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Properties

Label 40.0.13030509980...5936.4
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 $13640$ (GRH)
Class group $[2, 6820]$ (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, 12220, 0, 0, 0, 328758, 0, 0, 0, 1748889, 0, 0, 0, 1881661, 0, 0, 0, 834866, 0, 0, 0, 190246, 0, 0, 0, 24166, 0, 0, 0, 1728, 0, 0, 0, 65, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 + 65*x^36 + 1728*x^32 + 24166*x^28 + 190246*x^24 + 834866*x^20 + 1881661*x^16 + 1748889*x^12 + 328758*x^8 + 12220*x^4 + 1)
 
gp: K = bnfinit(x^40 + 65*x^36 + 1728*x^32 + 24166*x^28 + 190246*x^24 + 834866*x^20 + 1881661*x^16 + 1748889*x^12 + 328758*x^8 + 12220*x^4 + 1, 1)
 

Normalized defining polynomial

\( x^{40} + 65 x^{36} + 1728 x^{32} + 24166 x^{28} + 190246 x^{24} + 834866 x^{20} + 1881661 x^{16} + 1748889 x^{12} + 328758 x^{8} + 12220 x^{4} + 1 \)

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:  \(130305099804548492884220428175380349368393046678311823693003457545895936=2^{80}\cdot 3^{20}\cdot 11^{36}\)
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$$\chi_{264}(1,·)$, $\chi_{264}(131,·)$, $\chi_{264}(133,·)$, $\chi_{264}(263,·)$, $\chi_{264}(17,·)$, $\chi_{264}(107,·)$, $\chi_{264}(149,·)$, $\chi_{264}(25,·)$, $\chi_{264}(157,·)$, $\chi_{264}(31,·)$, $\chi_{264}(161,·)$, $\chi_{264}(35,·)$, $\chi_{264}(37,·)$, $\chi_{264}(167,·)$, $\chi_{264}(41,·)$, $\chi_{264}(173,·)$, $\chi_{264}(29,·)$, $\chi_{264}(49,·)$, $\chi_{264}(181,·)$, $\chi_{264}(163,·)$, $\chi_{264}(95,·)$, $\chi_{264}(65,·)$, $\chi_{264}(67,·)$, $\chi_{264}(197,·)$, $\chi_{264}(199,·)$, $\chi_{264}(83,·)$, $\chi_{264}(215,·)$, $\chi_{264}(169,·)$, $\chi_{264}(91,·)$, $\chi_{264}(223,·)$, $\chi_{264}(97,·)$, $\chi_{264}(227,·)$, $\chi_{264}(101,·)$, $\chi_{264}(103,·)$, $\chi_{264}(233,·)$, $\chi_{264}(235,·)$, $\chi_{264}(239,·)$, $\chi_{264}(229,·)$, $\chi_{264}(115,·)$, $\chi_{264}(247,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $\frac{1}{3932854626532852909940501} a^{36} - \frac{1037115215397826204386482}{3932854626532852909940501} a^{32} + \frac{1752222902408999748278426}{3932854626532852909940501} a^{28} - \frac{1613165176977480426299668}{3932854626532852909940501} a^{24} - \frac{206759667378976948375976}{3932854626532852909940501} a^{20} + \frac{382141438266401909036720}{3932854626532852909940501} a^{16} - \frac{66911533856392384577397}{3932854626532852909940501} a^{12} - \frac{1467580011646731050825778}{3932854626532852909940501} a^{8} - \frac{1933795443995376727065271}{3932854626532852909940501} a^{4} + \frac{846396170749556039797521}{3932854626532852909940501}$, $\frac{1}{3932854626532852909940501} a^{37} - \frac{1037115215397826204386482}{3932854626532852909940501} a^{33} + \frac{1752222902408999748278426}{3932854626532852909940501} a^{29} - \frac{1613165176977480426299668}{3932854626532852909940501} a^{25} - \frac{206759667378976948375976}{3932854626532852909940501} a^{21} + \frac{382141438266401909036720}{3932854626532852909940501} a^{17} - \frac{66911533856392384577397}{3932854626532852909940501} a^{13} - \frac{1467580011646731050825778}{3932854626532852909940501} a^{9} - \frac{1933795443995376727065271}{3932854626532852909940501} a^{5} + \frac{846396170749556039797521}{3932854626532852909940501} a$, $\frac{1}{3932854626532852909940501} a^{38} - \frac{1037115215397826204386482}{3932854626532852909940501} a^{34} + \frac{1752222902408999748278426}{3932854626532852909940501} a^{30} - \frac{1613165176977480426299668}{3932854626532852909940501} a^{26} - \frac{206759667378976948375976}{3932854626532852909940501} a^{22} + \frac{382141438266401909036720}{3932854626532852909940501} a^{18} - \frac{66911533856392384577397}{3932854626532852909940501} a^{14} - \frac{1467580011646731050825778}{3932854626532852909940501} a^{10} - \frac{1933795443995376727065271}{3932854626532852909940501} a^{6} + \frac{846396170749556039797521}{3932854626532852909940501} a^{2}$, $\frac{1}{3932854626532852909940501} a^{39} - \frac{1037115215397826204386482}{3932854626532852909940501} a^{35} + \frac{1752222902408999748278426}{3932854626532852909940501} a^{31} - \frac{1613165176977480426299668}{3932854626532852909940501} a^{27} - \frac{206759667378976948375976}{3932854626532852909940501} a^{23} + \frac{382141438266401909036720}{3932854626532852909940501} a^{19} - \frac{66911533856392384577397}{3932854626532852909940501} a^{15} - \frac{1467580011646731050825778}{3932854626532852909940501} a^{11} - \frac{1933795443995376727065271}{3932854626532852909940501} a^{7} + \frac{846396170749556039797521}{3932854626532852909940501} a^{3}$

