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Properties

Label 40.0.569...625.1
Degree $40$
Signature $[0, 20]$
Discriminant $5.698\times 10^{69}$
Root discriminant \(55.45\)
Ramified primes $5,11$
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 - 31*x^35 + 718*x^30 - 14725*x^25 + 282001*x^20 - 3578175*x^15 + 42397182*x^10 - 444816117*x^5 + 3486784401)
 
gp: K = bnfinit(y^40 - 31*y^35 + 718*y^30 - 14725*y^25 + 282001*y^20 - 3578175*y^15 + 42397182*y^10 - 444816117*y^5 + 3486784401, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^40 - 31*x^35 + 718*x^30 - 14725*x^25 + 282001*x^20 - 3578175*x^15 + 42397182*x^10 - 444816117*x^5 + 3486784401);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^40 - 31*x^35 + 718*x^30 - 14725*x^25 + 282001*x^20 - 3578175*x^15 + 42397182*x^10 - 444816117*x^5 + 3486784401)
 

\( x^{40} - 31 x^{35} + 718 x^{30} - 14725 x^{25} + 282001 x^{20} - 3578175 x^{15} + 42397182 x^{10} + \cdots + 3486784401 \) 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:   \(5698414109730419226674323998106663768936641645268537104129791259765625\) \(\medspace = 5^{70}\cdot 11^{20}\) 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:  \(55.45\)
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:  $5^{7/4}11^{1/2}\approx 55.449016844865795$
Ramified primes:   \(5\), \(11\) 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:  \(275=5^{2}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{275}(1,·)$, $\chi_{275}(131,·)$, $\chi_{275}(133,·)$, $\chi_{275}(263,·)$, $\chi_{275}(12,·)$, $\chi_{275}(142,·)$, $\chi_{275}(144,·)$, $\chi_{275}(274,·)$, $\chi_{275}(21,·)$, $\chi_{275}(23,·)$, $\chi_{275}(153,·)$, $\chi_{275}(32,·)$, $\chi_{275}(34,·)$, $\chi_{275}(164,·)$, $\chi_{275}(166,·)$, $\chi_{275}(43,·)$, $\chi_{275}(177,·)$, $\chi_{275}(54,·)$, $\chi_{275}(56,·)$, $\chi_{275}(186,·)$, $\chi_{275}(188,·)$, $\chi_{275}(67,·)$, $\chi_{275}(197,·)$, $\chi_{275}(199,·)$, $\chi_{275}(76,·)$, $\chi_{275}(78,·)$, $\chi_{275}(208,·)$, $\chi_{275}(87,·)$, $\chi_{275}(89,·)$, $\chi_{275}(219,·)$, $\chi_{275}(221,·)$, $\chi_{275}(98,·)$, $\chi_{275}(232,·)$, $\chi_{275}(109,·)$, $\chi_{275}(111,·)$, $\chi_{275}(241,·)$, $\chi_{275}(243,·)$, $\chi_{275}(122,·)$, $\chi_{275}(252,·)$, $\chi_{275}(254,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{524288}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{3}a^{21}-\frac{1}{3}a^{16}+\frac{1}{3}a^{11}-\frac{1}{3}a^{6}+\frac{1}{3}a$, $\frac{1}{9}a^{22}-\frac{4}{9}a^{17}-\frac{2}{9}a^{12}-\frac{1}{9}a^{7}+\frac{4}{9}a^{2}$, $\frac{1}{27}a^{23}-\frac{4}{27}a^{18}-\frac{11}{27}a^{13}-\frac{10}{27}a^{8}+\frac{13}{27}a^{3}$, $\frac{1}{81}a^{24}-\frac{31}{81}a^{19}-\frac{11}{81}a^{14}+\frac{17}{81}a^{9}+\frac{40}{81}a^{4}$, $\frac{1}{68526243}a^{25}+\frac{62}{243}a^{20}+\frac{22}{243}a^{15}+\frac{47}{243}a^{10}+\frac{1}{243}a^{5}-\frac{14725}{282001}$, $\frac{1}{205578729}a^{26}+\frac{62}{729}a^{21}+\frac{22}{729}a^{16}+\frac{290}{729}a^{11}+\frac{244}{729}a^{6}-\frac{296726}{846003}a$, $\frac{1}{616736187}a^{27}+