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Properties

Label 40.0.316...000.2
Degree $40$
Signature $[0, 20]$
Discriminant $3.165\times 10^{70}$
Root discriminant \(57.88\)
Ramified primes $2,5,11$
Class number not computed
Class group not computed
Galois group $C_2\times C_{20}$ (as 40T2)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^40 + 11*x^38 + 77*x^36 + 440*x^34 + 2244*x^32 + 9273*x^30 + 33033*x^28 + 104786*x^26 + 294635*x^24 + 688611*x^22 + 1416910*x^20 + 2574154*x^18 + 3942422*x^16 + 4573316*x^14 + 4694437*x^12 + 4152720*x^10 + 2752508*x^8 + 863819*x^6 + 263538*x^4 + 73205*x^2 + 14641)
 
gp: K = bnfinit(y^40 + 11*y^38 + 77*y^36 + 440*y^34 + 2244*y^32 + 9273*y^30 + 33033*y^28 + 104786*y^26 + 294635*y^24 + 688611*y^22 + 1416910*y^20 + 2574154*y^18 + 3942422*y^16 + 4573316*y^14 + 4694437*y^12 + 4152720*y^10 + 2752508*y^8 + 863819*y^6 + 263538*y^4 + 73205*y^2 + 14641, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^40 + 11*x^38 + 77*x^36 + 440*x^34 + 2244*x^32 + 9273*x^30 + 33033*x^28 + 104786*x^26 + 294635*x^24 + 688611*x^22 + 1416910*x^20 + 2574154*x^18 + 3942422*x^16 + 4573316*x^14 + 4694437*x^12 + 4152720*x^10 + 2752508*x^8 + 863819*x^6 + 263538*x^4 + 73205*x^2 + 14641);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^40 + 11*x^38 + 77*x^36 + 440*x^34 + 2244*x^32 + 9273*x^30 + 33033*x^28 + 104786*x^26 + 294635*x^24 + 688611*x^22 + 1416910*x^20 + 2574154*x^18 + 3942422*x^16 + 4573316*x^14 + 4694437*x^12 + 4152720*x^10 + 2752508*x^8 + 863819*x^6 + 263538*x^4 + 73205*x^2 + 14641)
 

\( x^{40} + 11 x^{38} + 77 x^{36} + 440 x^{34} + 2244 x^{32} + 9273 x^{30} + 33033 x^{28} + 104786 x^{26} + \cdots + 14641 \) Copy content Toggle raw display

