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Properties

Label 40.0.130...936.9
Degree $40$
Signature $[0, 20]$
Discriminant $1.303\times 10^{71}$
Root discriminant \(59.96\)
Ramified primes $2,3,11$
Class number not computed
Class group not computed
Galois group $C_2^2\times C_{10}$ (as 40T7)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^40 - 9*x^36 + 81*x^32 - 729*x^28 + 6561*x^24 - 59049*x^20 + 531441*x^16 - 4782969*x^12 + 43046721*x^8 - 387420489*x^4 + 3486784401)
 
gp: K = bnfinit(y^40 - 9*y^36 + 81*y^32 - 729*y^28 + 6561*y^24 - 59049*y^20 + 531441*y^16 - 4782969*y^12 + 43046721*y^8 - 387420489*y^4 + 3486784401, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^40 - 9*x^36 + 81*x^32 - 729*x^28 + 6561*x^24 - 59049*x^20 + 531441*x^16 - 4782969*x^12 + 43046721*x^8 - 387420489*x^4 + 3486784401);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^40 - 9*x^36 + 81*x^32 - 729*x^28 + 6561*x^24 - 59049*x^20 + 531441*x^16 - 4782969*x^12 + 43046721*x^8 - 387420489*x^4 + 3486784401)
 

\( x^{40} - 9 x^{36} + 81 x^{32} - 729 x^{28} + 6561 x^{24} - 59049 x^{20} + 531441 x^{16} + \cdots + 3486784401 \) 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:   \(130305099804548492884220428175380349368393046678311823693003457545895936\) \(\medspace = 2^{80}\cdot 3^{20}\cdot 11^{36}\) 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:  \(59.96\)
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:  $2^{2}3^{1/2}11^{9/10}\approx 59.96171354565991$
Ramified primes:   \(2\), \(3\), \(11\) 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:  \(264=2^{3}\cdot 3\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{264}(1,·)$, $\chi_{264}(131,·)$, $\chi_{264}(5,·)$, $\chi_{264}(7,·)$, $\chi_{264}(145,·)$, $\chi_{264}(149,·)$, $\chi_{264}(151,·)$, $\chi_{264}(25,·)$, $\chi_{264}(155,·)$, $\chi_{264}(29,·)$, $\chi_{264}(31,·)$, $\chi_{264}(35,·)$, $\chi_{264}(169,·)$, $\chi_{264}(173,·)$, $\chi_{264}(175,·)$, $\chi_{264}(49,·)$, $\chi_{264}(179,·)$, $\chi_{264}(53,·)$, $\chi_{264}(59,·)$, $\chi_{264}(193,·)$, $\chi_{264}(197,·)$, $\chi_{264}(199,·)$, $\chi_{264}(73,·)$, $\chi_{264}(203,·)$, $\chi_{264}(79,·)$, $\chi_{264}(83,·)$, $\chi_{264}(217,·)$, $\chi_{264}(221,·)$, $\chi_{264}(223,·)$, $\chi_{264}(97,·)$, $\chi_{264}(227,·)$, $\chi_{264}(101,·)$, $\chi_{264}(103,·)$, $\chi_{264}(107,·)$, $\chi_{264}(241,·)$, $\chi_{264}(245,·)$, $\chi_{264}(247,·)$, $\chi_{264}(251,·)$, $\chi_{264}(125,·)$, $\chi_{264}(127,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{524288}$

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

$1$, $a$, $\frac{1}{3}a^{2}$, $\frac{1}{3}a^{3}$, $\frac{1}{9}a^{4}$, $\frac{1}{9}a^{5}$, $\frac{1}{27}a^{6}$, $\frac{1}{27}a^{7}$, $\frac{1}{81}a^{8}$, $\frac{1}{81}a^{9}$, $\frac{1}{243}a^{10}$, $\frac{1}{243}a^{11}$, $\frac{1}{729}a^{12}$, $\frac{1}{729}a^{13}$, $\frac{1}{2187}a^{14}$, $\frac{1}{2187}a^{15}$, $\frac{1}{6561}a^{16}$, $\frac{1}{6561}a^{17}$, $\frac{1}{19683}a^{18}$, $\frac{1}{19683}a^{19}$, $\frac{1}{59049}a^{20}$, $\frac{1}{59049}a^{21}$, $\frac{1}{177147}a^{22}$, $\frac{1}{177147}a^{23}$, $\frac{1}{531441}a^{24}$, $\frac{1}{531441}a^{25}$, $\frac{1}{1594323}a^{26}$, $\frac{1}{1594323}a^{27}$, $\frac{1}{4782969}a^{28}$, $\frac{1}{4782969}a^{29}$, $\frac{1}{14348907}a^{30}$, $\frac{1}{14348907}a^{31}$, $\frac{1}{43046721}a^{32}$, $\frac{1}{43046721}a^{33}$, $\frac{1}{129140163}a^{34}$, $\frac{1}{129140163}a^{35}$, $\frac{1}{387420489}a^{36}$, $\frac{1}{387420489}a^{37}$, $\frac{1}{1162261467}a^{38}$, $\frac{1}{1162261467}a^{39}$ 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:   \( \frac{1}{27} a^{6} \)  (order $44$) 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 - 9*x^36 + 81*x^32 - 729*x^28 + 6561*x^24 - 59049*x^20 + 531441*x^16 - 4782969*x^12 + 43046721*x^8 - 387420489*x^4 + 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 - 9*x^36 + 81*x^32 - 729*x^28 + 6561*x^24 - 59049*x^20 + 531441*x^16 - 4782969*x^12 + 43046721*x^8 - 387420489*x^4 + 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 - 9*x^36 + 81*x^32 - 729*x^28 + 6561*x^24 - 59049*x^20 + 531441*x^16 - 4782969*x^12 + 43046721*x^8 - 387420489*x^4 + 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 - 9*x^36 + 81*x^32 - 729*x^28 + 6561*x^24 - 59049*x^20 + 531441*x^16 - 4782969*x^12 + 43046721*x^8 - 387420489*x^4 + 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

