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Properties

Label 40.0.91278848325...1136.1
Degree $40$
Signature $[0, 20]$
Discriminant $2^{60}\cdot 41^{39}$
Root discriminant $105.68$
Ramified primes $2, 41$
Class number Not computed
Class group Not computed
Galois group $C_{40}$ (as 40T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![42991616, 0, 1504706560, 0, 15724183552, 0, 77497761792, 0, 219576991744, 0, 399230894080, 0, 499038617600, 0, 449134755840, 0, 300523991040, 0, 152898170880, 0, 60067138560, 0, 18400012800, 0, 4416003072, 0, 830359552, 0, 121690624, 0, 13739264, 0, 1170960, 0, 72816, 0, 3116, 0, 82, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 + 82*x^38 + 3116*x^36 + 72816*x^34 + 1170960*x^32 + 13739264*x^30 + 121690624*x^28 + 830359552*x^26 + 4416003072*x^24 + 18400012800*x^22 + 60067138560*x^20 + 152898170880*x^18 + 300523991040*x^16 + 449134755840*x^14 + 499038617600*x^12 + 399230894080*x^10 + 219576991744*x^8 + 77497761792*x^6 + 15724183552*x^4 + 1504706560*x^2 + 42991616)
 
gp: K = bnfinit(x^40 + 82*x^38 + 3116*x^36 + 72816*x^34 + 1170960*x^32 + 13739264*x^30 + 121690624*x^28 + 830359552*x^26 + 4416003072*x^24 + 18400012800*x^22 + 60067138560*x^20 + 152898170880*x^18 + 300523991040*x^16 + 449134755840*x^14 + 499038617600*x^12 + 399230894080*x^10 + 219576991744*x^8 + 77497761792*x^6 + 15724183552*x^4 + 1504706560*x^2 + 42991616, 1)
 

Normalized defining polynomial

\( x^{40} + 82 x^{38} + 3116 x^{36} + 72816 x^{34} + 1170960 x^{32} + 13739264 x^{30} + 121690624 x^{28} + 830359552 x^{26} + 4416003072 x^{24} + 18400012800 x^{22} + 60067138560 x^{20} + 152898170880 x^{18} + 300523991040 x^{16} + 449134755840 x^{14} + 499038617600 x^{12} + 399230894080 x^{10} + 219576991744 x^{8} + 77497761792 x^{6} + 15724183552 x^{4} + 1504706560 x^{2} + 42991616 \)

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:  \(912788483257978497757884926199917783690257306123427760963531833190283833440731136=2^{60}\cdot 41^{39}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $105.68$
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:  $2, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(328=2^{3}\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{328}(1,·)$, $\chi_{328}(261,·)$, $\chi_{328}(9,·)$, $\chi_{328}(13,·)$, $\chi_{328}(149,·)$, $\chi_{328}(25,·)$, $\chi_{328}(29,·)$, $\chi_{328}(325,·)$, $\chi_{328}(33,·)$, $\chi_{328}(293,·)$, $\chi_{328}(305,·)$, $\chi_{328}(297,·)$, $\chi_{328}(301,·)$, $\chi_{328}(157,·)$, $\chi_{328}(49,·)$, $\chi_{328}(53,·)$, $\chi_{328}(73,·)$, $\chi_{328}(57,·)$, $\chi_{328}(317,·)$, $\chi_{328}(309,·)$, $\chi_{328}(181,·)$, $\chi_{328}(69,·)$, $\chi_{328}(289,·)$, $\chi_{328}(201,·)$, $\chi_{328}(81,·)$, $\chi_{328}(85,·)$, $\chi_{328}(185,·)$, $\chi_{328}(93,·)$, $\chi_{328}(101,·)$, $\chi_{328}(225,·)$, $\chi_{328}(113,·)$, $\chi_{328}(229,·)$, $\chi_{328}(209,·)$, $\chi_{328}(105,·)$, $\chi_{328}(109,·)$, $\chi_{328}(241,·)$, $\chi_{328}(117,·)$, $\chi_{328}(169,·)$, $\chi_{328}(121,·)$, $\chi_{328}(253,·)$$\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:  \( -1 \) (order $2$)
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

$C_{40}$ (as 40T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 40
The 40 conjugacy class representatives for $C_{40}$
Character table for $C_{40}$ is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 5.5.2825761.1, 8.0.797713505816576.4, 10.10.327381934393961.1, \(\Q(\zeta_{41})^+\)

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{5}$ $20^{2}$ $40$ $40$ $40$ $40$ $40$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{4}$ $40$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{4}$ R $20^{2}$ $40$ $40$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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
41Data not computed