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

Class group and class number

$C_{2}\times C_{6820}$, which has order $13640$ (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:  \( -\frac{2164259387924765787436}{3932854626532852909940501} a^{37} - \frac{140766837084166661524113}{3932854626532852909940501} a^{33} - \frac{3745696429925655953913055}{3932854626532852909940501} a^{29} - \frac{52457439488345567954384271}{3932854626532852909940501} a^{25} - \frac{413926932226812360755246562}{3932854626532852909940501} a^{21} - \frac{1824093968334464352510426563}{3932854626532852909940501} a^{17} - \frac{4147794503215809873388416510}{3932854626532852909940501} a^{13} - \frac{3951736500964096976304891366}{3932854626532852909940501} a^{9} - \frac{854166991648209791755489924}{3932854626532852909940501} a^{5} - \frac{41397226216524455234378597}{3932854626532852909940501} a \) (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:  \( 2307157166474863.5 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-33}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-66}) \), \(\Q(\sqrt{66}) \), \(\Q(i, \sqrt{33})\), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{66})\), \(\Q(\sqrt{2}, \sqrt{-33})\), \(\Q(\sqrt{-2}, \sqrt{-33})\), \(\Q(\sqrt{2}, \sqrt{33})\), \(\Q(\sqrt{-2}, \sqrt{33})\), \(\Q(\zeta_{11})^+\), 8.0.77720518656.8, 10.0.219503494144.1, 10.0.586732839846912.1, \(\Q(\zeta_{33})^+\), 10.10.7024111812608.1, 10.0.7024111812608.1, 10.0.18775450875101184.1, 10.10.18775450875101184.1, 20.0.344255425354822086003595935744.1, 20.0.50522262278163705147147943936.1, 20.0.360977976896857923653306611918700544.10, 20.0.360977976896857923653306611918700544.7, 20.0.360977976896857923653306611918700544.6, 20.20.352517555563337816067682238201856.1, 20.0.352517555563337816067682238201856.7

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 ${\href{/LocalNumberField/13.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{8}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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
3Data not computed
11Data not computed