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Properties

Label 40.0.24342527253...0000.1
Degree $40$
Signature $[0, 20]$
Discriminant $2^{80}\cdot 5^{20}\cdot 11^{32}$
Root discriminant $60.91$
Ramified primes $2, 5, 11$
Class number $24025$ (GRH)
Class group $[155, 155]$ (GRH)
Galois group $C_2^2\times C_{10}$ (as 40T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, 0, 1085, 0, 0, 0, 42966, 0, 0, 0, 496503, 0, 0, 0, 1558024, 0, 0, 0, 1781941, 0, 0, 0, 849466, 0, 0, 0, 161966, 0, 0, 0, 9273, 0, 0, 0, 175, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 + 175*x^36 + 9273*x^32 + 161966*x^28 + 849466*x^24 + 1781941*x^20 + 1558024*x^16 + 496503*x^12 + 42966*x^8 + 1085*x^4 + 1)
 
gp: K = bnfinit(x^40 + 175*x^36 + 9273*x^32 + 161966*x^28 + 849466*x^24 + 1781941*x^20 + 1558024*x^16 + 496503*x^12 + 42966*x^8 + 1085*x^4 + 1, 1)
 

Normalized defining polynomial

\( x^{40} + 175 x^{36} + 9273 x^{32} + 161966 x^{28} + 849466 x^{24} + 1781941 x^{20} + 1558024 x^{16} + 496503 x^{12} + 42966 x^{8} + 1085 x^{4} + 1 \)

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:  \(243425272531849218182083207491041757811794582804889600000000000000000000=2^{80}\cdot 5^{20}\cdot 11^{32}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.91$
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:  $2, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(440=2^{3}\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{440}(1,·)$, $\chi_{440}(389,·)$, $\chi_{440}(9,·)$, $\chi_{440}(269,·)$, $\chi_{440}(399,·)$, $\chi_{440}(401,·)$, $\chi_{440}(279,·)$, $\chi_{440}(411,·)$, $\chi_{440}(31,·)$, $\chi_{440}(289,·)$, $\chi_{440}(291,·)$, $\chi_{440}(421,·)$, $\chi_{440}(169,·)$, $\chi_{440}(71,·)$, $\chi_{440}(301,·)$, $\chi_{440}(49,·)$, $\chi_{440}(179,·)$, $\chi_{440}(309,·)$, $\chi_{440}(311,·)$, $\chi_{440}(159,·)$, $\chi_{440}(181,·)$, $\chi_{440}(69,·)$, $\chi_{440}(199,·)$, $\chi_{440}(201,·)$, $\chi_{440}(331,·)$, $\chi_{440}(141,·)$, $\chi_{440}(81,·)$, $\chi_{440}(339,·)$, $\chi_{440}(89,·)$, $\chi_{440}(91,·)$, $\chi_{440}(221,·)$, $\chi_{440}(419,·)$, $\chi_{440}(59,·)$, $\chi_{440}(229,·)$, $\chi_{440}(379,·)$, $\chi_{440}(361,·)$, $\chi_{440}(191,·)$, $\chi_{440}(111,·)$, $\chi_{440}(119,·)$, $\chi_{440}(251,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{3} a^{20} - \frac{1}{3} a^{16} + \frac{1}{3} a^{12} - \frac{1}{3} a^{8} - \frac{1}{3} a^{4} - \frac{1}{3}$, $\frac{1}{3} a^{21} - \frac{1}{3} a^{17} + \frac{1}{3} a^{13} - \frac{1}{3} a^{9} - \frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{3} a^{22} - \frac{1}{3} a^{18} + \frac{1}{3} a^{14} - \frac{1}{3} a^{10} - \frac{1}{3} a^{6} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{23} - \frac{1}{3} a^{19} + \frac{1}{3} a^{15} - \frac{1}{3} a^{11} - \frac{1}{3} a^{7} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{24} + \frac{1}{3} a^{8} + \frac{1}{3} a^{4} - \frac{1}{3}$, $\frac{1}{3} a^{25} + \frac{1}{3} a^{9} + \frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{3} a^{26} + \frac{1}{3} a^{10} + \frac{1}{3} a^{6} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{27} + \frac{1}{3} a^{11} + \frac{1}{3} a^{7} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{28} + \frac{1}{3} a^{12} + \frac{1}{3} a^{8} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{29} + \frac{1}{3} a^{13} + \frac{1}{3} a^{9} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{30} + \frac{1}{3} a^{14} + \frac{1}{3} a^{10} - \frac{1}{3} a^{6}$, $\frac{1}{3} a^{31} + \frac{1}{3} a^{15} + \frac{1}{3} a^{11} - \frac{1}{3} a^{7}$, $\frac{1}{269679} a^{32} - \frac{2840}{89893} a^{28} - \frac{36257}{269679} a^{24} - \frac{1414}{89893} a^{20} - \frac{115250}{269679} a^{16} - \frac{130310}{269679} a^{12} - \frac{7664}{89893} a^{8} + \frac{17023}{269679} a^{4} - \frac{56974}{269679}$, $\frac{1}{269679} a^{33} - \frac{2840}{89893} a^{29} - \frac{36257}{269679} a^{25} - \frac{1414}{89893} a^{21} - \frac{115250}{269679} a^{17} - \frac{130310}{269679} a^{13} - \frac{7664}{89893} a^{9} + \frac{17023}{269679} a^{5} - \frac{56974}{269679} a$, $\frac{1}{269679} a^{34} - \frac{2840}{89893} a^{30} - \frac{36257}{269679} a^{26} - \frac{1414}{89893} a^{22} - \frac{115250}{269679} a^{18} - \frac{130310}{269679} a^{14} - \frac{7664}{89893} a^{10} + \frac{17023}{269679} a^{6} - \frac{56974}{269679} a^{2}$, $\frac{1}{269679} a^{35} - \frac{2840}{89893} a^{31} - \frac{36257}{269679} a^{27} - \frac{1414}{89893} a^{23} - \frac{115250}{269679} a^{19} - \frac{130310}{269679} a^{15} - \frac{7664}{89893} a^{11} + \frac{17023}{269679} a^{7} - \frac{56974}{269679} a^{3}$, $\frac{1}{12333107020927611587963781} a^{36} + \frac{16392002085665156992}{12333107020927611587963781} a^{32} - \frac{926062360534987103046416}{12333107020927611587963781} a^{28} + \frac{1782878011745516536923664}{12333107020927611587963781} a^{24} + \frac{1216508152595936505528220}{12333107020927611587963781} a^{20} - \frac{3925411092666018839236006}{12333107020927611587963781} a^{16} + \frac{1505777891049793142796187}{12333107020927611587963781} a^{12} + \frac{5704433496969442786325374}{12333107020927611587963781} a^{8} + \frac{869477124971120933764461}{4111035673642537195987927} a^{4} + \frac{2855180718947302234205636}{12333107020927611587963781}$, $\frac{1}{12333107020927611587963781} a^{37} + \frac{16392002085665156992}{12333107020927611587963781} a^{33} - \frac{926062360534987103046416}{12333107020927611587963781} a^{29} + \frac{1782878011745516536923664}{12333107020927611587963781} a^{25} + \frac{1216508152595936505528220}{12333107020927611587963781} a^{21} - \frac{3925411092666018839236006}{12333107020927611587963781} a^{17} + \frac{1505777891049793142796187}{12333107020927611587963781} a^{13} + \frac{5704433496969442786325374}{12333107020927611587963781} a^{9} + \frac{869477124971120933764461}{4111035673642537195987927} a^{5} + \frac{2855180718947302234205636}{12333107020927611587963781} a$, $\frac{1}{12333107020927611587963781} a^{38} + \frac{16392002085665156992}{12333107020927611587963781} a^{34} - \frac{926062360534987103046416}{12333107020927611587963781} a^{30} + \frac{1782878011745516536923664}{12333107020927611587963781} a^{26} + \frac{1216508152595936505528220}{12333107020927611587963781} a^{22} - \frac{3925411092666018839236006}{12333107020927611587963781} a^{18} + \frac{1505777891049793142796187}{12333107020927611587963781} a^{14} + \frac{5704433496969442786325374}{12333107020927611587963781} a^{10} + \frac{869477124971120933764461}{4111035673642537195987927} a^{6} + \frac{2855180718947302234205636}{12333107020927611587963781} a^{2}$, $\frac{1}{12333107020927611587963781} a^{39} + \frac{16392002085665156992}{12333107020927611587963781} a^{35} - \frac{926062360534987103046416}{12333107020927611587963781} a^{31} + \frac{1782878011745516536923664}{12333107020927611587963781} a^{27} + \frac{1216508152595936505528220}{12333107020927611587963781} a^{23} - \frac{3925411092666018839236006}{12333107020927611587963781} a^{19} + \frac{1505777891049793142796187}{12333107020927611587963781} a^{15} + \frac{5704433496969442786325374}{12333107020927611587963781} a^{11} + \frac{869477124971120933764461}{4111035673642537195987927} a^{7} + \frac{2855180718947302234205636}{12333107020927611587963781} a^{3}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{155}\times C_{155}$, which has order $24025$ (assuming GRH)

