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Properties

Label 40.0.32414294902...0000.1
Degree $40$
Signature $[0, 20]$
Discriminant $2^{40}\cdot 5^{20}\cdot 11^{36}$
Root discriminant $38.71$
Ramified primes $2, 5, 11$
Class number $62$ (GRH)
Class group $[62]$ (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, -3, 0, 8, 0, -21, 0, 55, 0, -144, 0, 377, 0, -987, 0, 2584, 0, -6765, 0, 17711, 0, -6765, 0, 2584, 0, -987, 0, 377, 0, -144, 0, 55, 0, -21, 0, 8, 0, -3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - 3*x^38 + 8*x^36 - 21*x^34 + 55*x^32 - 144*x^30 + 377*x^28 - 987*x^26 + 2584*x^24 - 6765*x^22 + 17711*x^20 - 6765*x^18 + 2584*x^16 - 987*x^14 + 377*x^12 - 144*x^10 + 55*x^8 - 21*x^6 + 8*x^4 - 3*x^2 + 1)
 
gp: K = bnfinit(x^40 - 3*x^38 + 8*x^36 - 21*x^34 + 55*x^32 - 144*x^30 + 377*x^28 - 987*x^26 + 2584*x^24 - 6765*x^22 + 17711*x^20 - 6765*x^18 + 2584*x^16 - 987*x^14 + 377*x^12 - 144*x^10 + 55*x^8 - 21*x^6 + 8*x^4 - 3*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{40} - 3 x^{38} + 8 x^{36} - 21 x^{34} + 55 x^{32} - 144 x^{30} + 377 x^{28} - 987 x^{26} + 2584 x^{24} - 6765 x^{22} + 17711 x^{20} - 6765 x^{18} + 2584 x^{16} - 987 x^{14} + 377 x^{12} - 144 x^{10} + 55 x^{8} - 21 x^{6} + 8 x^{4} - 3 x^{2} + 1 \)

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:  \(3241429490243539842962382172914969829546393600000000000000000000=2^{40}\cdot 5^{20}\cdot 11^{36}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.71$
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:  \(220=2^{2}\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{220}(1,·)$, $\chi_{220}(131,·)$, $\chi_{220}(179,·)$, $\chi_{220}(129,·)$, $\chi_{220}(9,·)$, $\chi_{220}(139,·)$, $\chi_{220}(141,·)$, $\chi_{220}(19,·)$, $\chi_{220}(21,·)$, $\chi_{220}(151,·)$, $\chi_{220}(29,·)$, $\chi_{220}(159,·)$, $\chi_{220}(161,·)$, $\chi_{220}(91,·)$, $\chi_{220}(39,·)$, $\chi_{220}(41,·)$, $\chi_{220}(171,·)$, $\chi_{220}(49,·)$, $\chi_{220}(51,·)$, $\chi_{220}(181,·)$, $\chi_{220}(31,·)$, $\chi_{220}(61,·)$, $\chi_{220}(191,·)$, $\chi_{220}(69,·)$, $\chi_{220}(71,·)$, $\chi_{220}(201,·)$, $\chi_{220}(119,·)$, $\chi_{220}(79,·)$, $\chi_{220}(81,·)$, $\chi_{220}(211,·)$, $\chi_{220}(89,·)$, $\chi_{220}(219,·)$, $\chi_{220}(199,·)$, $\chi_{220}(59,·)$, $\chi_{220}(101,·)$, $\chi_{220}(109,·)$, $\chi_{220}(111,·)$, $\chi_{220}(189,·)$, $\chi_{220}(169,·)$, $\chi_{220}(149,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{17711} a^{22} - \frac{6765}{17711}$, $\frac{1}{17711} a^{23} - \frac{6765}{17711} a$, $\frac{1}{17711} a^{24} - \frac{6765}{17711} a^{2}$, $\frac{1}{17711} a^{25} - \frac{6765}{17711} a^{3}$, $\frac{1}{17711} a^{26} - \frac{6765}{17711} a^{4}$, $\frac{1}{17711} a^{27} - \frac{6765}{17711} a^{5}$, $\frac{1}{17711} a^{28} - \frac{6765}{17711} a^{6}$, $\frac{1}{17711} a^{29} - \frac{6765}{17711} a^{7}$, $\frac{1}{17711} a^{30} - \frac{6765}{17711} a^{8}$, $\frac{1}{17711} a^{31} - \frac{6765}{17711} a^{9}$, $\frac{1}{17711} a^{32} - \frac{6765}{17711} a^{10}$, $\frac{1}{17711} a^{33} - \frac{6765}{17711} a^{11}$, $\frac{1}{17711} a^{34} - \frac{6765}{17711} a^{12}$, $\frac{1}{17711} a^{35} - \frac{6765}{17711} a^{13}$, $\frac{1}{17711} a^{36} - \frac{6765}{17711} a^{14}$, $\frac{1}{17711} a^{37} - \frac{6765}{17711} a^{15}$, $\frac{1}{17711} a^{38} - \frac{6765}{17711} a^{16}$, $\frac{1}{17711} a^{39} - \frac{6765}{17711} a^{17}$

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

Class group and class number

$C_{62}$, which has order $62$ (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:  \( \frac{1597}{17711} a^{39} + \frac{63245986}{17711} a^{17} \) (order $44$)
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:  \( 645826241875575.0 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{55}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{11}) \), \(\Q(i, \sqrt{5})\), \(\Q(i, \sqrt{55})\), \(\Q(i, \sqrt{11})\), \(\Q(\sqrt{5}, \sqrt{-11})\), \(\Q(\sqrt{5}, \sqrt{11})\), \(\Q(\sqrt{-5}, \sqrt{11})\), \(\Q(\sqrt{-5}, \sqrt{-11})\), \(\Q(\zeta_{11})^+\), 8.0.2342560000.1, 10.0.219503494144.1, 10.10.669871503125.1, 10.0.685948419200000.1, 10.0.7368586534375.1, 10.10.7545432611200000.1, \(\Q(\zeta_{11})\), \(\Q(\zeta_{44})^+\), 20.0.470525233802978928640000000000.1, 20.0.56933553290160450365440000000000.2, \(\Q(\zeta_{44})\), 20.0.54296067514572573056640625.1, 20.20.56933553290160450365440000000000.1, 20.0.56933553290160450365440000000000.1, 20.0.56933553290160450365440000000000.3

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 ${\href{/LocalNumberField/13.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
11Data not computed