745982587*x^3 + 4168134528839675291*x^2 - 499943870456762867*x - 149270874547838371)
 
gp: K = bnfinit(y^41 - y^40 - 400*y^39 + 267*y^38 + 70434*y^37 - 25456*y^36 - 7231147*y^35 + 643597*y^34 + 483795648*y^33 + 68617520*y^32 - 22359647160*y^31 - 7251158053*y^30 + 739255603225*y^29 + 335708547372*y^28 - 17885796411269*y^27 - 9479269329758*y^26 + 321562652641396*y^25 + 177909044828159*y^24 - 4339729918343900*y^23 - 2288515231871121*y^22 + 44211233079107427*y^21 + 20209478217861146*y^20 - 340291820674520880*y^19 - 118610646663806737*y^18 + 1969458668208622674*y^17 + 410494333593592611*y^16 - 8467703539109471908*y^15 - 372233629251247247*y^14 + 26446416435739805624*y^13 - 3641024063367368116*y^12 - 57732002041736983193*y^11 + 19074905803558500162*y^10 + 82370621416601788582*y^9 - 43840068640516285889*y^8 - 67556477435299552660*y^7 + 51180521959924527335*y^6 + 23228035973272766493*y^5 - 26391119726982867791*y^4 + 202102108745982587*y^3 + 4168134528839675291*y^2 - 499943870456762867*y - 149270874547838371, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^41 - x^40 - 400*x^39 + 267*x^38 + 70434*x^37 - 25456*x^36 - 7231147*x^35 + 643597*x^34 + 483795648*x^33 + 68617520*x^32 - 22359647160*x^31 - 7251158053*x^30 + 739255603225*x^29 + 335708547372*x^28 - 17885796411269*x^27 - 9479269329758*x^26 + 321562652641396*x^25 + 177909044828159*x^24 - 4339729918343900*x^23 - 2288515231871121*x^22 + 44211233079107427*x^21 + 20209478217861146*x^20 - 340291820674520880*x^19 - 118610646663806737*x^18 + 1969458668208622674*x^17 + 410494333593592611*x^16 - 8467703539109471908*x^15 - 372233629251247247*x^14 + 26446416435739805624*x^13 - 3641024063367368116*x^12 - 57732002041736983193*x^11 + 19074905803558500162*x^10 + 82370621416601788582*x^9 - 43840068640516285889*x^8 - 67556477435299552660*x^7 + 51180521959924527335*x^6 + 23228035973272766493*x^5 - 26391119726982867791*x^4 + 202102108745982587*x^3 + 4168134528839675291*x^2 - 499943870456762867*x - 149270874547838371);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^41 - x^40 - 400*x^39 + 267*x^38 + 70434*x^37 - 25456*x^36 - 7231147*x^35 + 643597*x^34 + 483795648*x^33 + 68617520*x^32 - 22359647160*x^31 - 7251158053*x^30 + 739255603225*x^29 + 335708547372*x^28 - 17885796411269*x^27 - 9479269329758*x^26 + 321562652641396*x^25 + 177909044828159*x^24 - 4339729918343900*x^23 - 2288515231871121*x^22 + 44211233079107427*x^21 + 20209478217861146*x^20 - 340291820674520880*x^19 - 118610646663806737*x^18 + 1969458668208622674*x^17 + 410494333593592611*x^16 - 8467703539109471908*x^15 - 372233629251247247*x^14 + 26446416435739805624*x^13 - 3641024063367368116*x^12 - 57732002041736983193*x^11 + 19074905803558500162*x^10 + 82370621416601788582*x^9 - 43840068640516285889*x^8 - 67556477435299552660*x^7 + 51180521959924527335*x^6 + 23228035973272766493*x^5 - 26391119726982867791*x^4 + 202102108745982587*x^3 + 4168134528839675291*x^2 - 499943870456762867*x - 149270874547838371)
 

