Properties

Label 41.41.556...401.1
Degree $41$
Signature $[41, 0]$
Discriminant $5.569\times 10^{114}$
Root discriminant \(629.04\)
Ramified prime $739$
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 - 360*x^39 + 745*x^38 + 56451*x^37 - 169135*x^36 - 5037002*x^35 + 19221892*x^34 + 281906305*x^33 - 1293434183*x^32 - 10337722909*x^31 + 55807085286*x^30 + 252777087235*x^29 - 1613716714335*x^28 - 4112155095553*x^27 + 32113136009541*x^26 + 43177039971301*x^25 - 446781341742247*x^24 - 262923207247335*x^23 + 4379546308331567*x^22 + 474335495077459*x^21 - 30267239554430324*x^20 + 5902126403913719*x^19 + 146509496594642143*x^18 - 56363172819685690*x^17 - 489681930758553989*x^16 + 246552011225708759*x^15 + 1105369786197101758*x^14 - 636716564586328491*x^13 - 1632560051818863682*x^12 + 995239432812617607*x^11 + 1514536703235322325*x^10 - 898680726508319339*x^9 - 853943547109610470*x^8 + 424335389068850481*x^7 + 299252740781325936*x^6 - 91861302074915780*x^5 - 62873369979635247*x^4 + 4140602930707759*x^3 + 6001367177186495*x^2 + 926057723623551*x + 39010859256913)
 
gp: K = bnfinit(y^41 - y^40 - 360*y^39 + 745*y^38 + 56451*y^37 - 169135*y^36 - 5037002*y^35 + 19221892*y^34 + 281906305*y^33 - 1293434183*y^32 - 10337722909*y^31 + 55807085286*y^30 + 252777087235*y^29 - 1613716714335*y^28 - 4112155095553*y^27 + 32113136009541*y^26 + 43177039971301*y^25 - 446781341742247*y^24 - 262923207247335*y^23 + 4379546308331567*y^22 + 474335495077459*y^21 - 30267239554430324*y^20 + 5902126403913719*y^19 + 146509496594642143*y^18 - 56363172819685690*y^17 - 489681930758553989*y^16 + 246552011225708759*y^15 + 1105369786197101758*y^14 - 636716564586328491*y^13 - 1632560051818863682*y^12 + 995239432812617607*y^11 + 1514536703235322325*y^10 - 898680726508319339*y^9 - 853943547109610470*y^8 + 424335389068850481*y^7 + 299252740781325936*y^6 - 91861302074915780*y^5 - 62873369979635247*y^4 + 4140602930707759*y^3 + 6001367177186495*y^2 + 926057723623551*y + 39010859256913, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^41 - x^40 - 360*x^39 + 745*x^38 + 56451*x^37 - 169135*x^36 - 5037002*x^35 + 19221892*x^34 + 281906305*x^33 - 1293434183*x^32 - 10337722909*x^31 + 55807085286*x^30 + 252777087235*x^29 - 1613716714335*x^28 - 4112155095553*x^27 + 32113136009541*x^26 + 43177039971301*x^25 - 446781341742247*x^24 - 262923207247335*x^23 + 4379546308331567*x^22 + 474335495077459*x^21 - 30267239554430324*x^20 + 5902126403913719*x^19 + 146509496594642143*x^18 - 56363172819685690*x^17 - 489681930758553989*x^16 + 246552011225708759*x^15 + 1105369786197101758*x^14 - 636716564586328491*x^13 - 1632560051818863682*x^12 + 995239432812617607*x^11 + 1514536703235322325*x^10 - 898680726508319339*x^9 - 853943547109610470*x^8 + 424335389068850481*x^7 + 299252740781325936*x^6 - 91861302074915780*x^5 - 