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Properties

Label 40.40.3182696644...9536.1
Degree $40$
Signature $[40, 0]$
Discriminant $2^{60}\cdot 3^{20}\cdot 41^{39}$
Root discriminant $183.05$
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![149902496042582016, 0, -1748862453830123520, 0, 6091870880841596928, 0, -10008073589954052096, 0, 9452069501623271424, 0, -5728526970680770560, 0, 2386886237783654400, 0, -716065871335096320, 0, 159710770322288640, 0, -27085452276879360, 0, 3546904464829440, 0, -362167451942400, 0, 28973396155392, 0, -1815996340224, 0, 88712464896, 0, -3338641152, 0, 94847760, 0, -1966032, 0, 28044, 0, -246, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - 246*x^38 + 28044*x^36 - 1966032*x^34 + 94847760*x^32 - 3338641152*x^30 + 88712464896*x^28 - 1815996340224*x^26 + 28973396155392*x^24 - 362167451942400*x^22 + 3546904464829440*x^20 - 27085452276879360*x^18 + 159710770322288640*x^16 - 716065871335096320*x^14 + 2386886237783654400*x^12 - 5728526970680770560*x^10 + 9452069501623271424*x^8 - 10008073589954052096*x^6 + 6091870880841596928*x^4 - 1748862453830123520*x^2 + 149902496042582016)
 
gp: K = bnfinit(x^40 - 246*x^38 + 28044*x^36 - 1966032*x^34 + 94847760*x^32 - 3338641152*x^30 + 88712464896*x^28 - 1815996340224*x^26 + 28973396155392*x^24 - 362167451942400*x^22 + 3546904464829440*x^20 - 27085452276879360*x^18 + 159710770322288640*x^16 - 716065871335096320*x^14 + 2386886237783654400*x^12 - 5728526970680770560*x^10 + 9452069501623271424*x^8 - 10008073589954052096*x^6 + 6091870880841596928*x^4 - 1748862453830123520*x^2 + 149902496042582016, 1)
 

Normalized defining polynomial

\( x^{40} - 246 x^{38} + 28044 x^{36} - 1966032 x^{34} + 94847760 x^{32} - 3338641152 x^{30} + 88712464896 x^{28} - 1815996340224 x^{26} + 28973396155392 x^{24} - 362167451942400 x^{22} + 3546904464829440 x^{20} - 27085452276879360 x^{18} + 159710770322288640 x^{16} - 716065871335096320 x^{14} + 2386886237783654400 x^{12} - 5728526970680770560 x^{10} + 9452069501623271424 x^{8} - 10008073589954052096 x^{6} + 6091870880841596928 x^{4} - 1748862453830123520 x^{2} + 149902496042582016 \)

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:  \(3182696644836369084775606635426909535653681390667449697577999525834815735203623483039809536=2^{60}\cdot 3^{20}\cdot 41^{39}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $183.05$
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:  \(984=2^{3}\cdot 3\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{984}(149,·)$, $\chi_{984}(1,·)$, $\chi_{984}(773,·)$, $\chi_{984}(769,·)$, $\chi_{984}(653,·)$, $\chi_{984}(529,·)$, $\chi_{984}(917,·)$, $\chi_{984}(25,·)$, $\chi_{984}(409,·)$, $\chi_{984}(29,·)$, $\chi_{984}(581,·)$, $\chi_{984}(289,·)$, $\chi_{984}(293,·)$, $\chi_{984}(49,·)$, $\chi_{984}(553,·)$, $\chi_{984}(557,·)$, $\chi_{984}(413,·)$, $\chi_{984}(385,·)$, $\chi_{984}(433,·)$, $\chi_{984}(53,·)$, $\chi_{984}(841,·)$, $\chi_{984}(317,·)$, $\chi_{984}(437,·)$, $\chi_{984}(961,·)$, $\chi_{984}(965,·)$, $\chi_{984}(73,·)$, $\chi_{984}(337,·)$, $\chi_{984}(725,·)$, $\chi_{984}(625,·)$, $\chi_{984}(101,·)$, $\chi_{984}(865,·)$, $\chi_{984}(485,·)$, $\chi_{984}(361,·)$, $\chi_{984}(749,·)$, $\chi_{984}(241,·)$, $\chi_{984}(629,·)$, $\chi_{984}(169,·)$, $\chi_{984}(121,·)$, $\chi_{984}(509,·)$, $\chi_{984}(341,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $\frac{1}{6} a^{2}$, $\frac{1}{6} a^{3}$, $\frac{1}{36} a^{4}$, $\frac{1}{36} a^{5}$, $\frac{1}{216} a^{6}$, $\frac{1}{216} a^{7}$, $\frac{1}{1296} a^{8}$, $\frac{1}{1296} a^{9}$, $\frac{1}{7776} a^{10}$, $\frac{1}{7776} a^{11}$, $\frac{1}{46656} a^{12}$, $\frac{1}{46656} a^{13}$, $\frac{1}{279936} a^{14}$, $\frac{1}{279936} a^{15}$, $\frac{1}{1679616} a^{16}$, $\frac{1}{1679616} a^{17}$, $\frac{1}{10077696} a^{18}$, $\frac{1}{10077696} a^{19}$, $\frac{1}{60466176} a^{20}$, $\frac{1}{60466176} a^{21}$, $\frac{1}{362797056} a^{22}$, $\frac{1}{362797056} a^{23}$, $\frac{1}{2176782336} a^{24}$, $\frac{1}{2176782336} a^{25}$, $\frac{1}{13060694016} a^{26}$, $\frac{1}{13060694016} a^{27}$, $\frac{1}{78364164096} a^{28}$, $\frac{1}{78364164096} a^{29}$, $\frac{1}{470184984576} a^{30}$, $\frac{1}{470184984576} a^{31}$, $\frac{1}{2821109907456} a^{32}$, $\frac{1}{2821109907456} a^{33}$, $\frac{1}{16926659444736} a^{34}$, $\frac{1}{16926659444736} a^{35}$, $\frac{1}{101559956668416} a^{36}$, $\frac{1}{101559956668416} a^{37}$, $\frac{1}{609359740010496} a^{38}$, $\frac{1}{609359740010496} a^{39}$

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, 8.8.64614793971142656.1, 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 $20^{2}$ $40$ $40$ $40$ $40$ $40$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{8}$ $40$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{4}$ R $20^{2}$ $40$ $40$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{8}$

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