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Properties

Label 40.0.28789677222...5625.1
Degree $40$
Signature $[0, 20]$
Discriminant $5^{30}\cdot 11^{36}$
Root discriminant $28.94$
Ramified primes $5, 11$
Class number $10$ (GRH)
Class group $[10]$ (GRH)
Galois group $C_2\times C_{20}$ (as 40T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 0, 0, 0, 1, -1, 0, 0, 0, 1, 0, -1, 0, 0, 1, 0, -1, 0, 0, 1, 0, 0, -1, 0, 1, 0, 0, -1, 0, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - x^39 + x^35 - x^34 + x^30 - x^28 + x^25 - x^23 + x^20 - x^17 + x^15 - x^12 + x^10 - x^6 + x^5 - x + 1)
 
gp: K = bnfinit(x^40 - x^39 + x^35 - x^34 + x^30 - x^28 + x^25 - x^23 + x^20 - x^17 + x^15 - x^12 + x^10 - x^6 + x^5 - x + 1, 1)
 

Normalized defining polynomial

\( x^{40} - x^{39} + x^{35} - x^{34} + x^{30} - x^{28} + x^{25} - x^{23} + x^{20} - x^{17} + x^{15} - x^{12} + x^{10} - x^{6} + x^{5} - x + 1 \)

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:  \(28789677222138897176527746894292024300433695316314697265625=5^{30}\cdot 11^{36}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.94$
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:  $5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(55=5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{55}(1,·)$, $\chi_{55}(2,·)$, $\chi_{55}(3,·)$, $\chi_{55}(4,·)$, $\chi_{55}(6,·)$, $\chi_{55}(7,·)$, $\chi_{55}(8,·)$, $\chi_{55}(9,·)$, $\chi_{55}(12,·)$, $\chi_{55}(13,·)$, $\chi_{55}(14,·)$, $\chi_{55}(16,·)$, $\chi_{55}(17,·)$, $\chi_{55}(18,·)$, $\chi_{55}(19,·)$, $\chi_{55}(21,·)$, $\chi_{55}(23,·)$, $\chi_{55}(24,·)$, $\chi_{55}(26,·)$, $\chi_{55}(27,·)$, $\chi_{55}(28,·)$, $\chi_{55}(29,·)$, $\chi_{55}(31,·)$, $\chi_{55}(32,·)$, $\chi_{55}(34,·)$, $\chi_{55}(36,·)$, $\chi_{55}(37,·)$, $\chi_{55}(38,·)$, $\chi_{55}(39,·)$, $\chi_{55}(41,·)$, $\chi_{55}(42,·)$, $\chi_{55}(43,·)$, $\chi_{55}(46,·)$, $\chi_{55}(47,·)$, $\chi_{55}(48,·)$, $\chi_{55}(49,·)$, $\chi_{55}(51,·)$, $\chi_{55}(52,·)$, $\chi_{55}(53,·)$, $\chi_{55}(54,·)$$\rbrace$
This is a CM field.

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

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

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

Class group and class number

$C_{10}$, which has order $10$ (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:  \( -a \) (order $110$)
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:  \( 30395381425048.965 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{20}$ (as 40T2):

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 $C_2\times C_{20}$
Character table for $C_2\times C_{20}$ is not computed

Intermediate fields

\(\Q(\sqrt{-55}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{5}, \sqrt{-11})\), \(\Q(\zeta_{5})\), 4.4.15125.1, \(\Q(\zeta_{11})^+\), 8.0.228765625.1, 10.0.7368586534375.1, 10.10.669871503125.1, \(\Q(\zeta_{11})\), 20.0.54296067514572573056640625.1, 20.0.1402274470934209014892578125.1, \(\Q(\zeta_{55})^+\)

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 $20^{2}$ $20^{2}$ R $20^{2}$ R $20^{2}$ $20^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{8}$ $20^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{10}$ $20^{2}$ $20^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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
5Data not computed
$11$11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$