\frac{62}{2187}a^{22}+\frac{751}{2187}a^{17}+\frac{1019}{2187}a^{12}+\frac{244}{2187}a^{7}-\frac{1142729}{2538009}a^{2}$, $\frac{1}{1850208561}a^{28}+\frac{62}{6561}a^{23}-\frac{1436}{6561}a^{18}+\frac{3206}{6561}a^{13}+\frac{244}{6561}a^{8}-\frac{1142729}{7614027}a^{3}$, $\frac{1}{5550625683}a^{29}+\frac{62}{19683}a^{24}-\frac{1436}{19683}a^{19}+\frac{9767}{19683}a^{14}+\frac{6805}{19683}a^{9}+\frac{6471298}{22842081}a^{4}$, $\frac{1}{16651877049}a^{30}-\frac{31}{16651877049}a^{25}+\frac{6340}{59049}a^{20}+\frac{24590}{59049}a^{15}+\frac{1}{59049}a^{10}-\frac{14725}{68526243}a^{5}+\frac{718}{282001}$, $\frac{1}{49955631147}a^{31}-\frac{31}{49955631147}a^{26}+\frac{6340}{177147}a^{21}+\frac{83639}{177147}a^{16}-\frac{59048}{177147}a^{11}+\frac{68511518}{205578729}a^{6}-\frac{93761}{282001}a$, $\frac{1}{149866893441}a^{32}-\frac{31}{149866893441}a^{27}+\frac{6340}{531441}a^{22}+\frac{260786}{531441}a^{17}-\frac{59048}{531441}a^{12}+\frac{274090247}{616736187}a^{7}+\frac{188240}{846003}a^{2}$, $\frac{1}{449600680323}a^{33}-\frac{31}{449600680323}a^{28}+\frac{6340}{1594323}a^{23}+\frac{260786}{1594323}a^{18}-\frac{59048}{1594323}a^{13}+\frac{274090247}{1850208561}a^{8}+\frac{1034243}{2538009}a^{3}$, $\frac{1}{1348802040969}a^{34}-\frac{31}{1348802040969}a^{29}+\frac{6340}{4782969}a^{24}+\frac{1855109}{4782969}a^{19}-\frac{1653371}{4782969}a^{14}-\frac{1576118314}{5550625683}a^{9}-\frac{1503766}{7614027}a^{4}$, $\frac{1}{4046406122907}a^{35}-\frac{31}{4046406122907}a^{30}+\frac{718}{4046406122907}a^{25}-\frac{6529849}{14348907}a^{20}+\frac{1}{14348907}a^{15}-\frac{14725}{16651877049}a^{10}+\frac{718}{68526243}a^{5}-\frac{31}{282001}$, $\frac{1}{12139218368721}a^{36}-\frac{31}{12139218368721}a^{31}+\frac{718}{12139218368721}a^{26}-\frac{6529849}{43046721}a^{21}+\frac{14348908}{43046721}a^{16}-\frac{16651891774}{49955631147}a^{11}+\frac{68526961}{205578729}a^{6}-\frac{282032}{846003}a$, $\frac{1}{36417655106163}a^{37}-\frac{31}{36417655106163}a^{32}+\frac{718}{36417655106163}a^{27}-\frac{6529849}{129140163}a^{22}+\frac{14348908}{129140163}a^{17}-\frac{66607522921}{149866893441}a^{12}-\frac{137051768}{616736187}a^{7}-\frac{282032}{2538009}a^{2}$, $\frac{1}{109252965318489}a^{38}-\frac{31}{109252965318489}a^{33}+\frac{718}{109252965318489}a^{28}-\frac{6529849}{387420489}a^{23}-\frac{114791255}{387420489}a^{18}+\frac{83259370520}{449600680323}a^{13}+\frac{479684419}{1850208561}a^{8}-\frac{282032}{7614027}a^{3}$, $\frac{1}{327758895955467}a^{39}-\frac{31}{327758895955467}a^{34}+\frac{718}{327758895955467}a^{29}-\frac{6529849}{1162261467}a^{24}-\frac{502211744}{1162261467}a^{19}+\frac{532860050843}{1348802040969}a^{14}-\frac{1370524142}{5550625683}a^{9}-\frac{7896059}{22842081}a^{4}$ 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:   \( \frac{1079}{1348802040969} a^{39} + \frac{500858642}{1348802040969} a^{14} \)  (order $50$) 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 - 31*x^35 + 718*x^30 - 14725*x^25 + 282001*x^20 - 3578175*x^15 + 42397182*x^10 - 444816117*x^5 + 3486784401)
 
