↑

Properties

Label 40.0.20366967879...4096.1
Degree $40$
Signature $[0, 20]$
Discriminant $2^{80}\cdot 7^{20}\cdot 11^{32}$
Root discriminant $72.06$
Ramified primes $2, 7, 11$
Class number Not computed
Class group Not computed
Galois group $C_2^2\times C_{10}$ (as 40T7)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![19356878641, 0, 0, 0, -688869171, 0, 0, 0, -534900542, 0, 0, 0, -65359259, 0, 0, 0, 92377760, 0, 0, 0, -22941208, 0, 0, 0, 2662270, 0, 0, 0, -164960, 0, 0, 0, 5588, 0, 0, 0, -103, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - 103*x^36 + 5588*x^32 - 164960*x^28 + 2662270*x^24 - 22941208*x^20 + 92377760*x^16 - 65359259*x^12 - 534900542*x^8 - 688869171*x^4 + 19356878641)
 
gp: K = bnfinit(x^40 - 103*x^36 + 5588*x^32 - 164960*x^28 + 2662270*x^24 - 22941208*x^20 + 92377760*x^16 - 65359259*x^12 - 534900542*x^8 - 688869171*x^4 + 19356878641, 1)
 

Normalized defining polynomial

\( x^{40} - 103 x^{36} + 5588 x^{32} - 164960 x^{28} + 2662270 x^{24} - 22941208 x^{20} + 92377760 x^{16} - 65359259 x^{12} - 534900542 x^{8} - 688869171 x^{4} + 19356878641 \)

