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Properties

Label 40.0.12989284101...0000.1
Degree $40$
Signature $[0, 20]$
Discriminant $2^{40}\cdot 3^{20}\cdot 5^{68}$
Root discriminant $53.44$
Ramified primes $2, 3, 5$
Class number $605$ (GRH)
Class group $[11, 55]$ (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, -75, 0, 4850, 0, -52915, 0, 400970, 0, -1380878, 0, 3173350, 0, -5116175, 0, 6247290, 0, -5912800, 0, 4465008, 0, -2717025, 0, 1349050, 0, -547185, 0, 181750, 0, -49003, 0, 10625, 0, -1800, 0, 230, 0, -20, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - 20*x^38 + 230*x^36 - 1800*x^34 + 10625*x^32 - 49003*x^30 + 181750*x^28 - 547185*x^26 + 1349050*x^24 - 2717025*x^22 + 4465008*x^20 - 5912800*x^18 + 6247290*x^16 - 5116175*x^14 + 3173350*x^12 - 1380878*x^10 + 400970*x^8 - 52915*x^6 + 4850*x^4 - 75*x^2 + 1)
 
gp: K = bnfinit(x^40 - 20*x^38 + 230*x^36 - 1800*x^34 + 10625*x^32 - 49003*x^30 + 181750*x^28 - 547185*x^26 + 1349050*x^24 - 2717025*x^22 + 4465008*x^20 - 5912800*x^18 + 6247290*x^16 - 5116175*x^14 + 3173350*x^12 - 1380878*x^10 + 400970*x^8 - 52915*x^6 + 4850*x^4 - 75*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{40} - 20 x^{38} + 230 x^{36} - 1800 x^{34} + 10625 x^{32} - 49003 x^{30} + 181750 x^{28} - 547185 x^{26} + 1349050 x^{24} - 2717025 x^{22} + 4465008 x^{20} - 5912800 x^{18} + 6247290 x^{16} - 5116175 x^{14} + 3173350 x^{12} - 1380878 x^{10} + 400970 x^{8} - 52915 x^{6} + 4850 x^{4} - 75 x^{2} + 1 \)

