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Properties

Label 40.0.13030509980...5936.8
Degree $40$
Signature $[0, 20]$
Discriminant $2^{80}\cdot 3^{20}\cdot 11^{36}$
Root discriminant $59.96$
Ramified primes $2, 3, 11$
Class number $2728$ (GRH)
Class group $[2728]$ (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![14641, 0, 0, 0, -161051, 0, 0, 0, 1610510, 0, 0, 0, -1686377, 0, 0, 0, 1299056, 0, 0, 0, -380787, 0, 0, 0, 78650, 0, 0, 0, -8954, 0, 0, 0, 737, 0, 0, 0, -33, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - 33*x^36 + 737*x^32 - 8954*x^28 + 78650*x^24 - 380787*x^20 + 1299056*x^16 - 1686377*x^12 + 1610510*x^8 - 161051*x^4 + 14641)
 
gp: K = bnfinit(x^40 - 33*x^36 + 737*x^32 - 8954*x^28 + 78650*x^24 - 380787*x^20 + 1299056*x^16 - 1686377*x^12 + 1610510*x^8 - 161051*x^4 + 14641, 1)
 

Normalized defining polynomial

\( x^{40} - 33 x^{36} + 737 x^{32} - 8954 x^{28} + 78650 x^{24} - 380787 x^{20} + 1299056 x^{16} - 1686377 x^{12} + 1610510 x^{8} - 161051 x^{4} + 14641 \)

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:  \(130305099804548492884220428175380349368393046678311823693003457545895936=2^{80}\cdot 3^{20}\cdot 11^{36}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.96$
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:  $2, 3, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(264=2^{3}\cdot 3\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{264}(1,·)$, $\chi_{264}(131,·)$, $\chi_{264}(257,·)$, $\chi_{264}(137,·)$, $\chi_{264}(139,·)$, $\chi_{264}(13,·)$, $\chi_{264}(19,·)$, $\chi_{264}(149,·)$, $\chi_{264}(23,·)$, $\chi_{264}(25,·)$, $\chi_{264}(29,·)$, $\chi_{264}(31,·)$, $\chi_{264}(35,·)$, $\chi_{264}(169,·)$, $\chi_{264}(43,·)$, $\chi_{264}(173,·)$, $\chi_{264}(47,·)$, $\chi_{264}(49,·)$, $\chi_{264}(185,·)$, $\chi_{264}(61,·)$, $\chi_{264}(191,·)$, $\chi_{264}(197,·)$, $\chi_{264}(71,·)$, $\chi_{264}(119,·)$, $\chi_{264}(205,·)$, $\chi_{264}(83,·)$, $\chi_{264}(85,·)$, $\chi_{264}(89,·)$, $\chi_{264}(199,·)$, $\chi_{264}(223,·)$, $\chi_{264}(97,·)$, $\chi_{264}(227,·)$, $\chi_{264}(101,·)$, $\chi_{264}(103,·)$, $\chi_{264}(259,·)$, $\chi_{264}(107,·)$, $\chi_{264}(109,·)$, $\chi_{264}(113,·)$, $\chi_{264}(211,·)$, $\chi_{264}(247,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{11} a^{10}$, $\frac{1}{11} a^{11}$, $\frac{1}{11} a^{12}$, $\frac{1}{11} a^{13}$, $\frac{1}{11} a^{14}$, $\frac{1}{11} a^{15}$, $\frac{1}{11} a^{16}$, $\frac{1}{11} a^{17}$, $\frac{1}{11} a^{18}$, $\frac{1}{11} a^{19}$, $\frac{1}{121} a^{20}$, $\frac{1}{121} a^{21}$, $\frac{1}{121} a^{22}$, $\frac{1}{121} a^{23}$, $\frac{1}{121} a^{24}$, $\frac{1}{121} a^{25}$, $\frac{1}{121} a^{26}$, $\frac{1}{121} a^{27}$, $\frac{1}{13189} a^{28} + \frac{43}{13189} a^{24} + \frac{24}{13189} a^{20} + \frac{53}{1199} a^{16} - \frac{48}{1199} a^{12} - \frac{23}{109} a^{8} - \frac{27}{109} a^{4} + \frac{48}{109}$, $\frac{1}{13189} a^{29} + \frac{43}{13189} a^{25} + \frac{24}{13189} a^{21} + \frac{53}{1199} a^{17} - \frac{48}{1199} a^{13} - \frac{23}{109} a^{9} - \frac{27}{109} a^{5} + \frac{48}{109} a$, $\frac{1}{145079} a^{30} - \frac{6}{13189} a^{26} + \frac{2}{1199} a^{22} - \frac{15}{1199} a^{18} - \frac{4}{109} a^{14} - \frac{23}{1199} a^{10} - \frac{52}{109} a^{6} + \frac{44}{109} a^{2}$, $\frac{1}{145079} a^{31} - \frac{6}{13189} a^{27} + \frac{2}{1199} a^{23} - \frac{15}{1199} a^{19} - \frac{4}{109} a^{15} - \frac{23}{1199} a^{11} - \frac{52}{109} a^{7} + \frac{44}{109} a^{3}$, $\frac{1}{555217333} a^{32} - \frac{1271}{50474303} a^{28} - \frac{17914}{4588573} a^{24} - \frac{193916}{50474303} a^{20} - \frac{34607}{4588573} a^{16} - \frac{28901}{4588573} a^{12} + \frac{174667}{417143} a^{8} + \frac{1499}{417143} a^{4} + \frac{32802}{417143}$, $\frac{1}{555217333} a^{33} - \frac{1271}{50474303} a^{29} - \frac{17914}{4588573} a^{25} - \frac{193916}{50474303} a^{21} - \frac{34607}{4588573} a^{17} - \frac{28901}{4588573} a^{13} + \frac{174667}{417143} a^{9} + \frac{1499}{417143} a^{5} + \frac{32802}{417143} a$, $\frac{1}{555217333} a^{34} + \frac{1327}{555217333} a^{30} + \frac{128241}{50474303} a^{26} + \frac{142860}{50474303} a^{22} + \frac{152916}{4588573} a^{18} + \frac{131833}{4588573} a^{14} - \frac{9029}{417143} a^{10} + \frac{39769}{417143} a^{6} - \frac{127932}{417143} a^{2}$, $\frac{1}{555217333} a^{35} + \frac{1327}{555217333} a^{31} + \frac{128241}{50474303} a^{27} + \frac{142860}{50474303} a^{23} + \frac{152916}{4588573} a^{19} + \frac{131833}{4588573} a^{15} - \frac{9029}{417143} a^{11} + \frac{39769}{417143} a^{7} - \frac{127932}{417143} a^{3}$, $\frac{1}{977195276078659} a^{36} + \frac{15820}{977195276078659} a^{32} + \frac{652247789}{88835934188969} a^{28} - \frac{49834937375}{88835934188969} a^{24} + \frac{326488951231}{88835934188969} a^{20} - \frac{330541501756}{8075994017179} a^{16} - \frac{314178082332}{8075994017179} a^{12} + \frac{26337609990}{734181274289} a^{8} - \frac{181590528712}{734181274289} a^{4} - \frac{326842105055}{734181274289}$, $\frac{1}{977195276078659} a^{37} + \frac{15820}{977195276078659} a^{33} + \frac{652247789}{88835934188969} a^{29} - \frac{49834937375}{88835934188969} a^{25} + \frac{326488951231}{88835934188969} a^{21} - \frac{330541501756}{8075994017179} a^{17} - \frac{314178082332}{8075994017179} a^{13} + \frac{26337609990}{734181274289} a^{9} - \frac{181590528712}{734181274289} a^{5} - \frac{326842105055}{734181274289} a$, $\frac{1}{977195276078659} a^{38} + \frac{15820}{977195276078659} a^{34} + \frac{439117658}{977195276078659} a^{30} - \frac{9421289249}{88835934188969} a^{26} + \frac{178305574769}{88835934188969} a^{22} - \frac{229507381441}{8075994017179} a^{18} - \frac{17811329408}{8075994017179} a^{14} - \frac{289548579916}{8075994017179} a^{10} + \frac{168661088380}{734181274289} a^{6} + \frac{110972416310}{734181274289} a^{2}$, $\frac{1}{977195276078659} a^{39} + \frac{15820}{977195276078659} a^{35} + \frac{439117658}{977195276078659} a^{31} - \frac{9421289249}{88835934188969} a^{27} + \frac{178305574769}{88835934188969} a^{23} - \frac{229507381441}{8075994017179} a^{19} - \frac{17811329408}{8075994017179} a^{15} - \frac{289548579916}{8075994017179} a^{11} + \frac{168661088380}{734181274289} a^{7} + \frac{110972416310}{734181274289} a^{3}$

