↑

Properties

Label 40.0.10279259898...0625.1
Degree $40$
Signature $[0, 20]$
Discriminant $3^{20}\cdot 5^{20}\cdot 11^{36}$
Root discriminant $33.52$
Ramified primes $3, 5, 11$
Class number $22$ (GRH)
Class group $[22]$ (GRH)
Galois group $C_2^2\times C_{10}$ (as 40T7)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 1, -1, -4, -4, 4, 17, 17, -17, -72, -72, -127, 106, 504, 491, -496, -2088, -2091, 2090, 8856, 8855, -8856, 2090, 2091, -2088, 496, 491, -504, 106, 127, -72, 72, -17, -17, 17, -4, -4, 4, -1, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - x^39 - x^38 + 4*x^37 - 4*x^36 - 4*x^35 + 17*x^34 - 17*x^33 - 17*x^32 + 72*x^31 - 72*x^30 + 127*x^29 + 106*x^28 - 504*x^27 + 491*x^26 + 496*x^25 - 2088*x^24 + 2091*x^23 + 2090*x^22 - 8856*x^21 + 8855*x^20 + 8856*x^19 + 2090*x^18 - 2091*x^17 - 2088*x^16 - 496*x^15 + 491*x^14 + 504*x^13 + 106*x^12 - 127*x^11 - 72*x^10 - 72*x^9 - 17*x^8 + 17*x^7 + 17*x^6 + 4*x^5 - 4*x^4 - 4*x^3 - x^2 + x + 1)
 
gp: K = bnfinit(x^40 - x^39 - x^38 + 4*x^37 - 4*x^36 - 4*x^35 + 17*x^34 - 17*x^33 - 17*x^32 + 72*x^31 - 72*x^30 + 127*x^29 + 106*x^28 - 504*x^27 + 491*x^26 + 496*x^25 - 2088*x^24 + 2091*x^23 + 2090*x^22 - 8856*x^21 + 8855*x^20 + 8856*x^19 + 2090*x^18 - 2091*x^17 - 2088*x^16 - 496*x^15 + 491*x^14 + 504*x^13 + 106*x^12 - 127*x^11 - 72*x^10 - 72*x^9 - 17*x^8 + 17*x^7 + 17*x^6 + 4*x^5 - 4*x^4 - 4*x^3 - x^2 + x + 1, 1)
 

Normalized defining polynomial

\( x^{40} - x^{39} - x^{38} + 4 x^{37} - 4 x^{36} - 4 x^{35} + 17 x^{34} - 17 x^{33} - 17 x^{32} + 72 x^{31} - 72 x^{30} + 127 x^{29} + 106 x^{28} - 504 x^{27} + 491 x^{26} + 496 x^{25} - 2088 x^{24} + 2091 x^{23} + 2090 x^{22} - 8856 x^{21} + 8855 x^{20} + 8856 x^{19} + 2090 x^{18} - 2091 x^{17} - 2088 x^{16} - 496 x^{15} + 491 x^{14} + 504 x^{13} + 106 x^{12} - 127 x^{11} - 72 x^{10} - 72 x^{9} - 17 x^{8} + 17 x^{7} + 17 x^{6} + 4 x^{5} - 4 x^{4} - 4 x^{3} - x^{2} + x + 1 \)