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

Invariants

Degree:  $40$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 20]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(31654584865659568778929513407372752241664000000000000000000000000000000\) \(\medspace = 2^{40}\cdot 5^{30}\cdot 11^{36}\) 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:  \(57.88\)
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:  $2\cdot 5^{3/4}11^{9/10}\approx 57.87765351369302$
Ramified primes:   \(2\), \(5\), \(11\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $40$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(220=2^{2}\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{220}(1,·)$, $\chi_{220}(131,·)$, $\chi_{220}(133,·)$, $\chi_{220}(7,·)$, $\chi_{220}(9,·)$, $\chi_{220}(139,·)$, $\chi_{220}(141,·)$, $\chi_{220}(43,·)$, $\chi_{220}(19,·)$, $\chi_{220}(151,·)$, $\chi_{220}(157,·)$, $\chi_{220}(37,·)$, $\chi_{220}(39,·)$, $\chi_{220}(169,·)$, $\chi_{220}(171,·)$, $\chi_{220}(49,·)$, $\chi_{220}(51,·)$, $\chi_{220}(53,·)$, $\chi_{220}(137,·)$, $\chi_{220}(63,·)$, $\chi_{220}(181,·)$, $\chi_{220}(69,·)$, $\chi_{220}(201,·)$, $\chi_{220}(183,·)$, $\chi_{220}(79,·)$, $\chi_{220}(81,·)$, $\chi_{220}(83,·)$, $\chi_{220}(213,·)$, $\chi_{220}(87,·)$, $\chi_{220}(89,·)$, $\chi_{220}(219,·)$, $\chi_{220}(93,·)$, $\chi_{220}(107,·)$, $\chi_{220}(97,·)$, $\chi_{220}(167,·)$, $\chi_{220}(177,·)$, $\chi_{220}(113,·)$, $\chi_{220}(211,·)$, $\chi_{220}(123,·)$, $\chi_{220}(127,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{524288}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{11}a^{10}$, $\frac{1}{11}a^{11}$, $\frac{1}{11}a^{12}$, $\frac{1}{11}a^{13}$, $\frac{1}{11}a^{14}$, $\frac{1}{11}a^{15}$, $\frac{1}{11}a^{16}$, $\frac{1}{11}a^{17}$, $\frac{1}{121}a^{18}-\frac{1}{11}a^{8}$, $\frac{1}{121}a^{19}-\frac{1}{11}a^{9}$, $\frac{1}{121}a^{20}$, $\frac{1}{121}a^{21}$, $\frac{1}{121}a^{22}$, $\frac{1}{121}a^{23}$, $\frac{1}{121}a^{24}$, $\frac{1}{121}a^{25}$, $\frac{1}{1331}a^{26}-\frac{2}{121}a^{16}+\frac{1}{11}a^{6}$, $\frac{1}{1331}a^{27}-\frac{2}{121}a^{17}+\frac{1}{11}a^{7}$, $\frac{1}{1331}a^{28}-\frac{1}{11}a^{8}$, $\frac{1}{1331}a^{29}-\frac{1}{11}a^{9}$, $\frac{1}{1331}a^{30}$, $\frac{1}{1331}a^{31}$, $\frac{1}{1331}a^{32}$, $\frac{1}{1331}a^{33}$, $\frac{1}{774464896196321}a^{34}-\frac{3190709248}{70405899654211}a^{32}-\frac{1618603772}{70405899654211}a^{30}+\frac{16792576465}{70405899654211}a^{28}+\frac{10252294982}{70405899654211}a^{26}-\frac{17977069773}{70405899654211}a^{24}+\frac{2237044627}{6400536332201}a^{22}+\frac{14502562927}{6400536332201}a^{20}+\frac{598530065}{581866939291}a^{18}+\frac{288566388494}{6400536332201}a^{16}-\frac{286816002206}{6400536332201}a^{14}-\frac{18532336046}{581866939291}a^{12}+\frac{3055556333}{581866939291}a^{10}-\frac{161664475384}{581866939291}a^{8}+\frac{128503633979}{581866939291}a^{6}+\frac{127185291727}{581866939291}a^{4}-\frac{11798061945}{52896994481}a^{2}+\frac{7493800533}{52896994481}$, $\frac{1}{774464896196321}a^{35}-\frac{3190709248}{70405899654211}a^{33}-\frac{1618603772}{70405899654211}a^{31}+\frac{16792576465}{70405899654211}a^{29}+\frac{10252294982}{70405899654211}a^{27}-\frac{17977069773}{70405899654211}a^{25}+\frac{2237044