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 - 31*x^35 + 718*x^30 - 14725*x^25 + 282001*x^20 - 3578175*x^15 + 42397182*x^10 - 444816117*x^5 + 3486784401, 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 - 31*x^35 + 718*x^30 - 14725*x^25 + 282001*x^20 - 3578175*x^15 + 42397182*x^10 - 444816117*x^5 + 3486784401);
 
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 - 31*x^35 + 718*x^30 - 14725*x^25 + 282001*x^20 - 3578175*x^15 + 42397182*x^10 - 444816117*x^5 + 3486784401);
 
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 $C_2\times C_{20}$

Intermediate fields

\(\Q(\sqrt{-11}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{5}, \sqrt{-11})\), 4.4.15125.1, \(\Q(\zeta_{5})\), 5.5.390625.1, 8.0.228765625.1, 10.0.24574432373046875.1, \(\Q(\zeta_{25})^+\), 10.0.122872161865234375.1, 20.0.15097568161436356604099273681640625.2, 20.20.75487840807181783020496368408203125.1, \(\Q(\zeta_{25})\)

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 $20^{2}$ $20^{2}$ R ${\href{/padicField/7.4.0.1}{4} }^{10}$ R $20^{2}$ $20^{2}$ ${\href{/padicField/19.10.0.1}{10} }^{4}$ $20^{2}$ ${\href{/padicField/29.10.0.1}{10} }^{4}$ ${\href{/padicField/31.5.0.1}{5} }^{8}$ $20^{2}$ ${\href{/padicField/41.10.0.1}{10} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{10}$ $20^{2}$ $20^{2}$ ${\href{/padicField/59.10.0.1}{10} }^{4}$

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
\(5\) Copy content Toggle raw display Deg $20$$20$$1$$35$
Deg $20$$20$$1$$35$
\(11\) Copy content Toggle raw display 11.10.5.2$x^{10} + 55 x^{8} + 20 x^{7} + 1210 x^{6} - 202 x^{5} + 13410 x^{4} - 14080 x^{3} + 75585 x^{2} - 68970 x + 171252$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
11.10.5.2$x^{10} + 55 x^{8} + 20 x^{7} + 1210 x^{6} - 202 x^{5} + 13410 x^{4} - 14080 x^{3} + 75585 x^{2} - 68970 x + 171252$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
11.10.5.2$x^{10} + 55 x^{8} + 20 x^{7} + 1210 x^{6} - 202 x^{5} + 13410 x^{4} - 14080 x^{3} + 75585 x^{2} - 68970 x + 171252$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
11.10.5.2$x^{10} + 55 x^{8} + 20 x^{7} + 1210 x^{6} - 202 x^{5} + 13410 x^{4} - 14080 x^{3} + 75585 x^{2} - 68970 x + 171252$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$