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

Class group and class number

$C_{2728}$, which has order $2728$ (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:  \( -\frac{1720057043}{977195276078659} a^{38} + \frac{5002106944}{88835934188969} a^{34} - \frac{1210666196290}{977195276078659} a^{30} + \frac{1284644679822}{88835934188969} a^{26} - \frac{10905062085446}{88835934188969} a^{22} + \frac{4305094604522}{8075994017179} a^{18} - \frac{13178268444118}{8075994017179} a^{14} + \frac{5995963067436}{8075994017179} a^{10} - \frac{54554228858}{734181274289} a^{6} - \frac{2308120940327}{734181274289} a^{2} \) (order $12$)
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:  \( 24039412784024376 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{22}) \), \(\Q(\sqrt{-22}) \), \(\Q(\sqrt{-66}) \), \(\Q(\sqrt{66}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(i, \sqrt{22})\), \(\Q(i, \sqrt{66})\), \(\Q(\zeta_{12})\), \(\Q(\sqrt{-3}, \sqrt{22})\), \(\Q(\sqrt{3}, \sqrt{22})\), \(\Q(\sqrt{3}, \sqrt{-22})\), \(\Q(\sqrt{-3}, \sqrt{-22})\), \(\Q(\zeta_{11})^+\), 8.0.77720518656.9, 10.0.219503494144.1, 10.10.77265229938688.1, 10.0.77265229938688.1, 10.0.18775450875101184.1, 10.10.18775450875101184.1, 10.0.52089208083.1, 10.10.53339349076992.1, 20.0.6113193735657808322804901216256.4, 20.0.360977976896857923653306611918700544.10, 20.0.2845086159957207322343768064.1, 20.0.352517555563337816067682238201856.5, 20.20.360977976896857923653306611918700544.1, 20.0.360977976896857923653306611918700544.1, 20.0.352517555563337816067682238201856.4

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 ${\href{/LocalNumberField/5.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{4}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{8}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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
3Data not computed
11Data not computed