\( x^{41} - x^{40} - 400 x^{39} + 267 x^{38} + 70434 x^{37} - 25456 x^{36} - 7231147 x^{35} + \cdots - 14\!\cdots\!71 \) 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:  $[41, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(374\!\cdots\!801\) \(\medspace = 821^{40}\) 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:  \(697.05\)
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:  $821^{40/41}\approx 697.0462846642772$
Ramified primes:   \(821\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $41$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(821\)
Dirichlet character group:    $\lbrace$$\chi_{821}(1,·)$, $\chi_{821}(515,·)$, $\chi_{821}(132,·)$, $\chi_{821}(649,·)$, $\chi_{821}(651,·)$, $\chi_{821}(784,·)$, $\chi_{821}(658,·)$, $\chi_{821}(404,·)$, $\chi_{821}(284,·)$, $\chi_{821}(543,·)$, $\chi_{821}(35,·)$, $\chi_{821}(548,·)$, $\chi_{821}(165,·)$, $\chi_{821}(166,·)$, $\chi_{821}(297,·)$, $\chi_{821}(42,·)$, $\chi_{821}(685,·)$, $\chi_{821}(434,·)$, $\chi_{821}(563,·)$, $\chi_{821}(566,·)$, $\chi_{821}(183,·)$, $\chi_{821}(159,·)$, $\chi_{821}(63,·)$, $\chi_{821}(198,·)$, $\chi_{821}(28,·)$, $\chi_{821}(463,·)$, $\chi_{821}(249,·)$, $\chi_{821}(88,·)$, $\chi_{821}(412,·)$, $\chi_{821}(347,·)$, $\chi_{821}(606,·)$, $\chi_{821}(355,·)$, $\chi_{821}(617,·)$, $\chi_{821}(618,·)$, $\chi_{821}(110,·)$, $\chi_{821}(106,·)$, $\chi_{821}(505,·)$, $\chi_{821}(122,·)$, $\chi_{821}(426,·)$, $\chi_{821}(362,·)$, $\chi_{821}(639,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $\frac{1}{72\!\cdots\!21}a^{40}+\frac{19\!\cdots\!99}{72\!\cdots\!21}a^{39}-\frac{67\!\cdots\!87}{72\!\cdots\!21}a^{38}+\frac{18\!\cdots\!90}{72\!\cdots\!21}a^{37}+\frac{39\!\cdots\!92}{72\!\cdots\!21}a^{36}+\frac{62\!\cdots\!95}{72\!\cdots\!21}a^{35}-\frac{12\!\cdots\!01}{72\!\cdots\!21}a^{34}-\frac{35\!\cdots\!47}{72\!\cdots\!21}a^{33}+\frac{36\!\cdots\!04}{72\!\cdots\!21}a^{32}+\frac{10\!\cdots\!55}{72\!\cdots\!21}a^{31}-\frac{67\!\cdots\!36}{72\!\cdots\!21}a^{30}+\frac{22\!\cdots\!49}{72\!\cdots\!21}a^{29}+\frac{16\!\cdots\!28}{72\!\cdots\!21}a^{28}+\frac{22\!\cdots\!70}{72\!\cdots\!21}a^{27}+\frac{19\!\cdots\!61}{72\!\cdots\!21}a^{26}-\frac{22\!\cdots\!77}{72\!\cdots\!21}a^{25}-\frac{23\!\cdots\!85}{72\!\cdots\!21}a^{24}+\frac{10\!\cdots\!85}{72\!\cdots\!21}a^{23}-\frac{27\!\cdots\!29}{72\!\cdots\!21}a^{22}+\frac{15\!\cdots\!93}{72\!\cdots\!21}a^{21}-\frac{37\!\cdots\!50}{72\!\cdots\!21}a^{20}-\frac{18\!\cdots\!86}{72\!\cdots\!21}a^{19}-\frac{26\!\cdots\!71}{72\!\cdots\!21}a^{18}-\frac{10\!\cdots\!11}{72\!\cdots\!21}a^{17}-\frac{17\!\cdots\!00}{72\!\cdots\!21}a^{16}-\frac{28\!\cdots\!59}{72\!\cdots\!21}a^{15}-\frac{25\!\cdots\!07}{72\!\cdots\!21}a^{14}+\frac{18\!\cdots\!23}{72\!\cdots\!21}a^{13}-\frac{18\!\cdots\!25}{72\!\cdots\!21}a^{12}+\frac{33\!\cdots\!02}{72\!\cdots\!21}a^{11}+\frac{62\!\cdots\!73}{72\!\cdots\!21}a^{10}-\frac{58\!\cdots\!21}{72\!\cdots\!21}a^{9}-\frac{21\!\cdots\!32}{72\!\cdots\!21}a^{8}+\frac{34\!\cdots\!97}{72\!\cdots\!21}a^{7}+\frac{60\!\cdots\!44}{72\!\cdots\!21}a^{6}-\frac{92\!\cdots\!37}{72\!\cdots\!21}a^{5}+\frac{17\!\cdots\!81}{72\!\cdots\!21}a^{4}-\frac{15\!\cdots\!73}{72\!\cdots\!21}a^{3}+\frac{25\!\cdots\!60}{72\!\cdots\!21}a^{2}+\frac{17\!\cdots\!28}{72\!\cdots\!21}a+\frac{22\!\cdots\!42}{72\!\cdots\!21}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
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:  $40$
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 - x^40 - 400*x^39 + 267*x^38 + 70434*x^37 - 25456*x^36 - 7231147*x^35 + 643597*x^34 + 483795648*x^33 + 68617520*x^32 - 22359647160*x^31 - 7251158053*x^30 + 739255603225*x^29 + 335708547372*x^28 - 17885796411269*x^27 - 9479269329758*x^26 + 321562652641396*x^25 + 177909044828159*x^24 - 4339729918343900*x^23 - 2288515231871121*x^22 + 44211233079107427*x^21 + 20209478217861146*x^20 - 340291820674520880*x^19 - 118610646663806737*x^18 + 1969458668208622674*x^17 + 410494333593592611*x^16 - 8467703539109471908*x^15 - 372233629251247247*x^14 + 26446416435739805624*x^13 - 3641024063367368116*x^12 - 57732002041736983193*x^11 + 19074905803558500162*x^10 + 82370621416601788582*x^9 - 43840068640516285889*x^8 - 67556477435299552660*x^7 + 51180521959924527335*x^6 + 23228035973272766493*x^5 - 26391119726982867791*x^4 + 202102108745982587*x^3 + 4168134528839675291*x^2 - 499943870456762867*x - 149270874547838371)
 