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:  \(1298928410187363624572753906250000000000000000000000000000000000000000=2^{40}\cdot 3^{20}\cdot 5^{68}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.44$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(300=2^{2}\cdot 3\cdot 5^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{300}(1,·)$, $\chi_{300}(131,·)$, $\chi_{300}(11,·)$, $\chi_{300}(109,·)$, $\chi_{300}(269,·)$, $\chi_{300}(271,·)$, $\chi_{300}(19,·)$, $\chi_{300}(149,·)$, $\chi_{300}(151,·)$, $\chi_{300}(281,·)$, $\chi_{300}(29,·)$, $\chi_{300}(31,·)$, $\chi_{300}(161,·)$, $\chi_{300}(41,·)$, $\chi_{300}(71,·)$, $\chi_{300}(49,·)$, $\chi_{300}(179,·)$, $\chi_{300}(181,·)$, $\chi_{300}(59,·)$, $\chi_{300}(61,·)$, $\chi_{300}(191,·)$, $\chi_{300}(139,·)$, $\chi_{300}(199,·)$, $\chi_{300}(119,·)$, $\chi_{300}(79,·)$, $\chi_{300}(209,·)$, $\chi_{300}(211,·)$, $\chi_{300}(89,·)$, $\chi_{300}(91,·)$, $\chi_{300}(221,·)$, $\chi_{300}(101,·)$, $\chi_{300}(229,·)$, $\chi_{300}(259,·)$, $\chi_{300}(289,·)$, $\chi_{300}(239,·)$, $\chi_{300}(241,·)$, $\chi_{300}(169,·)$, $\chi_{300}(121,·)$, $\chi_{300}(251,·)$, $\chi_{300}(299,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $\frac{1}{7} a^{32} - \frac{2}{7} a^{30} + \frac{2}{7} a^{28} + \frac{3}{7} a^{24} + \frac{1}{7} a^{22} - \frac{2}{7} a^{18} - \frac{3}{7} a^{16} + \frac{3}{7} a^{14} + \frac{1}{7} a^{10} - \frac{1}{7} a^{8} + \frac{2}{7} a^{4} + \frac{3}{7} a^{2} - \frac{3}{7}$, $\frac{1}{7} a^{33} - \frac{2}{7} a^{31} + \frac{2}{7} a^{29} + \frac{3}{7} a^{25} + \frac{1}{7} a^{23} - \frac{2}{7} a^{19} - \frac{3}{7} a^{17} + \frac{3}{7} a^{15} + \frac{1}{7} a^{11} - \frac{1}{7} a^{9} + \frac{2}{7} a^{5} + \frac{3}{7} a^{3} - \frac{3}{7} a$, $\frac{1}{7} a^{34} - \frac{2}{7} a^{30} - \frac{3}{7} a^{28} + \frac{3}{7} a^{26} + \frac{2}{7} a^{22} - \frac{2}{7} a^{20} - \frac{3}{7} a^{16} - \frac{1}{7} a^{14} + \frac{1}{7} a^{12} + \frac{1}{7} a^{10} - \frac{2}{7} a^{8} + \frac{2}{7} a^{6} + \frac{3}{7} a^{2} + \frac{1}{7}$, $\frac{1}{7} a^{35} - \frac{2}{7} a^{31} - \frac{3}{7} a^{29} + \frac{3}{7} a^{27} + \frac{2}{7} a^{23} - \frac{2}{7} a^{21} - \frac{3}{7} a^{17} - \frac{1}{7} a^{15} + \frac{1}{7} a^{13} + \frac{1}{7} a^{11} - \frac{2}{7} a^{9} + \frac{2}{7} a^{7} + \frac{3}{7} a^{3} + \frac{1}{7} a$, $\frac{1}{1057} a^{36} - \frac{6}{151} a^{34} - \frac{3}{1057} a^{32} + \frac{503}{1057} a^{30} + \frac{463}{1057} a^{28} - \frac{2}{151} a^{26} + \frac{160}{1057} a^{24} - \frac{269}{1057} a^{22} + \frac{72}{151} a^{20} + \frac{328}{1057} a^{18} + \frac{457}{1057} a^{16} + \frac{390}{1057} a^{14} - \frac{104}{1057} a^{12} - \frac{465}{1057} a^{10} - \frac{123}{1057} a^{8} - \frac{22}{151} a^{6} + \frac{351}{1057} a^{4} + \frac{516}{1057} a^{2} - \frac{228}{1057}$, $\frac{1}{1057} a^{37} - \frac{6}{151} a^{35} - \frac{3}{1057} a^{33} + \frac{503}{1057} a^{31} + \frac{463}{1057} a^{29} - \frac{2}{151} a^{27} + \frac{160}{1057} a^{25} - \frac{269}{1057} a^{23} + \frac{72}{151} a^{21} + \frac{328}{1057} a^{19} + \frac{457}{1057} a^{17} + \frac{390}{1057} a^{15} - \frac{104}{1057} a^{13} - \frac{465}{1057} a^{11} - \frac{123}{1057} a^{9} - \frac{22}{151} a^{7} + \frac{351}{1057} a^{5} + \frac{516}{1057} a^{3} - \frac{228}{1057} a$, $\frac{1}{1882407816972786612108716536103085223451} a^{38} - \frac{15731247118461406791933737135185259}{268915402424683801729816648014726460493} a^{36} - \frac{4744286105323148867704494207657566058}{1882407816972786612108716536103085223451} a^{34} - \frac{101421087075471908145491953758925469246}{1882407816972786612108716536103085223451} a^{32} + \frac{36147641761880874056837769217242006963}{268915402424683801729816648014726460493} a^{30} + \frac{90711810938737594109164176240933737862}{1882407816972786612108716536103085223451} a^{28} - \frac{102427968760729823643591009972806236380}{268915402424683801729816648014726460493} a^{26} - \frac{806999825103606531467228369605338526322}{1882407816972786612108716536103085223451} a^{24} - \frac{155390767313901812670149206493216028115}{1882407816972786612108716536103085223451} a^{22} + \frac{747793050586371829378203965736102343313}{1882407816972786612108716536103085223451} a^{20} + \frac{839644775900602388959531304026463029302}{1882407816972786612108716536103085223451} a^{18} + \frac{121873705886970916263232665104303631433}{268915402424683801729816648014726460493} a^{16} - \frac{8253407650906769059279587305643488853}{268915402424683801729816648014726460493} a^{14} - \frac{71286438178095507523238246865516142017}{1882407816972786612108716536103085223451} a^{12} + \frac{24763010676266317482329328997879215813}{268915402424683801729816648014726460493} a^{10} - \frac{741789267759613724500456389127965086672}{1882407816972786612108716536103085223451} a^{8} + \frac{16638928978525817324564426604035131257}{1882407816972786612108716536103085223451} a^{6} - \frac{200473924963798758801207648123172309225}{1882407816972786612108716536103085223451} a^{4} - \frac{326728996832349125884585999598655670182}{1882407816972786612108716536103085223451} a^{2} + \frac{542588334364143556238487559651239261370}{1882407816972786612108716536103085223451}$, $\frac{1}{1882407816972786612108716536103085223451} a^{39} - \frac{15731247118461406791933737135185259}{268915402424683801729816648014726460493} a^{37} - \frac{4744286105323148867704494207657566058}{1882407816972786612108716536103085223451} a^{35} - \frac{101421087075471908145491953758925469246}{1882407816972786612108716536103085223451} a^{33} + \frac{36147641761880874056837769217242006963}{268915402424683801729816648014726460493} a^{31} + \frac{90711810938737594109164176240933737862}{1882407816972786612108716536103085223451} a^{29} - \frac{102427968760729823643591009972806236380}{268915402424683801729816648014726460493} a^{27} - \frac{806999825103606531467228369605338526322}{1882407816972786612108716536103085223451} a^{25} - \frac{155390767313901812670149206493216028115}{1882407816972786612108716536103085223451} a^{23} + \frac{747793050586371829378203965736102343313}{1882407816972786612108716536103085223451} a^{21} + \frac{839644775900602388959531304026463029302}{1882407816972786612108716536103085223451} a^{19} + \frac{121873705886970916263232665104303631433}{268915402424683801729816648014726460493} a^{17} - \frac{8253407650906769059279587305643488853}{268915402424683801729816648014726460493} a^{15} - \frac{71286438178095507523238246865516142017}{1882407816972786612108716536103085223451} a^{13} + \frac{24763010676266317482329328997879215813}{268915402424683801729816648014726460493} a^{11} - \frac{741789267759613724500456389127965086672}{1882407816972786612108716536103085223451} a^{9} + \frac{16638928978525817324564426604035131257}{1882407816972786612108716536103085223451} a^{7} - \frac{200473924963798758801207648123172309225}{1882407816972786612108716536103085223451} a^{5} - \frac{326728996832349125884585999598655670182}{1882407816972786612108716536103085223451} a^{3} + \frac{542588334364143556238487559651239261370}{1882407816972786612108716536103085223451} a$

