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Properties

Label 40.0.97656250000...0000.2
Degree $40$
Signature $[0, 20]$
Discriminant $2^{60}\cdot 5^{70}$
Root discriminant $47.29$
Ramified primes $2, 5$
Class number Not computed
Class group Not computed
Galois group $C_2\times C_{20}$ (as 40T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1048576, 0, 0, 0, 0, 0, 0, 0, 0, 0, 32768, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1024, 0, 0, 0, 0, 0, 0, 0, 0, 0, 32, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 + 32*x^30 + 1024*x^20 + 32768*x^10 + 1048576)
 
gp: K = bnfinit(x^40 + 32*x^30 + 1024*x^20 + 32768*x^10 + 1048576, 1)
 

Normalized defining polynomial

\( x^{40} + 32 x^{30} + 1024 x^{20} + 32768 x^{10} + 1048576 \)

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:  \(9765625000000000000000000000000000000000000000000000000000000000000=2^{60}\cdot 5^{70}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.29$
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:  $2, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(200=2^{3}\cdot 5^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{200}(1,·)$, $\chi_{200}(133,·)$, $\chi_{200}(129,·)$, $\chi_{200}(9,·)$, $\chi_{200}(13,·)$, $\chi_{200}(17,·)$, $\chi_{200}(21,·)$, $\chi_{200}(153,·)$, $\chi_{200}(157,·)$, $\chi_{200}(69,·)$, $\chi_{200}(33,·)$, $\chi_{200}(37,·)$, $\chi_{200}(49,·)$, $\chi_{200}(41,·)$, $\chi_{200}(173,·)$, $\chi_{200}(29,·)$, $\chi_{200}(177,·)$, $\chi_{200}(181,·)$, $\chi_{200}(137,·)$, $\chi_{200}(57,·)$, $\chi_{200}(61,·)$, $\chi_{200}(53,·)$, $\chi_{200}(193,·)$, $\chi_{200}(197,·)$, $\chi_{200}(161,·)$, $\chi_{200}(73,·)$, $\chi_{200}(77,·)$, $\chi_{200}(141,·)$, $\chi_{200}(81,·)$, $\chi_{200}(89,·)$, $\chi_{200}(93,·)$, $\chi_{200}(97,·)$, $\chi_{200}(101,·)$, $\chi_{200}(109,·)$, $\chi_{200}(189,·)$, $\chi_{200}(113,·)$, $\chi_{200}(117,·)$, $\chi_{200}(169,·)$, $\chi_{200}(121,·)$, $\chi_{200}(149,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{32} a^{11}$, $\frac{1}{64} a^{12}$, $\frac{1}{64} a^{13}$, $\frac{1}{128} a^{14}$, $\frac{1}{128} a^{15}$, $\frac{1}{256} a^{16}$, $\frac{1}{256} a^{17}$, $\frac{1}{512} a^{18}$, $\frac{1}{512} a^{19}$, $\frac{1}{1024} a^{20}$, $\frac{1}{1024} a^{21}$, $\frac{1}{2048} a^{22}$, $\frac{1}{2048} a^{23}$, $\frac{1}{4096} a^{24}$, $\frac{1}{4096} a^{25}$, $\frac{1}{8192} a^{26}$, $\frac{1}{8192} a^{27}$, $\frac{1}{16384} a^{28}$, $\frac{1}{16384} a^{29}$, $\frac{1}{32768} a^{30}$, $\frac{1}{32768} a^{31}$, $\frac{1}{65536} a^{32}$, $\frac{1}{65536} a^{33}$, $\frac{1}{131072} a^{34}$, $\frac{1}{131072} a^{35}$, $\frac{1}{262144} a^{36}$, $\frac{1}{262144} a^{37}$, $\frac{1}{524288} a^{38}$, $\frac{1}{524288} a^{39}$

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:  \( -\frac{1}{8192} a^{26} \) (order $50$)
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

$C_2\times C_{20}$ (as 40T2):

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 $C_2\times C_{20}$
Character table for $C_2\times C_{20}$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}, \sqrt{5})\), 4.0.8000.2, \(\Q(\zeta_{5})\), 5.5.390625.1, 8.0.64000000.2, 10.10.5000000000000000.1, \(\Q(\zeta_{25})^+\), 10.10.25000000000000000.1, 20.20.625000000000000000000000000000000.1, 20.0.3125000000000000000000000000000000.1, \(\Q(\zeta_{25})\)

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 $20^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{10}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{4}$ $20^{2}$ $20^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ $20^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{8}$ $20^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{10}$ $20^{2}$ $20^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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