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Properties

Label 40.0.324...000.1
Degree $40$
Signature $[0, 20]$
Discriminant $3.247\times 10^{70}$
Root discriminant \(57.91\)
Ramified primes $2,3,5$
Class number $237710$ (GRH)
Class group [237710] (GRH)
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 + 40*x^38 + 740*x^36 + 8400*x^34 + 65450*x^32 + 371009*x^30 + 1582270*x^28 + 5178645*x^26 + 13151125*x^24 + 26030250*x^22 + 40123776*x^20 + 47888645*x^18 + 43779420*x^16 + 30152050*x^14 + 15277900*x^12 + 5507149*x^10 + 1345115*x^8 + 206465*x^6 + 17450*x^4 + 600*x^2 + 1)
 
gp: K = bnfinit(y^40 + 40*y^38 + 740*y^36 + 8400*y^34 + 65450*y^32 + 371009*y^30 + 1582270*y^28 + 5178645*y^26 + 13151125*y^24 + 26030250*y^22 + 40123776*y^20 + 47888645*y^18 + 43779420*y^16 + 30152050*y^14 + 15277900*y^12 + 5507149*y^10 + 1345115*y^8 + 206465*y^6 + 17450*y^4 + 600*y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^40 + 40*x^38 + 740*x^36 + 8400*x^34 + 65450*x^32 + 371009*x^30 + 1582270*x^28 + 5178645*x^26 + 13151125*x^24 + 26030250*x^22 + 40123776*x^20 + 47888645*x^18 + 43779420*x^16 + 30152050*x^14 + 15277900*x^12 + 5507149*x^10 + 1345115*x^8 + 206465*x^6 + 17450*x^4 + 600*x^2 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^40 + 40*x^38 + 740*x^36 + 8400*x^34 + 65450*x^32 + 371009*x^30 + 1582270*x^28 + 5178645*x^26 + 13151125*x^24 + 26030250*x^22 + 40123776*x^20 + 47888645*x^18 + 43779420*x^16 + 30152050*x^14 + 15277900*x^12 + 5507149*x^10 + 1345115*x^8 + 206465*x^6 + 17450*x^4 + 600*x^2 + 1)
 

\( x^{40} + 40 x^{38} + 740 x^{36} + 8400 x^{34} + 65450 x^{32} + 371009 x^{30} + 1582270 x^{28} + 5178645 x^{26} + \cdots + 1 \) 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:   \(32473210254684090614318847656250000000000000000000000000000000000000000\) \(\medspace = 2^{40}\cdot 3^{20}\cdot 5^{70}\) 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:  \(57.91\)
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:  $2\cdot 3^{1/2}5^{7/4}\approx 57.91460926441345$
Ramified primes:   \(2\), \(3\), \(5\) 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:  \(300=2^{2}\cdot 3\cdot 5^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{300}(1,·)$, $\chi_{300}(259,·)$, $\chi_{300}(263,·)$, $\chi_{300}(137,·)$, $\chi_{300}(139,·)$, $\chi_{300}(143,·)$, $\chi_{300}(17,·)$, $\chi_{300}(19,·)$, $\chi_{300}(23,·)$, $\chi_{300}(287,·)$, $\chi_{300}(289,·)$, $\chi_{300}(91,·)$, $\chi_{300}(293,·)$, $\chi_{300}(167,·)$, $\chi_{300}(169,·)$, $\chi_{300}(257,·)$, $\chi_{300}(173,·)$, $\chi_{300}(47,·)$, $\chi_{300}(49,·)$, $\chi_{300}(53,·)$, $\chi_{300}(31,·)$, $\chi_{300}(151,·)$, $\chi_{300}(61,·)$, $\chi_{300}(181,·)$, $\chi_{300}(197,·)$, $\chi_{300}(199,·)$, $\chi_{300}(203,·)$, $\chi_{300}(77,·)$, $\chi_{300}(79,·)$, $\chi_{300}(83,·)$, $\chi_{300}(271,·)$, $\chi_{300}(227,·)$, $\chi_{300}(229,·)$, $\chi_{300}(233,·)$, $\chi_{300}(107,·)$, $\chi_{300}(109,·)$, $\chi_{300}(113,·)$, $\chi_{300}(211,·)$, $\chi_{300}(241,·)$, $\chi_{300}(121,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{524288}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$ 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

$C_{237710}$, which has order $237710$ (assuming GRH)

