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Properties

Label 40.0.10038339744...5625.2
Degree $40$
Signature $[0, 20]$
Discriminant $3^{20}\cdot 5^{30}\cdot 11^{36}$
Root discriminant $50.12$
Ramified primes $3, 5, 11$
Class number Not computed
Class group Not computed
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, 4, 20, 95, 455, 2176, 10409, 49790, 238165, 1139235, 5449401, -3931240, 4695244, -3532620, 3132335, -2365144, 1906140, -1435731, 1114445, -826140, 682811, -252030, 197315, -75531, 56355, -21989, 15680, -6060, 4114, -1475, 936, -210, 265, -65, 74, -19, 20, -5, 5, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - x^39 + 5*x^38 - 5*x^37 + 20*x^36 - 19*x^35 + 74*x^34 - 65*x^33 + 265*x^32 - 210*x^31 + 936*x^30 - 1475*x^29 + 4114*x^28 - 6060*x^27 + 15680*x^26 - 21989*x^25 + 56355*x^24 - 75531*x^23 + 197315*x^22 - 252030*x^21 + 682811*x^20 - 826140*x^19 + 1114445*x^18 - 1435731*x^17 + 1906140*x^16 - 2365144*x^15 + 3132335*x^14 - 3532620*x^13 + 4695244*x^12 - 3931240*x^11 + 5449401*x^10 + 1139235*x^9 + 238165*x^8 + 49790*x^7 + 10409*x^6 + 2176*x^5 + 455*x^4 + 95*x^3 + 20*x^2 + 4*x + 1)
 
gp: K = bnfinit(x^40 - x^39 + 5*x^38 - 5*x^37 + 20*x^36 - 19*x^35 + 74*x^34 - 65*x^33 + 265*x^32 - 210*x^31 + 936*x^30 - 1475*x^29 + 4114*x^28 - 6060*x^27 + 15680*x^26 - 21989*x^25 + 56355*x^24 - 75531*x^23 + 197315*x^22 - 252030*x^21 + 682811*x^20 - 826140*x^19 + 1114445*x^18 - 1435731*x^17 + 1906140*x^16 - 2365144*x^15 + 3132335*x^14 - 3532620*x^13 + 4695244*x^12 - 3931240*x^11 + 5449401*x^10 + 1139235*x^9 + 238165*x^8 + 49790*x^7 + 10409*x^6 + 2176*x^5 + 455*x^4 + 95*x^3 + 20*x^2 + 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{40} - x^{39} + 5 x^{38} - 5 x^{37} + 20 x^{36} - 19 x^{35} + 74 x^{34} - 65 x^{33} + 265 x^{32} - 210 x^{31} + 936 x^{30} - 1475 x^{29} + 4114 x^{28} - 6060 x^{27} + 15680 x^{26} - 21989 x^{25} + 56355 x^{24} - 75531 x^{23} + 197315 x^{22} - 252030 x^{21} + 682811 x^{20} - 826140 x^{19} + 1114445 x^{18} - 1435731 x^{17} + 1906140 x^{16} - 2365144 x^{15} + 3132335 x^{14} - 3532620 x^{13} + 4695244 x^{12} - 3931240 x^{11} + 5449401 x^{10} + 1139235 x^{9} + 238165 x^{8} + 49790 x^{7} + 10409 x^{6} + 2176 x^{5} + 455 x^{4} + 95 x^{3} + 20 x^{2} + 4 x + 1 \)