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:  \(10279259898673041257519092860384627329993230987644195556640625=3^{20}\cdot 5^{20}\cdot 11^{36}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.52$
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:  $3, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(165=3\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{165}(1,·)$, $\chi_{165}(131,·)$, $\chi_{165}(4,·)$, $\chi_{165}(134,·)$, $\chi_{165}(136,·)$, $\chi_{165}(139,·)$, $\chi_{165}(14,·)$, $\chi_{165}(16,·)$, $\chi_{165}(146,·)$, $\chi_{165}(19,·)$, $\chi_{165}(149,·)$, $\chi_{165}(151,·)$, $\chi_{165}(26,·)$, $\chi_{165}(29,·)$, $\chi_{165}(31,·)$, $\chi_{165}(161,·)$, $\chi_{165}(34,·)$, $\chi_{165}(164,·)$, $\chi_{165}(41,·)$, $\chi_{165}(46,·)$, $\chi_{165}(49,·)$, $\chi_{165}(56,·)$, $\chi_{165}(59,·)$, $\chi_{165}(61,·)$, $\chi_{165}(64,·)$, $\chi_{165}(71,·)$, $\chi_{165}(74,·)$, $\chi_{165}(76,·)$, $\chi_{165}(79,·)$, $\chi_{165}(86,·)$, $\chi_{165}(89,·)$, $\chi_{165}(91,·)$, $\chi_{165}(94,·)$, $\chi_{165}(101,·)$, $\chi_{165}(104,·)$, $\chi_{165}(106,·)$, $\chi_{165}(109,·)$, $\chi_{165}(116,·)$, $\chi_{165}(119,·)$, $\chi_{165}(124,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{89} a^{22} - \frac{34}{89} a^{11} - \frac{1}{89}$, $\frac{1}{89} a^{23} - \frac{34}{89} a^{12} - \frac{1}{89} a$, $\frac{1}{89} a^{24} - \frac{34}{89} a^{13} - \frac{1}{89} a^{2}$, $\frac{1}{89} a^{25} - \frac{34}{89} a^{14} - \frac{1}{89} a^{3}$, $\frac{1}{89} a^{26} - \frac{34}{89} a^{15} - \frac{1}{89} a^{4}$, $\frac{1}{89} a^{27} - \frac{34}{89} a^{16} - \frac{1}{89} a^{5}$, $\frac{1}{89} a^{28} - \frac{34}{89} a^{17} - \frac{1}{89} a^{6}$, $\frac{1}{89} a^{29} - \frac{34}{89} a^{18} - \frac{1}{89} a^{7}$, $\frac{1}{178} a^{30} - \frac{1}{178} a^{27} - \frac{1}{178} a^{24} - \frac{1}{2} a^{21} - \frac{17}{89} a^{19} - \frac{1}{2} a^{18} + \frac{17}{89} a^{16} - \frac{1}{2} a^{15} + \frac{17}{89} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{9} + \frac{44}{89} a^{8} - \frac{1}{2} a^{6} - \frac{44}{89} a^{5} - \frac{1}{2} a^{3} - \frac{44}{89} a^{2} - \frac{1}{2}$, $\frac{1}{3524578} a^{31} - \frac{2338}{1762289} a^{30} - \frac{7564}{1762289} a^{29} - \frac{15121}{3524578} a^{28} + \frac{549}{1762289} a^{27} + \frac{9346}{1762289} a^{26} + \frac{18721}{3524578} a^{25} - \frac{142}{1762289} a^{24} - \frac{9782}{1762289} a^{23} - \frac{19441}{3524578} a^{22} + \frac{3782}{19801} a^{21} - \frac{544608}{1762289} a^{20} - \frac{930287}{3524578} a^{19} - \frac{79422}{1762289} a^{18} + \frac{801737}{1762289} a^{17} + \frac{219611}{3524578} a^{16} + \frac{434731}{1762289} a^{15} - \frac{446417}{1762289} a^{14} - \frac{51843}{3524578} a^{13} - \frac{518608}{1762289} a^{12} + \frac{362537}{1762289} a^{11} - \frac{19441}{39602} a^{10} + \frac{3782}{1762289} a^{9} - \frac{45722}{1762289} a^{8} - \frac{1666171}{3524578} a^{7} + \frac{50948}{1762289} a^{6} - \frac{176769}{1762289} a^{5} - \frac{1424981}{3524578} a^{4} + \frac{167972}{1762289} a^{3} - \frac{752798}{1762289} a^{2} - \frac{237735}{3524578} a + \frac{762438}{1762289}$, $\frac{1}{3524578} a^{32} + \frac{1}{3524578} a^{30} + \frac{15125}{3524578} a^{29} - \frac{7564}{1762289} a^{28} - \frac{15121}{3524578} a^{27} - \frac{18703}{3524578} a^{26} + \frac{9346}{1762289} a^{25} + \frac{18721}{3524578} a^{24} + \frac{19517}{3524578} a^{23} - \frac{9782}{1762289} a^{22} + \frac{1089415}{3524578} a^{21} - \frac{12237}{39602} a^{20} - \frac{544608}{1762289} a^{19} - \frac{1603521}{3524578} a^{18} + \frac{1603445}{3524578} a^{17} + \frac{801737}{1762289} a^{16} + \frac{892845}{3524578} a^{15} - \frac{892827}{3524578} a^{14} - \frac{446417}{1762289} a^{13} - \frac{725077}{3524578} a^{12} + \frac{725073}{3524578} a^{11} - \frac{865125}{1762289} a^{10} - \frac{19441}{39602} a^{9} - \frac{1754725}{3524578} a^{8} + \frac{825522}{1762289} a^{7} - \frac{1666171}{3524578} a^{6} - \frac{1660393}{3524578} a^{5} + \frac{714276}{1762289} a^{4} - \frac{1424981}{3524578} a^{3} - \frac{1426345}{3524578} a^{2} + \frac{118446}{1762289} a - \frac{237735}{3524578}$, $\frac{1}{3524578} a^{33} + \frac{1346269}{3524578}$, $\frac{1}{3524578} a^{34} + \frac{1346269}{3524578} a$, $\frac{1}{3524578} a^{35} + \frac{1346269}{3524578} a^{2}$, $\frac{1}{3524578} a^{36} + \frac{1346269}{3524578} a^{3}$, $\frac{1}{3524578} a^{37} + \frac{1346269}{3524578} a^{4}$, $\frac{1}{3524578} a^{38} + \frac{1346269}{3524578} a^{5}$, $\frac{1}{3524578} a^{39} + \frac{1346269}{3524578} a^{6}$

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

Class group and class number

$C_{22}$, which has order $22$ (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:  \( \frac{3}{3524578} a^{37} - \frac{24157817}{3524578} a^{4} \) (order $66$)
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:  \( 138297149643015.1 \) (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

\(\Q(\sqrt{165}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{5}, \sqrt{33})\), \(\Q(\sqrt{-11}, \sqrt{-15})\), \(\Q(\sqrt{-3}, \sqrt{-55})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{5}, \sqrt{-11})\), \(\Q(\sqrt{-15}, \sqrt{33})\), \(\Q(\sqrt{-3}, \sqrt{-11})\), \(\Q(\zeta_{11})^+\), 8.0.741200625.1, 10.10.1790566527853125.1, 10.10.669871503125.1, \(\Q(\zeta_{33})^+\), 10.0.162778775259375.1, \(\Q(\zeta_{11})\), 10.0.52089208083.1, 10.0.7368586534375.1, 20.20.3206128490667995866421572265625.1, 20.0.3206128490667995866421572265625.3, 20.0.3206128490667995866421572265625.2, 20.0.26496929674942114598525390625.1, 20.0.54296067514572573056640625.1, 20.0.3206128490667995866421572265625.1, \(\Q(\zeta_{33})\)

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{4}$ R R ${\href{/LocalNumberField/7.10.0.1}{10} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{8}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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
3Data not computed
5Data not computed
11Data not computed