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Properties

Label 40.40.8301762893...0000.1
Degree $40$
Signature $[40, 0]$
Discriminant $2^{40}\cdot 5^{20}\cdot 41^{39}$
Root discriminant $167.10$
Ramified primes $2, 5, 41$
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![3910064697265625, 0, -54740905761718750, 0, 228816986083984375, 0, -451096343994140625, 0, 511242523193359375, 0, -371812744140625000, 0, 185906372070312500, 0, -66926293945312500, 0, 17912625732421875, 0, -3645376464843750, 0, 572844873046875, 0, -70190478515625, 0, 6738285937500, 0, -506811250000, 0, 29709625000, 0, -1341725000, 0, 45740625, 0, -1137750, 0, 19475, 0, -205, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - 205*x^38 + 19475*x^36 - 1137750*x^34 + 45740625*x^32 - 1341725000*x^30 + 29709625000*x^28 - 506811250000*x^26 + 6738285937500*x^24 - 70190478515625*x^22 + 572844873046875*x^20 - 3645376464843750*x^18 + 17912625732421875*x^16 - 66926293945312500*x^14 + 185906372070312500*x^12 - 371812744140625000*x^10 + 511242523193359375*x^8 - 451096343994140625*x^6 + 228816986083984375*x^4 - 54740905761718750*x^2 + 3910064697265625)
 
gp: K = bnfinit(x^40 - 205*x^38 + 19475*x^36 - 1137750*x^34 + 45740625*x^32 - 1341725000*x^30 + 29709625000*x^28 - 506811250000*x^26 + 6738285937500*x^24 - 70190478515625*x^22 + 572844873046875*x^20 - 3645376464843750*x^18 + 17912625732421875*x^16 - 66926293945312500*x^14 + 185906372070312500*x^12 - 371812744140625000*x^10 + 511242523193359375*x^8 - 451096343994140625*x^6 + 228816986083984375*x^4 - 54740905761718750*x^2 + 3910064697265625, 1)
 

Normalized defining polynomial

\( x^{40} - 205 x^{38} + 19475 x^{36} - 1137750 x^{34} + 45740625 x^{32} - 1341725000 x^{30} + 29709625000 x^{28} - 506811250000 x^{26} + 6738285937500 x^{24} - 70190478515625 x^{22} + 572844873046875 x^{20} - 3645376464843750 x^{18} + 17912625732421875 x^{16} - 66926293945312500 x^{14} + 185906372070312500 x^{12} - 371812744140625000 x^{10} + 511242523193359375 x^{8} - 451096343994140625 x^{6} + 228816986083984375 x^{4} - 54740905761718750 x^{2} + 3910064697265625 \)

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

Invariants

Degree:  $40$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[40, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(83017628936247865363646717578368932999206418219130479061598801598873600000000000000000000=2^{40}\cdot 5^{20}\cdot 41^{39}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $167.10$
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:  $2, 5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(820=2^{2}\cdot 5\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{820}(1,·)$, $\chi_{820}(259,·)$, $\chi_{820}(519,·)$, $\chi_{820}(781,·)$, $\chi_{820}(399,·)$, $\chi_{820}(401,·)$, $\chi_{820}(659,·)$, $\chi_{820}(21,·)$, $\chi_{820}(539,·)$, $\chi_{820}(541,·)$, $\chi_{820}(681,·)$, $\chi_{820}(299,·)$, $\chi_{820}(559,·)$, $\chi_{820}(179,·)$, $\chi_{820}(439,·)$, $\chi_{820}(441,·)$, $\chi_{820}(61,·)$, $\chi_{820}(701,·)$, $\chi_{820}(199,·)$, $\chi_{820}(201,·)$, $\chi_{820}(79,·)$, $\chi_{820}(499,·)$, $\chi_{820}(461,·)$, $\chi_{820}(141,·)$, $\chi_{820}(81,·)$, $\chi_{820}(339,·)$, $\chi_{820}(121,·)$, $\chi_{820}(219,·)$, $\chi_{820}(221,·)$, $\chi_{820}(719,·)$, $\chi_{820}(479,·)$, $\chi_{820}(99,·)$, $\chi_{820}(361,·)$, $\chi_{820}(239,·)$, $\chi_{820}(241,·)$, $\chi_{820}(19,·)$, $\chi_{820}(501,·)$, $\chi_{820}(761,·)$, $\chi_{820}(639,·)$, $\chi_{820}(661,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $\frac{1}{5} a^{2}$, $\frac{1}{5} a^{3}$, $\frac{1}{25} a^{4}$, $\frac{1}{25} a^{5}$, $\frac{1}{125} a^{6}$, $\frac{1}{125} a^{7}$, $\frac{1}{625} a^{8}$, $\frac{1}{625} a^{9}$, $\frac{1}{3125} a^{10}$, $\frac{1}{3125} a^{11}$, $\frac{1}{15625} a^{12}$, $\frac{1}{15625} a^{13}$, $\frac{1}{78125} a^{14}$, $\frac{1}{78125} a^{15}$, $\frac{1}{390625} a^{16}$, $\frac{1}{390625} a^{17}$, $\frac{1}{1953125} a^{18}$, $\frac{1}{1953125} a^{19}$, $\frac{1}{9765625} a^{20}$, $\frac{1}{9765625} a^{21}$, $\frac{1}{48828125} a^{22}$, $\frac{1}{48828125} a^{23}$, $\frac{1}{244140625} a^{24}$, $\frac{1}{244140625} a^{25}$, $\frac{1}{1220703125} a^{26}$, $\frac{1}{1220703125} a^{27}$, $\frac{1}{6103515625} a^{28}$, $\frac{1}{6103515625} a^{29}$, $\frac{1}{30517578125} a^{30}$, $\frac{1}{30517578125} a^{31}$, $\frac{1}{152587890625} a^{32}$, $\frac{1}{152587890625} a^{33}$, $\frac{1}{762939453125} a^{34}$, $\frac{1}{762939453125} a^{35}$, $\frac{1}{3814697265625} a^{36}$, $\frac{1}{3814697265625} a^{37}$, $\frac{1}{19073486328125} a^{38}$, $\frac{1}{19073486328125} a^{39}$

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:  $39$
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, 8.8.31160683820960000.1, 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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{5}$ R $40$ $40$ $40$ $40$ $40$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{4}$ $40$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{8}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{4}$ 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
5Data not computed
41Data not computed