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:  \( \frac{305129630405970959162}{37260142057183116579951} a^{37} + \frac{17799580877135952392828}{12420047352394372193317} a^{33} + \frac{2829650962622683518719896}{37260142057183116579951} a^{29} + \frac{16476744417685704651858369}{12420047352394372193317} a^{25} + \frac{86452717278717868054754281}{12420047352394372193317} a^{21} + \frac{544445021764559724516187729}{37260142057183116579951} a^{17} + \frac{158812680314314296283165743}{12420047352394372193317} a^{13} + \frac{50549700223384721143289054}{12420047352394372193317} a^{9} + \frac{12680131192389703024317364}{37260142057183116579951} a^{5} + \frac{223816153561166017405595}{37260142057183116579951} a \) (order $8$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3147005109105615.0 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_{10}$ (as 40T7):

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 $C_2^2\times C_{10}$
Character table for $C_2^2\times C_{10}$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{10}) \), \(\Q(i, \sqrt{5})\), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{10})\), \(\Q(\sqrt{2}, \sqrt{-5})\), \(\Q(\sqrt{-2}, \sqrt{-5})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(\zeta_{11})^+\), 8.0.40960000.1, 10.0.219503494144.1, 10.0.685948419200000.1, 10.10.669871503125.1, 10.10.7024111812608.1, 10.0.7024111812608.1, 10.0.21950349414400000.4, 10.10.21950349414400000.1, 20.0.470525233802978928640000000000.1, 20.0.50522262278163705147147943936.1, 20.0.493381467560192433077616640000000000.6, 20.0.493381467560192433077616640000000000.5, 20.0.493381467560192433077616640000000000.1, 20.20.481817839414250422927360000000000.1, 20.0.481817839414250422927360000000000.3

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{4}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$