$C_2^2\times C_{10}$ (as 40T7):

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 $C_2^2\times C_{10}$
Character table for $C_2^2\times C_{10}$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{-66}) \), \(\Q(\sqrt{66}) \), \(\Q(i, \sqrt{6})\), \(\Q(i, \sqrt{11})\), \(\Q(i, \sqrt{66})\), \(\Q(\sqrt{6}, \sqrt{-11})\), \(\Q(\sqrt{6}, \sqrt{11})\), \(\Q(\sqrt{-6}, \sqrt{-11})\), \(\Q(\sqrt{-6}, \sqrt{11})\), \(\Q(\zeta_{11})^+\), 8.0.77720518656.3, 10.0.219503494144.1, 10.10.1706859170463744.1, 10.0.1706859170463744.1, \(\Q(\zeta_{11})\), \(\Q(\zeta_{44})^+\), 10.0.18775450875101184.1, 10.10.18775450875101184.1, 20.0.2983289065263288625233938941476864.4, \(\Q(\zeta_{44})\), 20.0.360977976896857923653306611918700544.10, 20.0.352517555563337816067682238201856.9, 20.20.360977976896857923653306611918700544.2, 20.0.352517555563337816067682238201856.8, 20.0.360977976896857923653306611918700544.3

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 ${\href{/padicField/5.5.0.1}{5} }^{8}$ ${\href{/padicField/7.10.0.1}{10} }^{4}$ R ${\href{/padicField/13.10.0.1}{10} }^{4}$ ${\href{/padicField/17.10.0.1}{10} }^{4}$ ${\href{/padicField/19.10.0.1}{10} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{20}$ ${\href{/padicField/29.10.0.1}{10} }^{4}$ ${\href{/padicField/31.10.0.1}{10} }^{4}$ ${\href{/padicField/37.10.0.1}{10} }^{4}$ ${\href{/padicField/41.10.0.1}{10} }^{4}$ ${\href{/padicField/43.2.0.1}{2} }^{20}$ ${\href{/padicField/47.10.0.1}{10} }^{4}$ ${\href{/padicField/53.5.0.1}{5} }^{8}$ ${\href{/padicField/59.10.0.1}{10} }^{4}$

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$$4$$10$$80$
\(3\) Copy content Toggle raw display 3.20.10.1$x^{20} + 30 x^{18} + 409 x^{16} + 4 x^{15} + 3244 x^{14} - 60 x^{13} + 16162 x^{12} - 1250 x^{11} + 53008 x^{10} - 7102 x^{9} + 121150 x^{8} - 12140 x^{7} + 219264 x^{6} + 5736 x^{5} + 257465 x^{4} + 35250 x^{3} + 250183 x^{2} + 51502 x + 77812$$2$$10$$10$20T3$[\ ]_{2}^{10}$
3.20.10.1$x^{20} + 30 x^{18} + 409 x^{16} + 4 x^{15} + 3244 x^{14} - 60 x^{13} + 16162 x^{12} - 1250 x^{11} + 53008 x^{10} - 7102 x^{9} + 121150 x^{8} - 12140 x^{7} + 219264 x^{6} + 5736 x^{5} + 257465 x^{4} + 35250 x^{3} + 250183 x^{2} + 51502 x + 77812$$2$$10$$10$20T3$[\ ]_{2}^{10}$
\(11\) Copy content Toggle raw display 11.20.18.1$x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241675 x^{10} + 2355135790 x^{9} + 1953262935 x^{8} + 1157863560 x^{7} + 500734950 x^{6} + 191763012 x^{5} + 243790230 x^{4} + 806750280 x^{3} + 2014356815 x^{2} + 2999040310 x + 2009802620$$10$$2$$18$20T3$[\ ]_{10}^{2}$
11.20.18.1$x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241675 x^{10} + 2355135790 x^{9} + 1953262935 x^{8} + 1157863560 x^{7} + 500734950 x^{6} + 191763012 x^{5} + 243790230 x^{4} + 806750280 x^{3} + 2014356815 x^{2} + 2999040310 x + 2009802620$$10$$2$$18$20T3$[\ ]_{10}^{2}$