↑

Properties

Label 40.0.33988851691...0000.2
Degree $40$
Signature $[0, 20]$
Discriminant $2^{60}\cdot 5^{20}\cdot 11^{36}$
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)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1048576, 0, 1572864, 0, 2097152, 0, 2752512, 0, 3604480, 0, 4718592, 0, 6176768, 0, 8085504, 0, 10584064, 0, 13854720, 0, 18136064, 0, 3463680, 0, 661504, 0, 126336, 0, 24128, 0, 4608, 0, 880, 0, 168, 0, 32, 0, 6, 0, 1]);
 
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(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)
 

Normalized defining polynomial

\( x^{40} + 6 x^{38} + 32 x^{36} + 168 x^{34} + 880 x^{32} + 4608 x^{30} + 24128 x^{28} + 126336 x^{26} + 661504 x^{24} + 3463680 x^{22} + 18136064 x^{20} + 13854720 x^{18} + 10584064 x^{16} + 8085504 x^{14} + 6176768 x^{12} + 4718592 x^{10} + 3604480 x^{8} + 2752512 x^{6} + 2097152 x^{4} + 1572864 x^{2} + 1048576 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $40$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 20]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3398885169161610034374122849346487403986439215513600000000000000000000=2^{60}\cdot 5^{20}\cdot 11^{36}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $54.74$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
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.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{32} a^{11}$, $\frac{1}{64} a^{12}$, $\frac{1}{64} a^{13}$, $\frac{1}{128} a^{14}$, $\frac{1}{128} a^{15}$, $\frac{1}{256} a^{16}$, $\frac{1}{256} a^{17}$, $\frac{1}{512} a^{18}$, $\frac{1}{512} a^{19}$, $\frac{1}{1024} a^{20}$, $\frac{1}{1024} a^{21}$, $\frac{1}{36272128} a^{22} + \frac{6765}{17711}$, $\frac{1}{36272128} a^{23} + \frac{6765}{17711} a$, $\frac{1}{72544256} a^{24} + \frac{6765}{35422} a^{2}$, $\frac{1}{72544256} a^{25} + \frac{6765}{35422} a^{3}$, $\frac{1}{145088512} a^{26} + \frac{6765}{70844} a^{4}$, $\frac{1}{145088512} a^{27} + \frac{6765}{70844} a^{5}$, $\frac{1}{290177024} a^{28} + \frac{6765}{141688} a^{6}$, $\frac{1}{290177024} a^{29} + \frac{6765}{141688} a^{7}$, $\frac{1}{580354048} a^{30} + \frac{6765}{283376} a^{8}$, $\frac{1}{580354048} a^{31} + \frac{6765}{283376} a^{9}$, $\frac{1}{1160708096} a^{32} + \frac{6765}{566752} a^{10}$, $\frac{1}{1160708096} a^{33} + \frac{6765}{566752} a^{11}$, $\frac{1}{2321416192} a^{34} + \frac{6765}{1133504} a^{12}$, $\frac{1}{2321416192} a^{35} + \frac{6765}{1133504} a^{13}$, $\frac{1}{4642832384} a^{36} + \frac{6765}{2267008} a^{14}$, $\frac{1}{4642832384} a^{37} + \frac{6765}{2267008} a^{15}$, $\frac{1}{9285664768} a^{38} + \frac{6765}{4534016} a^{16}$, $\frac{1}{9285664768} a^{39} + \frac{6765}{4534016} a^{17}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $19$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{9}{145088512} a^{34} - \frac{5702887}{1133504} a^{12} \) (order $22$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

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

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
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

\(\Q(\sqrt{-22}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-110}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{5}, \sqrt{-22})\), \(\Q(\sqrt{10}, \sqrt{-22})\), \(\Q(\sqrt{2}, \sqrt{-11})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{5}, \sqrt{-11})\), \(\Q(\sqrt{10}, \sqrt{-11})\), \(\Q(\sqrt{2}, \sqrt{-55})\), \(\Q(\zeta_{11})^+\), 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.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.10.0.1}{10} }^{4}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{8}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
5Data not computed
11Data not computed