627}{6400536332201}a^{23}+\frac{14502562927}{6400536332201}a^{21}+\frac{598530065}{581866939291}a^{19}+\frac{288566388494}{6400536332201}a^{17}-\frac{286816002206}{6400536332201}a^{15}-\frac{18532336046}{581866939291}a^{13}+\frac{3055556333}{581866939291}a^{11}-\frac{161664475384}{581866939291}a^{9}+\frac{128503633979}{581866939291}a^{7}+\frac{127185291727}{581866939291}a^{5}-\frac{11798061945}{52896994481}a^{3}+\frac{7493800533}{52896994481}a$, $\frac{1}{774464896196321}a^{36}+\frac{16008996310}{70405899654211}a^{32}-\frac{4958158505}{70405899654211}a^{30}-\frac{18377658466}{70405899654211}a^{28}+\frac{7464166299}{70405899654211}a^{26}-\frac{9174883319}{6400536332201}a^{24}+\frac{18298437958}{6400536332201}a^{22}+\frac{15603799148}{6400536332201}a^{20}-\frac{1848375322}{581866939291}a^{18}-\frac{68172970267}{6400536332201}a^{16}-\frac{10106953870}{581866939291}a^{14}-\frac{2098182813}{52896994481}a^{12}+\frac{13017979267}{581866939291}a^{10}-\frac{198233377945}{581866939291}a^{8}-\frac{185892891824}{581866939291}a^{6}+\frac{14568552221}{52896994481}a^{4}+\frac{20239880052}{52896994481}a^{2}+\frac{23106908469}{52896994481}$, $\frac{1}{774464896196321}a^{37}+\frac{16008996310}{70405899654211}a^{33}-\frac{4958158505}{70405899654211}a^{31}-\frac{18377658466}{70405899654211}a^{29}+\frac{7464166299}{70405899654211}a^{27}-\frac{9174883319}{6400536332201}a^{25}+\frac{18298437958}{6400536332201}a^{23}+\frac{15603799148}{6400536332201}a^{21}-\frac{1848375322}{581866939291}a^{19}-\frac{68172970267}{6400536332201}a^{17}-\frac{10106953870}{581866939291}a^{15}-\frac{2098182813}{52896994481}a^{13}+\frac{13017979267}{581866939291}a^{11}-\frac{198233377945}{581866939291}a^{9}-\frac{185892891824}{581866939291}a^{7}+\frac{14568552221}{52896994481}a^{5}+\frac{20239880052}{52896994481}a^{3}+\frac{23106908469}{52896994481}a$, $\frac{1}{774464896196321}a^{38}-\frac{13287510312}{70405899654211}a^{32}-\frac{4106145961}{70405899654211}a^{30}-\frac{14089504255}{70405899654211}a^{28}+\frac{13731987822}{70405899654211}a^{26}+\frac{9374904263}{6400536332201}a^{24}-\frac{25218143065}{6400536332201}a^{22}+\frac{5056542771}{6400536332201}a^{20}-\frac{1384939810}{581866939291}a^{18}-\frac{168663881843}{6400536332201}a^{16}-\frac{21165014350}{581866939291}a^{14}+\frac{22839883305}{581866939291}a^{12}+\frac{3463587132}{581866939291}a^{10}-\frac{239027494430}{581866939291}a^{8}-\frac{83208508182}{581866939291}a^{6}+\frac{2968650789}{52896994481}a^{4}-\frac{11182737850}{52896994481}a^{2}-\frac{21109201013}{52896994481}$, $\frac{1}{774464896196321}a^{39}-\frac{13287510312}{70405899654211}a^{33}-\frac{4106145961}{70405899654211}a^{31}-\frac{14089504255}{70405899654211}a^{29}+\frac{13731987822}{70405899654211}a^{27}+\frac{9374904263}{6400536332201}a^{25}-\frac{25218143065}{6400536332201}a^{23}+\frac{5056542771}{6400536332201}a^{21}-\frac{1384939810}{581866939291}a^{19}-\frac{168663881843}{6400536332201}a^{17}-\frac{21165014350}{581866939291}a^{15}+\frac{22839883305}{581866939291}a^{13}+\frac{3463587132}{581866939291}a^{11}-\frac{239027494430}{581866939291}a^{9}-\frac{83208508182}{581866939291}a^{7}+\frac{2968650789}{52896994481}a^{5}-\frac{11182737850}{52896994481}a^{3}-\frac{21109201013}{52896994481}a$ Copy content Toggle raw display

