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Properties

Label 40.0.339...000.2
Degree $40$
Signature $[0, 20]$
Discriminant $3.399\times 10^{69}$
Root discriminant \(54.74\)
Ramified primes $2,5,11$
Class number not computed
Class group not computed
Galois group $C_2^2\times C_{10}$ (as 40T7)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^40 + 6*x^38 + 32*x^36 + 168*x^34 + 880*x^32 + 4608*x^30 + 24128*x^28 + 126336*x^26 + 661504*x^24 + 3463680*x^22 + 18136064*x^20 + 13854720*x^18 + 10584064*x^16 + 8085504*x^14 + 6176768*x^12 + 4718592*x^10 + 3604480*x^8 + 2752512*x^6 + 2097152*x^4 + 1572864*x^2 + 1048576)
 
gp: K = bnfinit(y^40 + 6*y^38 + 32*y^36 + 168*y^34 + 880*y^32 + 4608*y^30 + 24128*y^28 + 126336*y^26 + 661504*y^24 + 3463680*y^22 + 18136064*y^20 + 13854720*y^18 + 10584064*y^16 + 8085504*y^14 + 6176768*y^12 + 4718592*y^10 + 3604480*y^8 + 2752512*y^6 + 2097152*y^4 + 1572864*y^2 + 1048576, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^40 + 6*x^38 + 32*x^36 + 168*x^34 + 880*x^32 + 4608*x^30 + 24128*x^28 + 126336*x^26 + 661504*x^24 + 3463680*x^22 + 18136064*x^20 + 13854720*x^18 + 10584064*x^16 + 8085504*x^14 + 6176768*x^12 + 4718592*x^10 + 3604480*x^8 + 2752512*x^6 + 2097152*x^4 + 1572864*x^2 + 1048576);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^40 + 6*x^38 + 32*x^36 + 168*x^34 + 880*x^32 + 4608*x^30 + 24128*x^28 + 126336*x^26 + 661504*x^24 + 3463680*x^22 + 18136064*x^20 + 13854720*x^18 + 10584064*x^16 + 8085504*x^14 + 6176768*x^12 + 4718592*x^10 + 3604480*x^8 + 2752512*x^6 + 2097152*x^4 + 1572864*x^2 + 1048576)
 

\( x^{40} + 6 x^{38} + 32 x^{36} + 168 x^{34} + 880 x^{32} + 4608 x^{30} + 24128 x^{28} + 126336 x^{26} + \cdots + 1048576 \) 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:   \(3398885169161610034374122849346487403986439215513600000000000000000000\) \(\medspace = 2^{60}\cdot 5^{20}\cdot 11^{36}\) 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:  \(54.74\)
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:  $2^{3/2}5^{1/2}11^{9/10}\approx 54.73730515936835$
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:  \(440=2^{3}\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{440}(1,·)$, $\chi_{440}(389,·)$, $\chi_{440}(129,·)$, $\chi_{440}(9,·)$, $\chi_{440}(269,·)$, $\chi_{440}(401,·)$, $\chi_{440}(21,·)$, $\chi_{440}(281,·)$, $\chi_{440}(409,·)$, $\chi_{440}(29,·)$, $\chi_{440}(261,·)$, $\chi_{440}(289,·)$, $\chi_{440}(421,·)$, $\chi_{440}(369,·)$, $\chi_{440}(41,·)$, $\chi_{440}(301,·)$, $\chi_{440}(349,·)$, $\chi_{440}(49,·)$, $\chi_{440}(309,·)$, $\chi_{440}(329,·)$, $\chi_{440}(61,·)$, $\chi_{440}(181,·)$, $\chi_{440}(321,·)$, $\chi_{440}(69,·)$, $\chi_{440}(161,·)$, $\chi_{440}(201,·)$, $\chi_{440}(141,·)$, $\chi_{440}(81,·)$, $\chi_{440}(89,·)$, $\chi_{440}(221,·)$, $\chi_{440}(101,·)$, $\chi_{440}(229,·)$, $\chi_{440}(361,·)$, $\chi_{440}(109,·)$, $\chi_{440}(189,·)$, $\chi_{440}(241,·)$, $\chi_{440}(169,·)$, $\chi_{440}(249,·)$, $\chi_{440}(381,·)$, $\chi_{440}(149,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{524288}$

