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Properties

Label 40.0.40960000000...0000.1
Degree $40$
Signature $[0, 20]$
Discriminant $2^{80}\cdot 5^{68}$
Root discriminant $61.70$
Ramified primes $2, 5$
Class number $24805$ (GRH)
Class group $[11, 2255]$ (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, 4075, 0, 0, 0, 218435, 0, 0, 0, 747925, 0, 0, 0, 869360, 0, 0, 0, 457507, 0, 0, 0, 123375, 0, 0, 0, 18060, 0, 0, 0, 1450, 0, 0, 0, 60, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 + 60*x^36 + 1450*x^32 + 18060*x^28 + 123375*x^24 + 457507*x^20 + 869360*x^16 + 747925*x^12 + 218435*x^8 + 4075*x^4 + 1)
 
gp: K = bnfinit(x^40 + 60*x^36 + 1450*x^32 + 18060*x^28 + 123375*x^24 + 457507*x^20 + 869360*x^16 + 747925*x^12 + 218435*x^8 + 4075*x^4 + 1, 1)
 

Normalized defining polynomial

\( x^{40} + 60 x^{36} + 1450 x^{32} + 18060 x^{28} + 123375 x^{24} + 457507 x^{20} + 869360 x^{16} + 747925 x^{12} + 218435 x^{8} + 4075 x^{4} + 1 \)

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:  \(409600000000000000000000000000000000000000000000000000000000000000000000=2^{80}\cdot 5^{68}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $61.70$
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:  $2, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(200=2^{3}\cdot 5^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{200}(1,·)$, $\chi_{200}(131,·)$, $\chi_{200}(179,·)$, $\chi_{200}(129,·)$, $\chi_{200}(9,·)$, $\chi_{200}(11,·)$, $\chi_{200}(141,·)$, $\chi_{200}(19,·)$, $\chi_{200}(21,·)$, $\chi_{200}(151,·)$, $\chi_{200}(29,·)$, $\chi_{200}(159,·)$, $\chi_{200}(161,·)$, $\chi_{200}(39,·)$, $\chi_{200}(41,·)$, $\chi_{200}(171,·)$, $\chi_{200}(49,·)$, $\chi_{200}(51,·)$, $\chi_{200}(181,·)$, $\chi_{200}(31,·)$, $\chi_{200}(61,·)$, $\chi_{200}(191,·)$, $\chi_{200}(139,·)$, $\chi_{200}(69,·)$, $\chi_{200}(71,·)$, $\chi_{200}(119,·)$, $\chi_{200}(79,·)$, $\chi_{200}(81,·)$, $\chi_{200}(89,·)$, $\chi_{200}(91,·)$, $\chi_{200}(199,·)$, $\chi_{200}(99,·)$, $\chi_{200}(101,·)$, $\chi_{200}(109,·)$, $\chi_{200}(111,·)$, $\chi_{200}(59,·)$, $\chi_{200}(189,·)$, $\chi_{200}(169,·)$, $\chi_{200}(121,·)$, $\chi_{200}(149,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{7} a^{24} + \frac{2}{7} a^{20} + \frac{3}{7} a^{16} - \frac{2}{7} a^{12} - \frac{3}{7} a^{8} + \frac{2}{7} a^{4} - \frac{3}{7}$, $\frac{1}{7} a^{25} + \frac{2}{7} a^{21} + \frac{3}{7} a^{17} - \frac{2}{7} a^{13} - \frac{3}{7} a^{9} + \frac{2}{7} a^{5} - \frac{3}{7} a$, $\frac{1}{7} a^{26} + \frac{2}{7} a^{22} + \frac{3}{7} a^{18} - \frac{2}{7} a^{14} - \frac{3}{7} a^{10} + \frac{2}{7} a^{6} - \frac{3}{7} a^{2}$, $\frac{1}{7} a^{27} + \frac{2}{7} a^{23} + \frac{3}{7} a^{19} - \frac{2}{7} a^{15} - \frac{3}{7} a^{11} + \frac{2}{7} a^{7} - \frac{3}{7} a^{3}$, $\frac{1}{7} a^{28} - \frac{1}{7} a^{20} - \frac{1}{7} a^{16} + \frac{1}{7} a^{12} + \frac{1}{7} a^{8} - \frac{1}{7}$, $\frac{1}{7} a^{29} - \frac{1}{7} a^{21} - \frac{1}{7} a^{17} + \frac{1}{7} a^{13} + \frac{1}{7} a^{9} - \frac{1}{7} a$, $\frac{1}{7} a^{30} - \frac{1}{7} a^{22} - \frac{1}{7} a^{18} + \frac{1}{7} a^{14} + \frac{1}{7} a^{10} - \frac{1}{7} a^{2}$, $\frac{1}{7} a^{31} - \frac{1}{7} a^{23} - \frac{1}{7} a^{19} + \frac{1}{7} a^{15} + \frac{1}{7} a^{11} - \frac{1}{7} a^{3}$, $\frac{1}{7} a^{32} + \frac{1}{7} a^{20} - \frac{3}{7} a^{16} - \frac{1}{7} a^{12} - \frac{3}{7} a^{8} + \frac{1}{7} a^{4} - \frac{3}{7}$, $\frac{1}{7} a^{33} + \frac{1}{7} a^{21} - \frac{3}{7} a^{17} - \frac{1}{7} a^{13} - \frac{3}{7} a^{9} + \frac{1}{7} a^{5} - \frac{3}{7} a$, $\frac{1}{7} a^{34} + \frac{1}{7} a^{22} - \frac{3}{7} a^{18} - \frac{1}{7} a^{14} - \frac{3}{7} a^{10} + \frac{1}{7} a^{6} - \frac{3}{7} a^{2}$, $\frac{1}{7} a^{35} + \frac{1}{7} a^{23} - \frac{3}{7} a^{19} - \frac{1}{7} a^{15} - \frac{3}{7} a^{11} + \frac{1}{7} a^{7} - \frac{3}{7} a^{3}$, $\frac{1}{1173710522064219751801} a^{36} + \frac{5642206041008244573}{1173710522064219751801} a^{32} + \frac{55912877880291488300}{1173710522064219751801} a^{28} + \frac{24599963300249736804}{1173710522064219751801} a^{24} + \frac{349363673510104545123}{1173710522064219751801} a^{20} + \frac{30902009758157571149}{167672931723459964543} a^{16} + \frac{434536944347572511419}{1173710522064219751801} a^{12} + \frac{398952156474441534977}{1173710522064219751801} a^{8} + \frac{552337392491490080844}{1173710522064219751801} a^{4} + \frac{357398360527437043737}{1173710522064219751801}$, $\frac{1}{1173710522064219751801} a^{37} + \frac{5642206041008244573}{1173710522064219751801} a^{33} + \frac{55912877880291488300}{1173710522064219751801} a^{29} + \frac{24599963300249736804}{1173710522064219751801} a^{25} + \frac{349363673510104545123}{1173710522064219751801} a^{21} + \frac{30902009758157571149}{167672931723459964543} a^{17} + \frac{434536944347572511419}{1173710522064219751801} a^{13} + \frac{398952156474441534977}{1173710522064219751801} a^{9} + \frac{552337392491490080844}{1173710522064219751801} a^{5} + \frac{357398360527437043737}{1173710522064219751801} a$, $\frac{1}{1173710522064219751801} a^{38} + \frac{5642206041008244573}{1173710522064219751801} a^{34} + \frac{55912877880291488300}{1173710522064219751801} a^{30} + \frac{24599963300249736804}{1173710522064219751801} a^{26} + \frac{349363673510104545123}{1173710522064219751801} a^{22} + \frac{30902009758157571149}{167672931723459964543} a^{18} + \frac{434536944347572511419}{1173710522064219751801} a^{14} + \frac{398952156474441534977}{1173710522064219751801} a^{10} + \frac{552337392491490080844}{1173710522064219751801} a^{6} + \frac{357398360527437043737}{1173710522064219751801} a^{2}$, $\frac{1}{1173710522064219751801} a^{39} + \frac{5642206041008244573}{1173710522064219751801} a^{35} + \frac{55912877880291488300}{1173710522064219751801} a^{31} + \frac{24599963300249736804}{1173710522064219751801} a^{27} + \frac{349363673510104545123}{1173710522064219751801} a^{23} + \frac{30902009758157571149}{167672931723459964543} a^{19} + \frac{434536944347572511419}{1173710522064219751801} a^{15} + \frac{398952156474441534977}{1173710522064219751801} a^{11} + \frac{552337392491490080844}{1173710522064219751801} a^{7} + \frac{357398360527437043737}{1173710522064219751801} a^{3}$