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:   \( a^{25} + 25 a^{23} + 275 a^{21} + 1750 a^{19} + 7125 a^{17} + 19380 a^{15} + 35700 a^{13} + 44200 a^{11} + 35750 a^{9} + 17875 a^{7} + 5005 a^{5} + 650 a^{3} + 25 a \)  (order $4$) 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:   $a^{30}+30a^{28}+405a^{26}+3250a^{24}+17250a^{22}+63756a^{20}+168245a^{18}+319770a^{16}+436050a^{14}+419900a^{12}+277134a^{10}+119340a^{8}+30940a^{6}+4200a^{4}+225a^{2}+1$, $a^{38}+38a^{36}+665a^{34}+7107a^{32}+51864a^{30}+273761a^{28}+1080163a^{26}+3247100a^{24}+7511961a^{22}+13418867a^{20}+18466195a^{18}+19425220a^{16}+15407100a^{14}+9021389a^{12}+3778918a^{10}+1079649a^{8}+194837a^{6}+19330a^{4}+760a^{2}$, $a^{18}+18a^{16}+135a^{14}+546a^{12}+1287a^{10}+1782a^{8}+1386a^{6}+540a^{4}+81a^{2}+2$, $a^{34}+34a^{32}+527a^{30}+4930a^{28}+31059a^{26}+139230a^{24}+457470a^{22}+1118260a^{20}+2042975a^{18}+2778445a^{16}+2778430a^{14}+1998620a^{12}+999010a^{10}+328800a^{8}+65220a^{6}+6600a^{4}+225a^{2}$, $a^{36}+37a^{34}+628a^{32}+6479a^{30}+45385a^{28}+228375a^{26}+851761a^{24}+2395014a^{22}+5114647a^{20}+8293595a^{18}+10138969a^{16}+9211774a^{14}+6079606a^{12}+2817178a^{10}+871365a^{8}+166251a^{6}+17031a^{4}+684a^{2}+2$, $a^{38}+38a^{36}+665a^{34}+7107a^{32}+51864a^{30}+273760a^{28}+1080135a^{26}+3246750a^{24}+7509386a^{22}+13406492a^{20}+18425439a^{18}+19331362a^{16}+15255665a^{14}+8852494a^{12}+3652518a^{10}+1019291a^{8}+177898a^{6}+16920a^{4}+640a^{2}$, $a^{28}+28a^{26}+350a^{24}+2575a^{22}+12375a^{20}+40755a^{18}+93840a^{16}+151300a^{14}+168350a^{12}+125125a^{10}+58630a^{8}+15665a^{6}+1975a^{4}+75a^{2}$, $a^{24}+24a^{22}+252a^{20}+1520a^{18}+5814a^{16}+14688a^{14}+24752a^{12}+27456a^{10}+19305a^{8}+8008a^{6}+1716a^{4}+144a^{2}+2$, $a^{36}+36a^{34}+594a^{32}+5952a^{30}+40455a^{28}+197316a^{26}+712530a^{24}+1937520a^{22}+3996135a^{20}+6249100a^{18}+7354710a^{16}+6418656a^{14}+4056234a^{12}+1790712a^{10}+523260a^{8}+93024a^{6}+8721a^{4}+324a^{2}+2$, $a^{8}+8a^{6}+20a^{4}+16a^{2}+2$, $a^{38}+38a^{36}+665a^{34}+7106a^{32}+51832a^{30}+273296a^{28}+1076103a^{26}+3223350a^{24}+7413705a^{22}+13123110a^{20}+17809935a^{18}+18349630a^{16}+14115100a^{14}+7904455a^{12}+3105310a^{10}+810030a^{8}+128765a^{6}+10725a^{4}+325a^{2}$, $a^{26}+25a^{24}+275a^{22}+1750a^{20}+7125a^{18}+19380a^{16}+35700a^{14}+44200a^{12}+35750a^{10}+17875a^{8}+5005a^{6}+650a^{4}+25a^{2}$, $a^{36}+37a^{34}+628a^{32}+6479a^{30}+45385a^{28}+228375a^{26}+851761a^{24}+2395014a^{22}+5114647a^{20}+8293595a^{18}+10138969a^{16}+9211774a^{14}+6079606a^{12}+2817178a^{10}+871365a^{8}+166251a^{6}+17030a^{4}+680a^{2}$, $a^{36}+37a^{34}+628a^{32}+6479a^{30}+45385a^{28}+228376a^{26}+851786a^{24}+2395289a^{22}+5116397a^{20}+8300720a^{18}+10158350a^{16}+9247489a^{14}+6123896a^{12}+2853203a^{10}+889690a^{8}+171635a^{6}+17826a^{4}+729a^{2}+2$, $a^{36}+36a^{34}+594a^{32}+5952a^{30}+40455a^{28}+197316a^{26}+712530a^{24}+1937520a^{22}+3996135a^{20}+6249100a^{18}+7354710a^{16}+6418655a^{14}+4056220a^{12}+1790635a^{10}+523050a^{8}+92730a^{6}+8525a^{4}+275a^{2}$, $a^{32}+32a^{30}+464a^{28}+4032a^{26}+23400a^{24}+95680a^{22}+283360a^{20}+615295a^{18}+980610a^{16}+1136825a^{14}+940030a^{12}+536185a^{10}+199770a^{8}+44310a^{6}+4900a^{4}+175a^{2}$, $a^{2}+2$, $a^{38}+38a^{36}+665a^{34}+7106a^{32}+51832a^{30}+273296a^{28}+1076103a^{26}+3223350a^{24}+7413706a^{22}+13123132a^{20}+17810143a^{18}+18350734a^{16}+14118705a^{14}+7911918a^{12}+3115046a^{10}+817740a^{8}+132210a^{6}+11500a^{4}+401a^{2}+2$, $a^{16}+16a^{14}+104a^{12}+352a^{10}+660a^{8}+672a^{6}+336a^{4}+64a^{2}+2$ Copy content Toggle raw display (assuming GRH)
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:  \( 64293715230028.38 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{20}\cdot 64293715230028.38 \cdot 237710}{4\cdot\sqrt{32473210254684090614318847656250000000000000000000000000000000000000000}}\cr\approx \mathstrut & 0.194980301653947 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^40 + 40*x^38 + 740*x^36 + 8400*x^34 + 65450*x^32 + 371009*x^30 + 1582270*x^28 + 5178645*x^26 + 13151125*x^24 + 26030250*x^22 + 40123776*x^20 + 47888645*x^18 + 43779420*x^16 + 30152050*x^14 + 15277900*x^12 + 5507149*x^10 + 1345115*x^8 + 206465*x^6 + 17450*x^4 + 600*x^2 + 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^40 + 40*x^38 + 740*x^36 + 8400*x^34 + 65450*x^32 + 371009*x^30 + 1582270*x^28 + 5178645*x^26 + 13151125*x^24 + 26030250*x^22 + 40123776*x^20 + 47888645*x^18 + 43779420*x^16 + 30152050*x^14 + 15277900*x^12 + 5507149*x^10 + 1345115*x^8 + 206465*x^6 + 17450*x^4 + 600*x^2 + 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^40 + 40*x^38 + 740*x^36 + 8400*x^34 + 65450*x^32 + 371009*x^30 + 1582270*x^28 + 5178645*x^26 + 13151125*x^24 + 26030250*x^22 + 40123776*x^20 + 47888645*x^18 + 43779420*x^16 + 30152050*x^14 + 15277900*x^12 + 5507149*x^10 + 1345115*x^8 + 206465*x^6 + 17450*x^4 + 600*x^2 + 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^40 + 40*x^38 + 740*x^36 + 8400*x^34 + 65450*x^32 + 371009*x^30 + 1582270*x^28 + 5178645*x^26 + 13151125*x^24 + 26030250*x^22 + 40123776*x^20 + 47888645*x^18 + 43779420*x^16 + 30152050*x^14 + 15277900*x^12 + 5507149*x^10 + 1345115*x^8 + 206465*x^6 + 17450*x^4 + 600*x^2 + 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