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

Class group and class number

$C_{11}\times C_{55}$, which has order $605$ (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{60982801767390441554071367337047629250}{1882407816972786612108716536103085223451} a^{39} - \frac{1213794782372652920631719432053599379000}{1882407816972786612108716536103085223451} a^{37} + \frac{13908790224576361797650905026009808260985}{1882407816972786612108716536103085223451} a^{35} - \frac{108420547835145750095820544489227150733350}{1882407816972786612108716536103085223451} a^{33} + \frac{637388685179548060392678477680689857748000}{1882407816972786612108716536103085223451} a^{31} - \frac{2926049530117748539545923781139053315403625}{1882407816972786612108716536103085223451} a^{29} + \frac{1542339861080187372716350765320337278678875}{268915402424683801729816648014726460493} a^{27} - \frac{32303772672325487853334158331446881664055027}{1882407816972786612108716536103085223451} a^{25} + \frac{79063454558652641543575994694241142847422025}{1882407816972786612108716536103085223451} a^{23} - \frac{157793697230969287360314156678657208042636100}{1882407816972786612108716536103085223451} a^{21} + \frac{256394810026294995797732570712351494708372375}{1882407816972786612108716536103085223451} a^{19} - \frac{334490812535455175588602189457092734162481750}{1882407816972786612108716536103085223451} a^{17} + \frac{346486687276069007771254411843705604979062880}{1882407816972786612108716536103085223451} a^{15} - \frac{275642671629382105640711330056786688957004850}{1882407816972786612108716536103085223451} a^{13} + \frac{163845274629375733432108102813684197243671150}{1882407816972786612108716536103085223451} a^{11} - \frac{65896043620177710925924643221862793322828875}{1882407816972786612108716536103085223451} a^{9} + \frac{16546845794841566857231826268588526889049000}{1882407816972786612108716536103085223451} a^{7} - \frac{969383617920261423673130666850022214047330}{1882407816972786612108716536103085223451} a^{5} + \frac{15039422341862280240063950700811829009375}{1882407816972786612108716536103085223451} a^{3} + \frac{14813235805684446519030135164153967765650}{1882407816972786612108716536103085223451} a \) (order $12$)
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:  \( 12293887094064384 \) (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

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\zeta_{12})\), \(\Q(\sqrt{3}, \sqrt{-5})\), \(\Q(\sqrt{-3}, \sqrt{-5})\), \(\Q(i, \sqrt{15})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(i, \sqrt{5})\), 5.5.390625.1, 8.0.12960000.1, 10.10.37968750000000000.1, 10.10.189843750000000000.1, \(\Q(\zeta_{25})^+\), 10.0.37078857421875.1, 10.0.156250000000000.1, 10.0.781250000000000.1, 10.0.185394287109375.1, 20.20.36040649414062500000000000000000000.1, 20.0.1441625976562500000000000000000000.1, 20.0.36040649414062500000000000000000000.4, 20.0.36040649414062500000000000000000000.1, 20.0.36040649414062500000000000000000000.3, 20.0.34371041692793369293212890625.1, 20.0.610351562500000000000000000000.1

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 R R ${\href{/LocalNumberField/7.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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
3Data not computed
5Data not computed