Properties

Label 40.0.287...000.5
Degree $40$
Signature $(0, 20)$
Discriminant $2.879\times 10^{98}$
Root discriminant \(289.39\)
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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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^40 - 332849*x^30 + 77400539601*x^20 - 8633245841018249*x^10 + 672749994932560009201)
 
Copy content gp:K = bnfinit(y^40 - 332849*y^30 + 77400539601*y^20 - 8633245841018249*y^10 + 672749994932560009201, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^40 - 332849*x^30 + 77400539601*x^20 - 8633245841018249*x^10 + 672749994932560009201);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^40 - 332849*x^30 + 77400539601*x^20 - 8633245841018249*x^10 + 672749994932560009201)
 

\( x^{40} - 332849x^{30} + 77400539601x^{20} - 8633245841018249x^{10} + 672749994932560009201 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $40$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $(0, 20)$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(287\!\cdots\!000\) \(\medspace = 2^{40}\cdot 5^{70}\cdot 11^{36}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(289.39\)
Copy content comment:Root discriminant
 
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $2\cdot 5^{7/4}11^{9/10}\approx 289.3882675684651$
Ramified primes:   \(2\), \(5\), \(11\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant(OK))
 
Discriminant root field:  \(\Q\)
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:   $C_2\times C_{20}$
Copy content comment:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1100=2^{2}\cdot 5^{2}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{1100}(1,·)$, $\chi_{1100}(811,·)$, $\chi_{1100}(773,·)$, $\chi_{1100}(1047,·)$, $\chi_{1100}(269,·)$, $\chi_{1100}(533,·)$, $\chi_{1100}(919,·)$, $\chi_{1100}(793,·)$, $\chi_{1100}(1057,·)$, $\chi_{1100}(289,·)$, $\chi_{1100}(359,·)$, $\chi_{1100}(37,·)$, $\chi_{1100}(1063,·)$, $\chi_{1100}(43,·)$, $\chi_{1100}(1003,·)$, $\chi_{1100}(307,·)$, $\chi_{1100}(53,·)$, $\chi_{1100}(567,·)$, $\chi_{1100}(831,·)$, $\chi_{1100}(741,·)$, $\chi_{1100}(181,·)$, $\chi_{1100}(327,·)$, $\chi_{1100}(1099,·)$, $\chi_{1100}(79,·)$, $\chi_{1100}(83,·)$, $\chi_{1100}(609,·)$, $\chi_{1100}(213,·)$, $\chi_{1100}(377,·)$, $\chi_{1100}(861,·)$, $\chi_{1100}(351,·)$, $\chi_{1100}(97,·)$, $\chi_{1100}(229,·)$, $\chi_{1100}(871,·)$, $\chi_{1100}(491,·)$, $\chi_{1100}(749,·)$, $\chi_{1100}(239,·)$, $\chi_{1100}(723,·)$, $\chi_{1100}(887,·)$, $\chi_{1100}(1017,·)$, $\chi_{1100}(1021,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{524288}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{33}a^{10}-\frac{1}{3}$, $\frac{1}{33}a^{11}-\frac{1}{3}a$, $\frac{1}{33}a^{12}-\frac{1}{3}a^{2}$, $\frac{1}{33}a^{13}-\frac{1}{3}a^{3}$, $\frac{1}{363}a^{14}+\frac{2}{33}a^{4}$, $\frac{1}{363}a^{15}+\frac{2}{33}a^{5}$, $\frac{1}{363}a^{16}+\frac{2}{33}a^{6}$, $\frac{1}{3993}a^{17}-\frac{130}{363}a^{7}$, $\frac{1}{3993}a^{18}-\frac{130}{363}a^{8}$, $\frac{1}{3993}a^{19}-\frac{130}{363}a^{9}$, $\frac{1}{55474749}a^{20}-\frac{29654}{5043159}a^{10}-\frac{428}{3789}$, $\frac{1}{55474749}a^{21}-\frac{29654}{5043159}a^{11}-\frac{428}{3789}a$, $\frac{1}{610222239}a^{22}-\frac{488123}{55474749}a^{12}+\frac{10939}{41679}a^{2}$, $\frac{1}{6712444629}a^{23}+\frac{6236089}{610222239}a^{13}-\frac{2954}{458469}a^{3}$, $\frac{1}{6712444629}a^{24}-\frac{488123}{610222239}a^{14}-\frac{114098}{458469}a^{4}$, $\frac{1}{73836890919}a^{25}+\frac{6236089}{6712444629}a^{15}+\frac{913984}{5043159}a^{5}$, $\frac{1}{812205800109}a^{26}+\frac{98694004}{73836890919}a^{16}-\frac{22773581}{55474749}a^{6}$, $\frac{1}{812205800109}a^{27}+\frac{6236089}{73836890919}a^{17}+\frac{21086620}{55474749}a^{7}$, $\frac{1}{8934263801199}a^{28}+\frac{98694004}{812205800109}a^{18}-\frac{189197828}{610222239}a^{8}$, $\frac{1}{98276901813189}a^{29}+\frac{912323656}{8934263801199}a^{19}+\frac{157099090}{6712444629}a^{9}$, $\frac{1}{79\cdots 83}a^{30}+\frac{54860509921}{24\cdots 51}a^{20}+\frac{28250289536679}{20\cdots 69}a^{10}+\frac{77030738}{2786722353}$, $\frac{1}{79\cdots 83}a^{31}+\frac{54860509921}{24\cdots 51}a^{21}+\frac{28250289536679}{20\cdots 69}a^{11}+\frac{77030738}{2786722353}a$, $\frac{1}{87\cdots 13}a^{32}+\frac{54860509921}{26\cdots 61}a^{22}-\frac{646634114857079}{66\cdots 77}a^{12}-\frac{10140951223}{30653945883}a^{2}$, $\frac{1}{96\cdots 43}a^{33}+\frac{54860509921}{29\cdots 71}a^{23}-\frac{10\cdots 24}{73\cdots 47}a^{13}-\frac{112320770833}{337193404713}a^{3}$, $\frac{1}{10\cdots 73}a^{34}-\frac{1248081859676}{32\cdots 81}a^{24}-\frac{10\cdots 56}{80\cdots 17}a^{14}+\frac{118992629483}{3709127451843}a^{4}$, $\frac{1}{11\cdots 03}a^{35}-\frac{1248081859676}{35\cdots 91}a^{25}+\frac{11\cdots 03}{88\cdots 87}a^{15}-\frac{18201849026590}{40800401970273}a^{5}$, $\frac{1}{12\cdots 33}a^{36}-\frac{1248081859676}{38\cdots 01}a^{26}-\frac{32\cdots 31}{32\cdots 19}a^{16}-\frac{28092855564838}{448804421673003}a^{6}$, $\frac{1}{14\cdots 63}a^{37}+\frac{17861739561080}{42\cdots 11}a^{27}-\frac{33\cdots 69}{32\cdots 81}a^{17}-\frac{819468972983230}{49\cdots 33}a^{7}$, $\frac{1}{15\cdots 93}a^{38}+\frac{17861739561080}{46\cdots 21}a^{28}-\frac{43\cdots 54}{35\cdots 91}a^{18}+\frac{30\cdots 87}{54\cdots 63}a^{8}$, $\frac{1}{17\cdots 23}a^{39}+\frac{17861739561080}{51\cdots 31}a^{29}-\frac{39\cdots 02}{38\cdots 01}a^{19}+\frac{13\cdots 70}{59\cdots 93}a^{9}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  not computed
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  not computed
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 
Relative class number:   data not computed

