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Properties

Label 40.0.30352560470...8336.1
Degree $40$
Signature $[0, 20]$
Discriminant $2^{40}\cdot 3^{20}\cdot 41^{39}$
Root discriminant $129.44$
Ramified primes $2, 3, 41$
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![142958160441, 0, 3335690410290, 0, 23238643191687, 0, 76355541915543, 0, 144227134729359, 0, 174820769368920, 0, 145683974474100, 0, 87410384684460, 0, 38991887285715, 0, 13225318494570, 0, 3463773891435, 0, 707358304575, 0, 113177328732, 0, 14187471408, 0, 1386132264, 0, 104332536, 0, 5927985, 0, 245754, 0, 7011, 0, 123, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 + 123*x^38 + 7011*x^36 + 245754*x^34 + 5927985*x^32 + 104332536*x^30 + 1386132264*x^28 + 14187471408*x^26 + 113177328732*x^24 + 707358304575*x^22 + 3463773891435*x^20 + 13225318494570*x^18 + 38991887285715*x^16 + 87410384684460*x^14 + 145683974474100*x^12 + 174820769368920*x^10 + 144227134729359*x^8 + 76355541915543*x^6 + 23238643191687*x^4 + 3335690410290*x^2 + 142958160441)
 
gp: K = bnfinit(x^40 + 123*x^38 + 7011*x^36 + 245754*x^34 + 5927985*x^32 + 104332536*x^30 + 1386132264*x^28 + 14187471408*x^26 + 113177328732*x^24 + 707358304575*x^22 + 3463773891435*x^20 + 13225318494570*x^18 + 38991887285715*x^16 + 87410384684460*x^14 + 145683974474100*x^12 + 174820769368920*x^10 + 144227134729359*x^8 + 76355541915543*x^6 + 23238643191687*x^4 + 3335690410290*x^2 + 142958160441, 1)
 

Normalized defining polynomial

\( x^{40} + 123 x^{38} + 7011 x^{36} + 245754 x^{34} + 5927985 x^{32} + 104332536 x^{30} + 1386132264 x^{28} + 14187471408 x^{26} + 113177328732 x^{24} + 707358304575 x^{22} + 3463773891435 x^{20} + 13225318494570 x^{18} + 38991887285715 x^{16} + 87410384684460 x^{14} + 145683974474100 x^{12} + 174820769368920 x^{10} + 144227134729359 x^{8} + 76355541915543 x^{6} + 23238643191687 x^{4} + 3335690410290 x^{2} + 142958160441 \)

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:  \(3035256047092789730811697612215909515050584211986016938760756994089904532626746638336=2^{40}\cdot 3^{20}\cdot 41^{39}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $129.44$
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:  $2, 3, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(492=2^{2}\cdot 3\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{492}(1,·)$, $\chi_{492}(133,·)$, $\chi_{492}(263,·)$, $\chi_{492}(11,·)$, $\chi_{492}(275,·)$, $\chi_{492}(277,·)$, $\chi_{492}(407,·)$, $\chi_{492}(409,·)$, $\chi_{492}(289,·)$, $\chi_{492}(35,·)$, $\chi_{492}(37,·)$, $\chi_{492}(167,·)$, $\chi_{492}(169,·)$, $\chi_{492}(299,·)$, $\chi_{492}(47,·)$, $\chi_{492}(385,·)$, $\chi_{492}(433,·)$, $\chi_{492}(179,·)$, $\chi_{492}(311,·)$, $\chi_{492}(95,·)$, $\chi_{492}(61,·)$, $\chi_{492}(191,·)$, $\chi_{492}(395,·)$, $\chi_{492}(49,·)$, $\chi_{492}(71,·)$, $\chi_{492}(73,·)$, $\chi_{492}(335,·)$, $\chi_{492}(337,·)$, $\chi_{492}(469,·)$, $\chi_{492}(347,·)$, $\chi_{492}(349,·)$, $\chi_{492}(479,·)$, $\chi_{492}(227,·)$, $\chi_{492}(361,·)$, $\chi_{492}(239,·)$, $\chi_{492}(241,·)$, $\chi_{492}(25,·)$, $\chi_{492}(373,·)$, $\chi_{492}(121,·)$, $\chi_{492}(383,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{9} a^{4}$, $\frac{1}{9} a^{5}$, $\frac{1}{27} a^{6}$, $\frac{1}{27} a^{7}$, $\frac{1}{81} a^{8}$, $\frac{1}{81} a^{9}$, $\frac{1}{243} a^{10}$, $\frac{1}{243} a^{11}$, $\frac{1}{729} a^{12}$, $\frac{1}{729} a^{13}$, $\frac{1}{2187} a^{14}$, $\frac{1}{2187} a^{15}$, $\frac{1}{6561} a^{16}$, $\frac{1}{6561} a^{17}$, $\frac{1}{19683} a^{18}$, $\frac{1}{19683} a^{19}$, $\frac{1}{59049} a^{20}$, $\frac{1}{59049} a^{21}$, $\frac{1}{177147} a^{22}$, $\frac{1}{177147} a^{23}$, $\frac{1}{531441} a^{24}$, $\frac{1}{531441} a^{25}$, $\frac{1}{1594323} a^{26}$, $\frac{1}{1594323} a^{27}$, $\frac{1}{4782969} a^{28}$, $\frac{1}{4782969} a^{29}$, $\frac{1}{14348907} a^{30}$, $\frac{1}{14348907} a^{31}$, $\frac{1}{43046721} a^{32}$, $\frac{1}{43046721} a^{33}$, $\frac{1}{129140163} a^{34}$, $\frac{1}{129140163} a^{35}$, $\frac{1}{387420489} a^{36}$, $\frac{1}{387420489} a^{37}$, $\frac{1}{1162261467} a^{38}$, $\frac{1}{1162261467} a^{39}$

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, 8.0.4038424623196416.1, 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 R $20^{2}$ $40$ $40$ $40$ $40$ $40$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{4}$ $40$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{8}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{8}$ R $20^{2}$ $40$ $40$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{8}$

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
$3$3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
41Data not computed