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

Monogenic:  Not computed
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:  $19$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -\frac{1555976772}{774464896196321} a^{38} - \frac{14392785141}{774464896196321} a^{36} - \frac{91802629548}{774464896196321} a^{34} - \frac{4098794166}{6400536332201} a^{32} - \frac{221726690010}{70405899654211} a^{30} - \frac{831280590441}{70405899654211} a^{28} - \frac{2758357822563}{70405899654211} a^{26} - \frac{8205443507142}{70405899654211} a^{24} - \frac{175640306498}{581866939291} a^{22} - \frac{3802807230768}{6400536332201} a^{20} - \frac{7123261662216}{6400536332201} a^{18} - \frac{11337624749178}{6400536332201} a^{16} - \frac{13330441999917}{6400536332201} a^{14} - \frac{623510550556}{581866939291} a^{12} - \frac{1140530973876}{581866939291} a^{10} - \frac{781489333737}{581866939291} a^{8} - \frac{245455335783}{581866939291} a^{6} - \frac{75075879249}{581866939291} a^{4} - \frac{168705162564}{52896994481} a^{2} - \frac{388994193}{52896994481} \)  (order $10$) 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^40 + 11*x^38 + 77*x^36 + 440*x^34 + 2244*x^32 + 9273*x^30 + 33033*x^28 + 104786*x^26 + 294635*x^24 + 688611*x^22 + 1416910*x^20 + 2574154*x^18 + 3942422*x^16 + 4573316*x^14 + 4694437*x^12 + 4152720*x^10 + 2752508*x^8 + 863819*x^6 + 263538*x^4 + 73205*x^2 + 14641)
 
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^40 + 11*x^38 + 77*x^36 + 440*x^34 + 2244*x^32 + 9273*x^30 + 33033*x^28 + 104786*x^26 + 294635*x^24 + 688611*x^22 + 1416910*x^20 + 2574154*x^18 + 3942422*x^16 + 4573316*x^14 + 4694437*x^12 + 4152720*x^10 + 2752508*x^8 + 863819*x^6 + 263538*x^4 + 73205*x^2 + 14641, 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^40 + 11*x^38 + 77*x^36 + 440*x^34 + 2244*x^32 + 9273*x^30 + 33033*x^28 + 104786*x^26 + 294635*x^24 + 688611*x^22 + 1416910*x^20 + 2574154*x^18 + 3942422*x^16 + 4573316*x^14 + 4694437*x^12 + 4152720*x^10 + 2752508*x^8 + 863819*x^6 + 263538*x^4 + 73205*x^2 + 14641);
 
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^40 + 11*x^38 + 77*x^36 + 440*x^34 + 2244*x^32 + 9273*x^30 + 33033*x^28 + 104786*x^26 + 294635*x^24 + 688611*x^22 + 1416910*x^20 + 2574154*x^18 + 3942422*x^16 + 4573316*x^14 + 4694437*x^12 + 4152720*x^10 + 2752508*x^8 + 863819*x^6 + 263538*x^4 + 73205*x^2 + 14641);
 
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

$C_2\times C_{20}$ (as 40T2):

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)
 
An abelian group of order 40
The 40 conjugacy class representatives for $C_2\times C_{20}$
Character table for $C_2\times C_{20}$ is not computed

Intermediate fields

\(\Q(\sqrt{11}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{55}) \), \(\Q(\sqrt{5}, \sqrt{11})\), \(\Q(\zeta_{5})\), 4.0.242000.2, \(\Q(\zeta_{11})^+\), 8.0.58564000000.3, \(\Q(\zeta_{44})^+\), 10.10.669871503125.1, 10.10.7545432611200000.1, 20.20.56933553290160450365440000000000.1, 20.0.1402274470934209014892578125.1, 20.0.177917354031751407392000000000000000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

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 R $20^{2}$ R $20^{2}$ R $20^{2}$ $20^{2}$ ${\href{/padicField/19.10.0.1}{10} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{10}$ ${\href{/padicField/29.10.0.1}{10} }^{4}$ ${\href{/padicField/31.10.0.1}{10} }^{4}$ $20^{2}$ ${\href{/padicField/41.10.0.1}{10} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{10}$ $20^{2}$ $20^{2}$ ${\href{/padicField/59.10.0.1}{10} }^{4}$

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
\(2\) Copy content Toggle raw display Deg $40$$2$$20$$40$
\(5\) Copy content Toggle raw display Deg $20$$4$$5$$15$
Deg $20$$4$$5$$15$
\(11\) Copy content Toggle raw display 11.10.9.1$x^{10} + 110$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} + 110$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} + 110$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} + 110$$10$$1$$9$$C_{10}$$[\ ]_{10}$