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:  \(203669678791643247319458007269651380031543505379155973278080554134033924096=2^{80}\cdot 7^{20}\cdot 11^{32}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $72.06$
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:  $2, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(616=2^{3}\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{616}(1,·)$, $\chi_{616}(265,·)$, $\chi_{616}(267,·)$, $\chi_{616}(141,·)$, $\chi_{616}(15,·)$, $\chi_{616}(531,·)$, $\chi_{616}(405,·)$, $\chi_{616}(279,·)$, $\chi_{616}(27,·)$, $\chi_{616}(155,·)$, $\chi_{616}(421,·)$, $\chi_{616}(295,·)$, $\chi_{616}(169,·)$, $\chi_{616}(559,·)$, $\chi_{616}(433,·)$, $\chi_{616}(309,·)$, $\chi_{616}(573,·)$, $\chi_{616}(181,·)$, $\chi_{616}(449,·)$, $\chi_{616}(323,·)$, $\chi_{616}(69,·)$, $\chi_{616}(71,·)$, $\chi_{616}(587,·)$, $\chi_{616}(335,·)$, $\chi_{616}(419,·)$, $\chi_{616}(463,·)$, $\chi_{616}(603,·)$, $\chi_{616}(477,·)$, $\chi_{616}(223,·)$, $\chi_{616}(97,·)$, $\chi_{616}(251,·)$, $\chi_{616}(379,·)$, $\chi_{616}(489,·)$, $\chi_{616}(225,·)$, $\chi_{616}(111,·)$, $\chi_{616}(113,·)$, $\chi_{616}(377,·)$, $\chi_{616}(575,·)$, $\chi_{616}(125,·)$, $\chi_{616}(533,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{23} a^{24} + \frac{2}{23} a^{20} + \frac{11}{23} a^{16} + \frac{5}{23} a^{12} + \frac{9}{23} a^{8} - \frac{6}{23} a^{4} + \frac{3}{23}$, $\frac{1}{23} a^{25} + \frac{2}{23} a^{21} + \frac{11}{23} a^{17} + \frac{5}{23} a^{13} + \frac{9}{23} a^{9} - \frac{6}{23} a^{5} + \frac{3}{23} a$, $\frac{1}{23} a^{26} + \frac{2}{23} a^{22} + \frac{11}{23} a^{18} + \frac{5}{23} a^{14} + \frac{9}{23} a^{10} - \frac{6}{23} a^{6} + \frac{3}{23} a^{2}$, $\frac{1}{23} a^{27} + \frac{2}{23} a^{23} + \frac{11}{23} a^{19} + \frac{5}{23} a^{15} + \frac{9}{23} a^{11} - \frac{6}{23} a^{7} + \frac{3}{23} a^{3}$, $\frac{1}{23} a^{28} + \frac{7}{23} a^{20} + \frac{6}{23} a^{16} - \frac{1}{23} a^{12} - \frac{1}{23} a^{8} - \frac{8}{23} a^{4} - \frac{6}{23}$, $\frac{1}{23} a^{29} + \frac{7}{23} a^{21} + \frac{6}{23} a^{17} - \frac{1}{23} a^{13} - \frac{1}{23} a^{9} - \frac{8}{23} a^{5} - \frac{6}{23} a$, $\frac{1}{23} a^{30} + \frac{7}{23} a^{22} + \frac{6}{23} a^{18} - \frac{1}{23} a^{14} - \frac{1}{23} a^{10} - \frac{8}{23} a^{6} - \frac{6}{23} a^{2}$, $\frac{1}{23} a^{31} + \frac{7}{23} a^{23} + \frac{6}{23} a^{19} - \frac{1}{23} a^{15} - \frac{1}{23} a^{11} - \frac{8}{23} a^{7} - \frac{6}{23} a^{3}$, $\frac{1}{23} a^{32} - \frac{8}{23} a^{20} - \frac{9}{23} a^{16} + \frac{10}{23} a^{12} - \frac{2}{23} a^{8} - \frac{10}{23} a^{4} + \frac{2}{23}$, $\frac{1}{23} a^{33} - \frac{8}{23} a^{21} - \frac{9}{23} a^{17} + \frac{10}{23} a^{13} - \frac{2}{23} a^{9} - \frac{10}{23} a^{5} + \frac{2}{23} a$, $\frac{1}{23} a^{34} - \frac{8}{23} a^{22} - \frac{9}{23} a^{18} + \frac{10}{23} a^{14} - \frac{2}{23} a^{10} - \frac{10}{23} a^{6} + \frac{2}{23} a^{2}$, $\frac{1}{23} a^{35} - \frac{8}{23} a^{23} - \frac{9}{23} a^{19} + \frac{10}{23} a^{15} - \frac{2}{23} a^{11} - \frac{10}{23} a^{7} + \frac{2}{23} a^{3}$, $\frac{1}{504865755838624879384946830837208236933177} a^{36} + \frac{97227810530169528365222534175331655915}{504865755838624879384946830837208236933177} a^{32} - \frac{50928543632721181178578921730800738048}{4631795925124998893439879181992736118653} a^{28} - \frac{86580058795471976078992721971174854269}{21950685036461951277606383949443836388399} a^{24} - \frac{54605902244223838787787250286982970047860}{504865755838624879384946830837208236933177} a^{20} - \frac{112820361167668064323318996328396274890546}{504865755838624879384946830837208236933177} a^{16} + \frac{146371985391617614839235122650520474211403}{504865755838624879384946830837208236933177} a^{12} + \frac{223235101106977153706470631700377577496189}{504865755838624879384946830837208236933177} a^{8} - \frac{82358439571229192704024456325470937712003}{504865755838624879384946830837208236933177} a^{4} - \frac{4960586456183972640137394588245588657543}{21950685036461951277606383949443836388399}$, $\frac{1}{188314926927807080010585167902278672376075021} a^{37} + \frac{116438295750386444582449741928970581349591}{8187605518600307826547181213142550972872827} a^{33} - \frac{20793318990931629269191950910654792921581}{1727659880071624587253074934883290572257569} a^{29} - \frac{3931163962878985134141359559555783735171608}{188314926927807080010585167902278672376075021} a^{25} - \frac{90601181677649772858914121041742808072193735}{188314926927807080010585167902278672376075021} a^{21} - \frac{19912338264056348116724277318726736697226444}{188314926927807080010585167902278672376075021} a^{17} + \frac{88300323091822813945706473063616967410021787}{188314926927807080010585167902278672376075021} a^{13} + \frac{38000364048857995302467057408693220001930868}{188314926927807080010585167902278672376075021} a^{9} - \frac{41239892882937387838215994361532664165960128}{188314926927807080010585167902278672376075021} a^{5} - \frac{57098071843147456887389332808285847803407293}{188314926927807080010585167902278672376075021} a$, $\frac{1}{70241467744072040843948267627549944796275982833} a^{38} + \frac{166430191174265044756339968327217342828497133}{70241467744072040843948267627549944796275982833} a^{34} + \frac{8542390434407555455376483813293481086963761}{644417135266715971045396950711467383452073237} a^{30} - \frac{249559329520888219930556795953832312921356418}{70241467744072040843948267627549944796275982833} a^{26} - \frac{11553248907718080730024967819441314170094151535}{70241467744072040843948267627549944796275982833} a^{22} + \frac{34924788015122057455586645140373680815523999192}{70241467744072040843948267627549944796275982833} a^{18} - \frac{9663137849561143807471986351789161241281515170}{70241467744072040843948267627549944796275982833} a^{14} - \frac{15297384772289518563820403354807304752188874103}{70241467744072040843948267627549944796275982833} a^{10} + \frac{13042553725840354518984179584240263790484817418}{70241467744072040843948267627549944796275982833} a^{6} + \frac{11258172754862477959400815103754719596706839621}{70241467744072040843948267627549944796275982833} a^{2}$, $\frac{1}{26200067468538871234792703825076129409010941596709} a^{39} + \frac{299456162318089917336362002033540025605221815291}{26200067468538871234792703825076129409010941596709} a^{35} - \frac{2933361922739730573230131334652101095542066234}{240367591454485057199933062615377334027623317401} a^{31} - \frac{278161453447371136776421528713651439985143723279}{26200067468538871234792703825076129409010941596709} a^{27} - \frac{8788682740058285271404256322670682242191710441189}{26200067468538871234792703825076129409010941596709} a^{23} + \frac{8726542927129427633189359239401553806476456483658}{26200067468538871234792703825076129409010941596709} a^{19} - \frac{6795599717298607872296735058891614262864117769732}{26200067468538871234792703825076129409010941596709} a^{15} + \frac{10200255206702535552001699388564956405836644281392}{26200067468538871234792703825076129409010941596709} a^{11} - \frac{10266643551776180927251879682778069048568861191968}{26200067468538871234792703825076129409010941596709} a^{7} + \frac{6757493053044216313797736605941051591552082756060}{26200067468538871234792703825076129409010941596709} a^{3}$