62873369979635247*x^4 + 4140602930707759*x^3 + 6001367177186495*x^2 + 926057723623551*x + 39010859256913);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^41 - x^40 - 360*x^39 + 745*x^38 + 56451*x^37 - 169135*x^36 - 5037002*x^35 + 19221892*x^34 + 281906305*x^33 - 1293434183*x^32 - 10337722909*x^31 + 55807085286*x^30 + 252777087235*x^29 - 1613716714335*x^28 - 4112155095553*x^27 + 32113136009541*x^26 + 43177039971301*x^25 - 446781341742247*x^24 - 262923207247335*x^23 + 4379546308331567*x^22 + 474335495077459*x^21 - 30267239554430324*x^20 + 5902126403913719*x^19 + 146509496594642143*x^18 - 56363172819685690*x^17 - 489681930758553989*x^16 + 246552011225708759*x^15 + 1105369786197101758*x^14 - 636716564586328491*x^13 - 1632560051818863682*x^12 + 995239432812617607*x^11 + 1514536703235322325*x^10 - 898680726508319339*x^9 - 853943547109610470*x^8 + 424335389068850481*x^7 + 299252740781325936*x^6 - 91861302074915780*x^5 - 62873369979635247*x^4 + 4140602930707759*x^3 + 6001367177186495*x^2 + 926057723623551*x + 39010859256913)
 

\( x^{41} - x^{40} - 360 x^{39} + 745 x^{38} + 56451 x^{37} - 169135 x^{36} - 5037002 x^{35} + \cdots + 39010859256913 \) 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:   \(556\!\cdots\!401\) \(\medspace = 739^{40}\) 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:  \(629.04\)
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:  $739^{40/41}\approx 629.0388966174669$
Ramified primes:   \(739\) 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:  \(739\)
Dirichlet character group:    $\lbrace$$\chi_{739}(1,·)$, $\chi_{739}(130,·)$, $\chi_{739}(133,·)$, $\chi_{739}(642,·)$, $\chi_{739}(270,·)$, $\chi_{739}(400,·)$, $\chi_{739}(401,·)$, $\chi_{739}(20,·)$, $\chi_{739}(277,·)$, $\chi_{739}(151,·)$, $\chi_{739}(538,·)$, $\chi_{739}(283,·)$, $\chi_{739}(541,·)$, $\chi_{739}(414,·)$, $\chi_{739}(416,·)$, $\chi_{739}(37,·)$, $\chi_{739}(687,·)$, $\chi_{739}(689,·)$, $\chi_{739}(692,·)$, $\chi_{739}(438,·)$, $\chi_{739}(57,·)$, $\chi_{739}(443,·)$, $\chi_{739}(191,·)$, $\chi_{739}(64,·)$, $\chi_{739}(579,·)$, $\chi_{739}(474,·)$, $\chi_{739}(731,·)$, $\chi_{739}(478,·)$, $\chi_{739}(293,·)$, $\chi_{739}(610,·)$, $\chi_{739}(227,·)$, $\chi_{739}(612,·)$, $\chi_{739}(487,·)$, $\chi_{739}(106,·)$, $\chi_{739}(367,·)$, $\chi_{739}(495,·)$, $\chi_{739}(630,·)$, $\chi_{739}(631,·)$, $\chi_{739}(376,·)$, $\chi_{739}(125,·)$, $\chi_{739}(383,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $\frac{1}{317}a^{37}-\frac{58}{317}a^{36}+\frac{145}{317}a^{35}-\frac{138}{317}a^{34}-\frac{7}{317}a^{33}+\frac{136}{317}a^{32}+\frac{93}{317}a^{31}+\frac{65}{317}a^{30}+\frac{148}{317}a^{29}-\frac{107}{317}a^{28}+\frac{63}{317}a^{27}-\frac{128}{317}a^{26}-\frac{102}{317}a^{25}-\frac{27}{317}a^{24}+\frac{8}{317}a^{23}+\frac{119}{317}a^{22}+\frac{156}{317}a^{21}-\frac{63}{317}a^{20}+\frac{46}{317}a^{19}+\frac{144}{317}a^{18}-\frac{21}{317}a^{17}-\frac{146}{317}a^{16}-\frac{108}{317}a^{15}+\frac{140}{317}a^{14}-\frac{52}{317}a^{13}+\frac{85}{317}a^{12}-\frac{17}{317}a^{11}-\frac{44}{317}a^{10}+\frac{14}{317}a^{9}+\frac{149}{317}a^{8}+\frac{140}{317}a^{7}+\frac{16}{317}a^{6}+\frac{143}{317}a^{5}+\frac{154}{317}a^{4}+\frac{90}{317}a^{3}-\frac{119}{317}a^{2}-\frac{115}{317}a-\frac{83}{317}$, $\frac{1}{26166131}a^{38}+\frac{37358}{26166131}a^{37}+\frac{3891374}{26166131}a^{36}+\frac{5387142}{26166131}a^{35}+\frac{4749809}{26166131}a^{34}-\frac{9097200}{26166131}a^{33}-\frac{10707124}{26166131}a^{32}-\frac{11016974}{26166131}a^{31}+\frac{10592085}{26166131}a^{30}-\frac{6067275}{26166131}a^{29}+\frac{4158984}{26166131}a^{28}+\frac{8459647}{26166131}a^{27}+\frac{9792650}{26166131}a^{26}-\frac{5182095}{26166131}a^{25}-\frac{8105635}{26166131}a^{24}-\frac{12874756}{26166131}a^{23}-\frac{7494119}{26166131}a^{22}-\frac{473369}{26166131}a^{21}-\frac{8122441}{26166131}a^{20}+\frac{7722724}{26166131}a^{19}+\frac{11574138}{26166131}a^{18}+\frac{5239337}{26166131}a^{17}+\frac{3925428}{26166131}a^{16}-\frac{4688736}{26166131}a^{15}+\frac{8551472}{26166131}a^{14}-\frac{12457267}{26166131}a^{13}+\frac{3657428}{26166131}a^{12}+\frac{5572329}{26166131}a^{11}+\frac{898269}{26166131}a^{10}-\frac{9005681}{26166131}a^{9}+\frac{10235658}{26166131}a^{8}-\frac{12802531}{26166131}a^{7}-\frac{2098554}{26166131}a^{6}-\frac{3757085}{26166131}a^{5}+\frac{9962087}{26166131}a^{4}+\frac{5793322}{26166131}a^{3}-\frac{10303488}{26166131}a^{2}-\frac{1070157}{26166131}a+\frac{11318923}{26166131}$, $\frac{1}{26166131}a^{39}+\frac{28733}{26166131}a^{37}+\frac{11038887}{26166131}a^{36}+\frac{7251057}{26166131}a^{35}+\frac{3101986}{26166131}a^{34}-\frac{11720881}{26166131}a^{33}+\frac{3474426}{26166131}a^{32}+\frac{5231860}{26166131}a^{31}-\frac{13021595}{26166131}a^{30}-\frac{11444247}{26166131}a^{29}+\frac{5017813}{26166131}a^{28}+\frac{6849210}{26166131}a^{27}+\frac{9899751}{26166131}a^{26}-\frac{1402}{82543}a^{25}+\frac{80637}{26166131}a^{24}-\frac{4725004}{26166131}a^{23}+\frac{924671}{26166131}a^{22}+\frac{1476786}{26166131}a^{21}-\frac{766573}{26166131}a^{20}+\frac{6134853}{26166131}a^{19}-\frac{5573440}{26166131}a^{18}-\frac{3905994}{26166131}a^{17}-\frac{6254848}{26166131}a^{16}+\frac{2714026}{26166131}a^{15}-\frac{3013070}{26166131}a^{14}-\frac{4440802}{26166131}a^{13}-\frac{12941740}{26166131}a^{12}+\frac{1874256}{26166131}a^{11}-\frac{3977382}{26166131}a^{10}-\frac{8808215}{26166131}a^{9}+\frac{607091}{26166131}a^{8}-\frac{2812211}{26166131}a^{7}+\frac{1037658}{26166131}a^{6}-\frac{12415895}{26166131}a^{5}+\frac{6907910}{26166131}a^{4}+\frac{9916841}{26166131}a^{3}-\frac{854601}{26166131}a^{2}-\frac{11661988}{26166131}a-\frac{54518}{26166131}$, $\frac{1}{12\!