$C_2\times C_{20}$ (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(i, \sqrt{5})\), 4.0.18000.1, \(\Q(\zeta_{15})^+\), 5.5.390625.1, 8.0.324000000.1, 10.0.156250000000000.1, \(\Q(\zeta_{25})^+\), 10.0.781250000000000.1, 20.0.610351562500000000000000000000.1, 20.0.180203247070312500000000000000000000.1, \(\Q(\zeta_{75})^+\)

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 R R ${\href{/padicField/7.4.0.1}{4} }^{10}$ ${\href{/padicField/11.10.0.1}{10} }^{4}$ $20^{2}$ $20^{2}$ ${\href{/padicField/19.10.0.1}{10} }^{4}$ $20^{2}$ ${\href{/padicField/29.5.0.1}{5} }^{8}$ ${\href{/padicField/31.10.0.1}{10} }^{4}$ $20^{2}$ ${\href{/padicField/41.10.0.1}{10} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{10}$ $20^{2}$ $20^{2}$ ${\href{/padicField/59.10.0.1}{10} }^{4}$

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$
\(3\) Copy content Toggle raw display 3.20.10.2$x^{20} + 162 x^{12} - 486 x^{10} + 1458 x^{8} - 19683 x^{2} + 118098$$2$$10$$10$20T1$[\ ]_{2}^{10}$
3.20.10.2$x^{20} + 162 x^{12} - 486 x^{10} + 1458 x^{8} - 19683 x^{2} + 118098$$2$$10$$10$20T1$[\ ]_{2}^{10}$
\(5\) Copy content Toggle raw display Deg $20$$20$$1$$35$
Deg $20$$20$$1$$35$