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $19$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( \frac{14440}{795084410063446867683} a^{30} - \frac{3197455736}{24093466971619602051} a^{20} + \frac{120946078475}{6033926113603707} a^{10} - \frac{7837033360}{2786722353} \)  (order $10$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:  not computed
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  not computed
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{0}\cdot(2\pi)^{20}\cdot R \cdot h}{10\cdot\sqrt{287896772221388971765277468942920243004336953163146972656250000000000000000000000000000000000000000}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^40 - 332849*x^30 + 77400539601*x^20 - 8633245841018249*x^10 + 672749994932560009201) 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))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^40 - 332849*x^30 + 77400539601*x^20 - 8633245841018249*x^10 + 672749994932560009201, 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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^40 - 332849*x^30 + 77400539601*x^20 - 8633245841018249*x^10 + 672749994932560009201); 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)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = polynomial_ring(QQ); K, a = number_field(x^40 - 332849*x^30 + 77400539601*x^20 - 8633245841018249*x^10 + 672749994932560009201); 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):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
An abelian group of order 40
The 40 conjugacy class representatives for $C_2\times C_{20}$
Character table for $C_2\times C_{20}$

Intermediate fields

\(\Q(\sqrt{11}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{55}) \), \(\Q(\sqrt{5}, \sqrt{11})\), \(\Q(\zeta_{5})\), \(\Q(\sqrt{-110 +22 \sqrt{5}})\), 5.5.5719140625.2, 8.0.58564000000.3, 10.10.368429326718750000000000.1, 10.10.163542847442626953125.1, 10.10.1842146633593750000000000.4, 20.20.3393504219660785816040039062500000000000000000000.4, 20.0.133731314748211766709573566913604736328125.4, 20.0.16967521098303929080200195312500000000000000000000.3

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

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content 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 ${\href{/padicField/3.4.0.1}{4} }^{10}$ R $20^{2}$ R ${\href{/padicField/13.4.0.1}{4} }^{10}$ $20^{2}$ ${\href{/padicField/19.10.0.1}{10} }^{4}$ $20^{2}$ ${\href{/padicField/29.10.0.1}{10} }^{4}$ ${\href{/padicField/31.10.0.1}{10} }^{4}$ $20^{2}$ ${\href{/padicField/41.10.0.1}{10} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{10}$ ${\href{/padicField/47.4.0.1}{4} }^{10}$ $20^{2}$ ${\href{/padicField/59.10.0.1}{10} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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 5.1.20.35a1.4$x^{20} + 20 x^{16} + 80$$20$$1$$35$20T1$$[2]_{4}$$
5.1.20.35a1.4$x^{20} + 20 x^{16} + 80$$20$$1$$35$20T1$$[2]_{4}$$
\(11\) Copy content Toggle raw display 11.1.10.9a1.2$x^{10} + 22$$10$$1$$9$$C_{10}$$$[\ ]_{10}$$
11.1.10.9a1.2$x^{10} + 22$$10$$1$$9$$C_{10}$$$[\ ]_{10}$$
11.1.10.9a1.2$x^{10} + 22$$10$$1$$9$$C_{10}$$$[\ ]_{10}$$
11.1.10.9a1.2$x^{10} + 22$$10$$1$$9$$C_{10}$$$[\ ]_{10}$$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)