\cdots\!89}a^{40}-\frac{20\!\cdots\!39}{12\!\cdots\!89}a^{39}-\frac{19\!\cdots\!98}{12\!\cdots\!89}a^{38}-\frac{66\!\cdots\!85}{12\!\cdots\!89}a^{37}-\frac{39\!\cdots\!29}{12\!\cdots\!89}a^{36}+\frac{18\!\cdots\!50}{12\!\cdots\!89}a^{35}-\frac{38\!\cdots\!69}{12\!\cdots\!89}a^{34}-\frac{37\!\cdots\!74}{12\!\cdots\!89}a^{33}+\frac{18\!\cdots\!07}{12\!\cdots\!89}a^{32}-\frac{29\!\cdots\!17}{12\!\cdots\!89}a^{31}-\frac{39\!\cdots\!01}{12\!\cdots\!89}a^{30}-\frac{23\!\cdots\!44}{12\!\cdots\!89}a^{29}+\frac{14\!\cdots\!10}{12\!\cdots\!89}a^{28}+\frac{47\!\cdots\!95}{12\!\cdots\!89}a^{27}-\frac{27\!\cdots\!13}{12\!\cdots\!89}a^{26}+\frac{22\!\cdots\!12}{12\!\cdots\!89}a^{25}-\frac{32\!\cdots\!85}{12\!\cdots\!89}a^{24}+\frac{52\!\cdots\!29}{12\!\cdots\!89}a^{23}-\frac{14\!\cdots\!39}{12\!\cdots\!89}a^{22}+\frac{52\!\cdots\!90}{12\!\cdots\!89}a^{21}+\frac{25\!\cdots\!44}{12\!\cdots\!89}a^{20}-\frac{58\!\cdots\!06}{12\!\cdots\!89}a^{19}-\frac{53\!\cdots\!91}{12\!\cdots\!89}a^{18}+\frac{44\!\cdots\!18}{12\!\cdots\!89}a^{17}+\frac{22\!\cdots\!12}{12\!\cdots\!89}a^{16}+\frac{20\!\cdots\!33}{12\!\cdots\!89}a^{15}-\frac{51\!\cdots\!59}{12\!\cdots\!89}a^{14}+\frac{57\!\cdots\!13}{12\!\cdots\!89}a^{13}-\frac{88\!\cdots\!55}{12\!\cdots\!89}a^{12}+\frac{21\!\cdots\!48}{12\!\cdots\!89}a^{11}+\frac{54\!\cdots\!67}{12\!\cdots\!89}a^{10}-\frac{41\!\cdots\!48}{12\!\cdots\!89}a^{9}+\frac{21\!\cdots\!97}{12\!\cdots\!89}a^{8}-\frac{85\!\cdots\!60}{12\!\cdots\!89}a^{7}-\frac{31\!\cdots\!53}{12\!\cdots\!89}a^{6}-\frac{12\!\cdots\!59}{12\!\cdots\!89}a^{5}-\frac{86\!\cdots\!37}{12\!\cdots\!89}a^{4}-\frac{54\!\cdots\!37}{12\!\cdots\!89}a^{3}-\frac{21\!\cdots\!72}{12\!\cdots\!89}a^{2}+\frac{35\!\cdots\!94}{12\!\cdots\!89}a-\frac{89\!\cdots\!72}{12\!\cdots\!89}$ 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 - 360*x^39 + 745*x^38 + 56451*x^37 - 169135*x^36 - 5037002*x^35 + 19221892*x^34 + 281906305*x^33 - 1293434183*x^32 - 10337722909*x^31 + 55807085286*x^30 + 252777087235*x^29 - 1613716714335*x^28 - 4112155095553*x^27 + 32113136009541*x^26 + 43177039971301*x^25 - 446781341742247*x^24 - 262923207247335*x^23 + 4379546308331567*x^22 + 474335495077459*x^21 - 30267239554430324*x^20 + 5902126403913719*x^19 + 146509496594642143*x^18 - 56363172819685690*x^17 - 489681930758553989*x^16 + 246552011225708759*x^15 + 1105369786197101758*x^14 - 636716564586328491*x^13 - 1632560051818863682*x^12 + 995239432812617607*x^11 + 1514536703235322325*x^10 - 898680726508319339*x^9 - 853943547109610470*x^8 + 424335389068850481*x^7 + 299252740781325936*x^6 - 91861302074915780*x^5 - 62873369979635247*x^4 + 4140602930707759*x^3 + 6001367177186495*x^2 + 926057723623551*x + 39010859256913)
 