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

Class group and class number

$C_{11}\times C_{2255}$, which has order $24805$ (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:  \( -\frac{155380917881805618250}{1173710522064219751801} a^{39} - \frac{9322745074008328324216}{1173710522064219751801} a^{35} - \frac{225295688002714930840740}{1173710522064219751801} a^{31} - \frac{2806017343545754208553400}{1173710522064219751801} a^{27} - \frac{19168074731518164940493765}{1173710522064219751801} a^{23} - \frac{10153369144579111620294350}{167672931723459964543} a^{19} - \frac{135027253103346561052117154}{1173710522064219751801} a^{15} - \frac{116103928625932626925593090}{1173710522064219751801} a^{11} - \frac{33839405690677968704638925}{1173710522064219751801} a^{7} - \frac{601046104243551107175290}{1173710522064219751801} a^{3} \) (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:  \( 3995169264157994.0 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{10}) \), \(\Q(i, \sqrt{5})\), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{10})\), \(\Q(\sqrt{2}, \sqrt{-5})\), \(\Q(\sqrt{-2}, \sqrt{-5})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{-2}, \sqrt{5})\), 5.5.390625.1, 8.0.40960000.1, 10.0.156250000000000.1, 10.0.781250000000000.1, \(\Q(\zeta_{25})^+\), 10.10.5000000000000000.1, 10.0.5000000000000000.1, 10.0.25000000000000000.1, 10.10.25000000000000000.1, 20.0.610351562500000000000000000000.1, 20.0.25600000000000000000000000000000000.1, 20.0.640000000000000000000000000000000000.1, 20.0.640000000000000000000000000000000000.4, 20.0.640000000000000000000000000000000000.3, 20.20.625000000000000000000000000000000.1, 20.0.625000000000000000000000000000000.1

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 ${\href{/LocalNumberField/7.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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