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^40 - 400*x^39 + 267*x^38 + 70434*x^37 - 25456*x^36 - 7231147*x^35 + 643597*x^34 + 483795648*x^33 + 68617520*x^32 - 22359647160*x^31 - 7251158053*x^30 + 739255603225*x^29 + 335708547372*x^28 - 17885796411269*x^27 - 9479269329758*x^26 + 321562652641396*x^25 + 177909044828159*x^24 - 4339729918343900*x^23 - 2288515231871121*x^22 + 44211233079107427*x^21 + 20209478217861146*x^20 - 340291820674520880*x^19 - 118610646663806737*x^18 + 1969458668208622674*x^17 + 410494333593592611*x^16 - 8467703539109471908*x^15 - 372233629251247247*x^14 + 26446416435739805624*x^13 - 3641024063367368116*x^12 - 57732002041736983193*x^11 + 19074905803558500162*x^10 + 82370621416601788582*x^9 - 43840068640516285889*x^8 - 67556477435299552660*x^7 + 51180521959924527335*x^6 + 23228035973272766493*x^5 - 26391119726982867791*x^4 + 202102108745982587*x^3 + 4168134528839675291*x^2 - 499943870456762867*x - 149270874547838371, 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^40 - 400*x^39 + 267*x^38 + 70434*x^37 - 25456*x^36 - 7231147*x^35 + 643597*x^34 + 483795648*x^33 + 68617520*x^32 - 22359647160*x^31 - 7251158053*x^30 + 739255603225*x^29 + 335708547372*x^28 - 17885796411269*x^27 - 9479269329758*x^26 + 321562652641396*x^25 + 177909044828159*x^24 - 4339729918343900*x^23 - 2288515231871121*x^22 + 44211233079107427*x^21 + 20209478217861146*x^20 - 340291820674520880*x^19 - 118610646663806737*x^18 + 1969458668208622674*x^17 + 410494333593592611*x^16 - 8467703539109471908*x^15 - 372233629251247247*x^14 + 26446416435739805624*x^13 - 3641024063367368116*x^12 - 57732002041736983193*x^11 + 19074905803558500162*x^10 + 82370621416601788582*x^9 - 43840068640516285889*x^8 - 67556477435299552660*x^7 + 51180521959924527335*x^6 + 23228035973272766493*x^5 - 26391119726982867791*x^4 + 202102108745982587*x^3 + 4168134528839675291*x^2 - 499943870456762867*x - 149270874547838371);
 
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^40 - 400*x^39 + 267*x^38 + 70434*x^37 - 25456*x^36 - 7231147*x^35 + 643597*x^34 + 483795648*x^33 + 68617520*x^32 - 22359647160*x^31 - 7251158053*x^30 + 739255603225*x^29 + 335708547372*x^28 - 17885796411269*x^27 - 9479269329758*x^26 + 321562652641396*x^25 + 177909044828159*x^24 - 4339729918343900*x^23 - 2288515231871121*x^22 + 44211233079107427*x^21 + 20209478217861146*x^20 - 340291820674520880*x^19 - 118610646663806737*x^18 + 1969458668208622674*x^17 + 410494333593592611*x^16 - 8467703539109471908*x^15 - 372233629251247247*x^14 + 26446416435739805624*x^13 - 3641024063367368116*x^12 - 57732002041736983193*x^11 + 19074905803558500162*x^10 + 82370621416601788582*x^9 - 43840068640516285889*x^8 - 67556477435299552660*x^7 + 51180521959924527335*x^6 + 23228035973272766493*x^5 - 26391119726982867791*x^4 + 202102108745982587*x^3 + 4168134528839675291*x^2 - 499943870456762867*x - 149270874547838371);
 
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

$C_{41}$ (as 41T1):

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 cyclic group of order 41
The 41 conjugacy class representatives for $C_{41}$
Character table for $C_{41}$

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 $41$ $41$ $41$ $41$ $41$ $41$ $41$ $41$ $41$ $41$ $41$ $41$ $41$ $41$ $41$ $41$ $41$

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
\(821\) Copy content Toggle raw display Deg $41$$41$$1$$40$