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 - 360*x^39 + 745*x^38 + 56451*x^37 - 169135*x^36 - 5037002*x^35 + 19221892*x^34 + 281906305*x^33 - 1293434183*x^32 - 10337722909*x^31 + 55807085286*x^30 + 252777087235*x^29 - 1613716714335*x^28 - 4112155095553*x^27 + 32113136009541*x^26 + 43177039971301*x^25 - 446781341742247*x^24 - 262923207247335*x^23 + 4379546308331567*x^22 + 474335495077459*x^21 - 30267239554430324*x^20 + 5902126403913719*x^19 + 146509496594642143*x^18 - 56363172819685690*x^17 - 489681930758553989*x^16 + 246552011225708759*x^15 + 1105369786197101758*x^14 - 636716564586328491*x^13 - 1632560051818863682*x^12 + 995239432812617607*x^11 + 1514536703235322325*x^10 - 898680726508319339*x^9 - 853943547109610470*x^8 + 424335389068850481*x^7 + 299252740781325936*x^6 - 91861302074915780*x^5 - 62873369979635247*x^4 + 4140602930707759*x^3 + 6001367177186495*x^2 + 926057723623551*x + 39010859256913, 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 - 360*x^39 + 745*x^38 + 56451*x^37 - 169135*x^36 - 5037002*x^35 + 19221892*x^34 + 281906305*x^33 - 1293434183*x^32 - 10337722909*x^31 + 55807085286*x^30 + 252777087235*x^29 - 1613716714335*x^28 - 4112155095553*x^27 + 32113136009541*x^26 + 43177039971301*x^25 - 446781341742247*x^24 - 262923207247335*x^23 + 4379546308331567*x^22 + 474335495077459*x^21 - 30267239554430324*x^20 + 5902126403913719*x^19 + 146509496594642143*x^18 - 56363172819685690*x^17 - 489681930758553989*x^16 + 246552011225708759*x^15 + 1105369786197101758*x^14 - 636716564586328491*x^13 - 1632560051818863682*x^12 + 995239432812617607*x^11 + 1514536703235322325*x^10 - 898680726508319339*x^9 - 853943547109610470*x^8 + 424335389068850481*x^7 + 299252740781325936*x^6 - 91861302074915780*x^5 - 62873369979635247*x^4 + 4140602930707759*x^3 + 6001367177186495*x^2 + 926057723623551*x + 39010859256913);
 
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 - 360*x^39 + 745*x^38 + 56451*x^37 - 169135*x^36 - 5037002*x^35 + 19221892*x^34 + 281906305*x^33 - 1293434183*x^32 - 10337722909*x^31 + 55807085286*x^30 + 252777087235*x^29 - 1613716714335*x^28 - 4112155095553*x^27 + 32113136009541*x^26 + 43177039971301*x^25 - 446781341742247*x^24 - 262923207247335*x^23 + 4379546308331567*x^22 + 474335495077459*x^21 - 30267239554430324*x^20 + 5902126403913719*x^19 + 146509496594642143*x^18 - 56363172819685690*x^17 - 489681930758553989*x^16 + 246552011225708759*x^15 + 1105369786197101758*x^14 - 636716564586328491*x^13 - 1632560051818863682*x^12 + 995239432812617607*x^11 + 1514536703235322325*x^10 - 898680726508319339*x^9 - 853943547109610470*x^8 + 424335389068850481*x^7 + 299252740781325936*x^6 - 91861302074915780*x^5 - 62873369979635247*x^4 + 4140602930707759*x^3 + 6001367177186495*x^2 + 926057723623551*x + 39010859256913);
 
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
\(739\) Copy content Toggle raw display Deg $41$$41$$1$$40$