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

$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{4}a^{5}$, $\frac{1}{8}a^{6}$, $\frac{1}{8}a^{7}$, $\frac{1}{16}a^{8}$, $\frac{1}{16}a^{9}$, $\frac{1}{32}a^{10}$, $\frac{1}{32}a^{11}$, $\frac{1}{64}a^{12}$, $\frac{1}{64}a^{13}$, $\frac{1}{128}a^{14}$, $\frac{1}{128}a^{15}$, $\frac{1}{256}a^{16}$, $\frac{1}{256}a^{17}$, $\frac{1}{512}a^{18}$, $\frac{1}{512}a^{19}$, $\frac{1}{1024}a^{20}$, $\frac{1}{1024}a^{21}$, $\frac{1}{36272128}a^{22}+\frac{6765}{17711}$, $\frac{1}{36272128}a^{23}+\frac{6765}{17711}a$, $\frac{1}{72544256}a^{24}+\frac{6765}{35422}a^{2}$, $\frac{1}{72544256}a^{25}+\frac{6765}{35422}a^{3}$, $\frac{1}{145088512}a^{26}+\frac{6765}{70844}a^{4}$, $\frac{1}{145088512}a^{27}+\frac{6765}{70844}a^{5}$, $\frac{1}{290177024}a^{28}+\frac{6765}{141688}a^{6}$, $\frac{1}{290177024}a^{29}+\frac{6765}{141688}a^{7}$, $\frac{1}{580354048}a^{30}+\frac{6765}{283376}a^{8}$, $\frac{1}{580354048}a^{31}+\frac{6765}{283376}a^{9}$, $\frac{1}{1160708096}a^{32}+\frac{6765}{566752}a^{10}$, $\frac{1}{1160708096}a^{33}+\frac{6765}{566752}a^{11}$, $\frac{1}{2321416192}a^{34}+\frac{6765}{1133504}a^{12}$, $\frac{1}{2321416192}a^{35}+\frac{6765}{1133504}a^{13}$, $\frac{1}{4642832384}a^{36}+\frac{6765}{2267008}a^{14}$, $\frac{1}{4642832384}a^{37}+\frac{6765}{2267008}a^{15}$, $\frac{1}{9285664768}a^{38}+\frac{6765}{4534016}a^{16}$, $\frac{1}{9285664768}a^{39}+\frac{6765}{4534016}a^{17}$ 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:   \( \frac{9}{145088512} a^{34} - \frac{5702887}{1133504} a^{12} \)  (order $22$) 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 + 6*x^38 + 32*x^36 + 168*x^34 + 880*x^32 + 4608*x^30 + 24128*x^28 + 126336*x^26 + 661504*x^24 + 3463680*x^22 + 18136064*x^20 + 13854720*x^18 + 10584064*x^16 + 8085504*x^14 + 6176768*x^12 + 4718592*x^10 + 3604480*x^8 + 2752512*x^6 + 2097152*x^4 + 1572864*x^2 + 1048576)
 
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 + 6*x^38 + 32*x^36 + 168*x^34 + 880*x^32 + 4608*x^30 + 24128*x^28 + 126336*x^26 + 661504*x^24 + 3463680*x^22 + 18136064*x^20 + 13854720*x^18 + 10584064*x^16 + 8085504*x^14 + 6176768*x^12 + 4718592*x^10 + 3604480*x^8 + 2752512*x^6 + 2097152*x^4 + 1572864*x^2 + 1048576, 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 + 6*x^38 + 32*x^36 + 168*x^34 + 880*x^32 + 4608*x^30 + 24128*x^28 + 126336*x^26 + 661504*x^24 + 3463680*x^22 + 18136064*x^20 + 13854720*x^18 + 10584064*x^16 + 8085504*x^14 + 6176768*x^12 + 4718592*x^10 + 3604480*x^8 + 2752512*x^6 + 2097152*x^4 + 1572864*x^2 + 1048576);
 
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 + 6*x^38 + 32*x^36 + 168*x^34 + 880*x^32 + 4608*x^30 + 24128*x^28 + 126336*x^26 + 661504*x^24 + 3463680*x^22 + 18136064*x^20 + 13854720*x^18 + 10584064*x^16 + 8085504*x^14 + 6176768*x^12 + 4718592*x^10 + 3604480*x^8 + 2752512*x^6 + 2097152*x^4 + 1572864*x^2 + 1048576);
 
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 $C_2^2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{-22}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-110}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{5}, \sqrt{-22})\), \(\Q(\sqrt{10}, \sqrt{-22})\), \(\Q(\sqrt{2}, \sqrt{-11})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{5}, \sqrt{-11})\), \(\Q(\sqrt{10}, \sqrt{-11})\), \(\Q(\sqrt{2}, \sqrt{-55})\), \(\Q(\zeta_{11})^+\), 8.0.37480960000.9, 10.0.77265229938688.1, 10.10.669871503125.1, 10.0.241453843558400000.1, 10.10.21950349414400000.1, 10.0.7368586534375.1, 10.10.7024111812608.1, \(\Q(\zeta_{11})\), 20.0.58299958569124301174210560000000000.8, 20.0.58299958569124301174210560000000000.7, 20.0.5969915757478328440239161344.6, 20.20.481817839414250422927360000000000.1, 20.0.54296067514572573056640625.1, 20.0.58299958569124301174210560000000000.10, 20.0.58299958569124301174210560000000000.5

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 ${\href{/padicField/3.10.0.1}{10} }^{4}$ R ${\href{/padicField/7.10.0.1}{10} }^{4}$ R ${\href{/padicField/13.10.0.1}{10} }^{4}$ ${\href{/padicField/17.10.0.1}{10} }^{4}$ ${\href{/padicField/19.10.0.1}{10} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{20}$ ${\href{/padicField/29.10.0.1}{10} }^{4}$ ${\href{/padicField/31.5.0.1}{5} }^{8}$ ${\href{/padicField/37.10.0.1}{10} }^{4}$ ${\href{/padicField/41.10.0.1}{10} }^{4}$ ${\href{/padicField/43.2.0.1}{2} }^{20}$ ${\href{/padicField/47.10.0.1}{10} }^{4}$ ${\href{/padicField/53.10.0.1}{10} }^{4}$ ${\href{/padicField/59.10.0.1}{10} }^{4}$

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 $20$$2$$10$$30$
Deg $20$$2$$10$$30$
\(5\) Copy content Toggle raw display Deg $20$$2$$10$$10$
Deg $20$$2$$10$$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}$