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:  \(100383397447978918530459891214693626269465146363712847232818603515625=3^{20}\cdot 5^{30}\cdot 11^{36}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.12$
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$$\chi_{165}(128,·)$, $\chi_{165}(1,·)$, $\chi_{165}(2,·)$, $\chi_{165}(4,·)$, $\chi_{165}(8,·)$, $\chi_{165}(137,·)$, $\chi_{165}(139,·)$, $\chi_{165}(16,·)$, $\chi_{165}(17,·)$, $\chi_{165}(19,·)$, $\chi_{165}(23,·)$, $\chi_{165}(152,·)$, $\chi_{165}(158,·)$, $\chi_{165}(31,·)$, $\chi_{165}(32,·)$, $\chi_{165}(34,·)$, $\chi_{165}(38,·)$, $\chi_{165}(46,·)$, $\chi_{165}(47,·)$, $\chi_{165}(49,·)$, $\chi_{165}(53,·)$, $\chi_{165}(151,·)$, $\chi_{165}(61,·)$, $\chi_{165}(62,·)$, $\chi_{165}(64,·)$, $\chi_{165}(68,·)$, $\chi_{165}(76,·)$, $\chi_{165}(79,·)$, $\chi_{165}(83,·)$, $\chi_{165}(91,·)$, $\chi_{165}(92,·)$, $\chi_{165}(122,·)$, $\chi_{165}(94,·)$, $\chi_{165}(98,·)$, $\chi_{165}(106,·)$, $\chi_{165}(107,·)$, $\chi_{165}(109,·)$, $\chi_{165}(113,·)$, $\chi_{165}(136,·)$, $\chi_{165}(124,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $\frac{1}{2526205995232921} a^{31} + \frac{969862058041036}{2526205995232921} a^{30} + \frac{241303777751570}{2526205995232921} a^{29} + \frac{1111938459179657}{2526205995232921} a^{28} + \frac{1206518888757845}{2526205995232921} a^{27} + \frac{710382005693106}{2526205995232921} a^{26} + \frac{743525622606569}{2526205995232921} a^{25} + \frac{759727505551736}{2526205995232921} a^{24} - \frac{1203027871576723}{2526205995232921} a^{23} - \frac{1072959607181926}{2526205995232921} a^{22} + \frac{1082438515708330}{2526205995232921} a^{21} + \frac{95503815941798}{2526205995232921} a^{20} + \frac{25370031947195}{2526205995232921} a^{19} - \frac{528579085352603}{2526205995232921} a^{18} - \frac{70364038799755}{2526205995232921} a^{17} + \frac{488230004469246}{2526205995232921} a^{16} - \frac{383166537792952}{2526205995232921} a^{15} + \frac{57003490590598}{2526205995232921} a^{14} + \frac{433614414914333}{2526205995232921} a^{13} + \frac{299709387039926}{2526205995232921} a^{12} - \frac{480277222460171}{2526205995232921} a^{11} + \frac{830362944453688}{2526205995232921} a^{10} - \frac{14059251761729}{2526205995232921} a^{9} - \frac{484226564561075}{2526205995232921} a^{8} + \frac{126960133769841}{2526205995232921} a^{7} - \frac{424260348281824}{2526205995232921} a^{6} - \frac{990746123121383}{2526205995232921} a^{5} + \frac{285771829634526}{2526205995232921} a^{4} - \frac{1020345858551353}{2526205995232921} a^{3} + \frac{1150915028118670}{2526205995232921} a^{2} - \frac{572259025431674}{2526205995232921} a + \frac{808763874066416}{2526205995232921}$, $\frac{1}{2526205995232921} a^{32} + \frac{1235679720585025}{2526205995232921} a^{30} - \frac{613072852992611}{2526205995232921} a^{29} + \frac{503379744866868}{2526205995232921} a^{28} - \frac{539158269730133}{2526205995232921} a^{27} - \frac{1135293883357864}{2526205995232921} a^{26} - \frac{920953358335506}{2526205995232921} a^{25} - \frac{1227823008417508}{2526205995232921} a^{24} + \frac{1120610297072924}{2526205995232921} a^{23} - \frac{1001803895028353}{2526205995232921} a^{22} - \frac{95503815942622}{2526205995232921} a^{21} + \frac{209142208610269}{2526205995232921} a^{20} + \frac{639739986408395}{2526205995232921} a^{19} - \frac{693236095240533}{2526205995232921} a^{18} - \frac{868143299208031}{2526205995232921} a^{17} + \frac{445016655269210}{2526205995232921} a^{16} + \frac{248343766226902}{2526205995232921} a^{15} - \frac{1247614246741653}{2526205995232921} a^{14} - \frac{163212758531710}{2526205995232921} a^{13} + \frac{773526171669338}{2526205995232921} a^{12} + \frac{913440026709071}{2526205995232921} a^{11} - \frac{304794668595476}{2526205995232921} a^{10} - \frac{916762311003590}{2526205995232921} a^{9} - \frac{1098690672453044}{2526205995232921} a^{8} + \frac{23573903759522}{2526205995232921} a^{7} - \frac{201973183291819}{2526205995232921} a^{6} - \frac{655525585245758}{2526205995232921} a^{5} + \frac{1040619139569614}{2526205995232921} a^{4} + \frac{558223872986398}{2526205995232921} a^{3} + \frac{1184123527600845}{2526205995232921} a^{2} + \frac{814362117403119}{2526205995232921} a + \frac{61996354910397}{2526205995232921}$, $\frac{1}{2526205995232921} a^{33} + \frac{936795771304776}{2526205995232921} a^{22} - \frac{554016545945649}{2526205995232921} a^{11} + \frac{204895633579702}{2526205995232921}$, $\frac{1}{2526205995232921} a^{34} + \frac{936795771304776}{2526205995232921} a^{23} - \frac{554016545945649}{2526205995232921} a^{12} + \frac{204895633579702}{2526205995232921} a$, $\frac{1}{2526205995232921} a^{35} + \frac{936795771304776}{2526205995232921} a^{24} - \frac{554016545945649}{2526205995232921} a^{13} + \frac{204895633579702}{2526205995232921} a^{2}$, $\frac{1}{2526205995232921} a^{36} + \frac{936795771304776}{2526205995232921} a^{25} - \frac{554016545945649}{2526205995232921} a^{14} + \frac{204895633579702}{2526205995232921} a^{3}$, $\frac{1}{2526205995232921} a^{37} + \frac{936795771304776}{2526205995232921} a^{26} - \frac{554016545945649}{2526205995232921} a^{15} + \frac{204895633579702}{2526205995232921} a^{4}$, $\frac{1}{2526205995232921} a^{38} + \frac{936795771304776}{2526205995232921} a^{27} - \frac{554016545945649}{2526205995232921} a^{16} + \frac{204895633579702}{2526205995232921} a^{5}$, $\frac{1}{2526205995232921} a^{39} + \frac{936795771304776}{2526205995232921} a^{28} - \frac{554016545945649}{2526205995232921} a^{17} + \frac{204895633579702}{2526205995232921} a^{6}$

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

Class group and class number

Not computed

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:  \( \frac{1008753747971}{2526205995232921} a^{39} - \frac{831212771199384}{2526205995232921} a^{28} - \frac{1248337057599278259}{2526205995232921} a^{17} - \frac{30260868834152543254}{2526205995232921} a^{6} \) (order $22$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
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

\(\Q(\sqrt{-11}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{5}, \sqrt{-11})\), 4.0.136125.2, \(\Q(\zeta_{15})^+\), \(\Q(\zeta_{11})^+\), 8.0.18530015625.1, \(\Q(\zeta_{11})\), 10.10.669871503125.1, 10.0.7368586534375.1, 20.0.54296067514572573056640625.1, 20.0.10019151533337487082567413330078125.1, 20.20.82802905234194108120391845703125.1

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 $20^{2}$ R R $20^{2}$ R $20^{2}$ $20^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{8}$ $20^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{10}$ $20^{2}$ $20^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{8}$

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