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:  \( \frac{61982565822024272246802953045}{4666217037116853354915957336124846483} a^{37} - \frac{5603665687321713249498294291785}{4666217037116853354915957336124846483} a^{33} + \frac{2530217119958042628969840986308}{42809330615750948210238140698393087} a^{29} - \frac{6751610933089392959325265388303797}{4666217037116853354915957336124846483} a^{25} + \frac{79982965391177819431109366225707675}{4666217037116853354915957336124846483} a^{21} - \frac{414018267261194009709396568033951761}{4666217037116853354915957336124846483} a^{17} + \frac{485841504990697262675098812540945766}{4666217037116853354915957336124846483} a^{13} + \frac{2494422731367515759218082924200581003}{4666217037116853354915957336124846483} a^{9} - \frac{4630609918613553147171189444264968455}{4666217037116853354915957336124846483} a^{5} - \frac{96162234377440979772421299141419186725}{4666217037116853354915957336124846483} a \) (order $8$)
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

$C_2^2\times C_{10}$ (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{14}) \), \(\Q(\sqrt{-14}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(i, \sqrt{14})\), \(\Q(i, \sqrt{7})\), \(\Q(\zeta_{8})\), \(\Q(\sqrt{2}, \sqrt{7})\), \(\Q(\sqrt{-2}, \sqrt{-7})\), \(\Q(\sqrt{-2}, \sqrt{7})\), \(\Q(\sqrt{2}, \sqrt{-7})\), \(\Q(\zeta_{11})^+\), 8.0.157351936.1, 10.0.219503494144.1, 10.10.118054247234502656.1, 10.0.118054247234502656.1, 10.10.3689195226078208.1, 10.0.3602729712967.1, 10.10.7024111812608.1, 10.0.7024111812608.1, 20.0.14271288617067599874203197103159640064.3, 20.0.13610161416118240236476132491264.1, 20.0.50522262278163705147147943936.1, 20.20.14271288617067599874203197103159640064.1, 20.0.13936805290105078002151559671054336.2, 20.0.14271288617067599874203197103159640064.2, 20.0.13936805290105078002151559671054336.6

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{4}$ R 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
7Data not computed
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$