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Properties

Label 40.0.56984141097...5625.1
Degree $40$
Signature $[0, 20]$
Discriminant $5^{70}\cdot 11^{20}$
Root discriminant $55.45$
Ramified primes $5, 11$
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![3486784401, 0, 0, 0, 0, -444816117, 0, 0, 0, 0, 42397182, 0, 0, 0, 0, -3578175, 0, 0, 0, 0, 282001, 0, 0, 0, 0, -14725, 0, 0, 0, 0, 718, 0, 0, 0, 0, -31, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - 31*x^35 + 718*x^30 - 14725*x^25 + 282001*x^20 - 3578175*x^15 + 42397182*x^10 - 444816117*x^5 + 3486784401)
 
gp: K = bnfinit(x^40 - 31*x^35 + 718*x^30 - 14725*x^25 + 282001*x^20 - 3578175*x^15 + 42397182*x^10 - 444816117*x^5 + 3486784401, 1)
 

Normalized defining polynomial

\( x^{40} - 31 x^{35} + 718 x^{30} - 14725 x^{25} + 282001 x^{20} - 3578175 x^{15} + 42397182 x^{10} - 444816117 x^{5} + 3486784401 \)

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:  \(5698414109730419226674323998106663768936641645268537104129791259765625=5^{70}\cdot 11^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.45$
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:  \(275=5^{2}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{275}(1,·)$, $\chi_{275}(131,·)$, $\chi_{275}(133,·)$, $\chi_{275}(263,·)$, $\chi_{275}(12,·)$, $\chi_{275}(142,·)$, $\chi_{275}(144,·)$, $\chi_{275}(274,·)$, $\chi_{275}(21,·)$, $\chi_{275}(23,·)$, $\chi_{275}(153,·)$, $\chi_{275}(32,·)$, $\chi_{275}(34,·)$, $\chi_{275}(164,·)$, $\chi_{275}(166,·)$, $\chi_{275}(43,·)$, $\chi_{275}(177,·)$, $\chi_{275}(54,·)$, $\chi_{275}(56,·)$, $\chi_{275}(186,·)$, $\chi_{275}(188,·)$, $\chi_{275}(67,·)$, $\chi_{275}(197,·)$, $\chi_{275}(199,·)$, $\chi_{275}(76,·)$, $\chi_{275}(78,·)$, $\chi_{275}(208,·)$, $\chi_{275}(87,·)$, $\chi_{275}(89,·)$, $\chi_{275}(219,·)$, $\chi_{275}(221,·)$, $\chi_{275}(98,·)$, $\chi_{275}(232,·)$, $\chi_{275}(109,·)$, $\chi_{275}(111,·)$, $\chi_{275}(241,·)$, $\chi_{275}(243,·)$, $\chi_{275}(122,·)$, $\chi_{275}(252,·)$, $\chi_{275}(254,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{3} a^{21} - \frac{1}{3} a^{16} + \frac{1}{3} a^{11} - \frac{1}{3} a^{6} + \frac{1}{3} a$, $\frac{1}{9} a^{22} - \frac{4}{9} a^{17} - \frac{2}{9} a^{12} - \frac{1}{9} a^{7} + \frac{4}{9} a^{2}$, $\frac{1}{27} a^{23} - \frac{4}{27} a^{18} - \frac{11}{27} a^{13} - \frac{10}{27} a^{8} + \frac{13}{27} a^{3}$, $\frac{1}{81} a^{24} - \frac{31}{81} a^{19} - \frac{11}{81} a^{14} + \frac{17}{81} a^{9} + \frac{40}{81} a^{4}$, $\frac{1}{68526243} a^{25} + \frac{62}{243} a^{20} + \frac{22}{243} a^{15} + \frac{47}{243} a^{10} + \frac{1}{243} a^{5} - \frac{14725}{282001}$, $\frac{1}{205578729} a^{26} + \frac{62}{729} a^{21} + \frac{22}{729} a^{16} + \frac{290}{729} a^{11} + \frac{244}{729} a^{6} - \frac{296726}{846003} a$, $\frac{1}{616736187} a^{27} + \frac{62}{2187} a^{22} + \frac{751}{2187} a^{17} + \frac{1019}{2187} a^{12} + \frac{244}{2187} a^{7} - \frac{1142729}{2538009} a^{2}$, $\frac{1}{1850208561} a^{28} + \frac{62}{6561} a^{23} - \frac{1436}{6561} a^{18} + \frac{3206}{6561} a^{13} + \frac{244}{6561} a^{8} - \frac{1142729}{7614027} a^{3}$, $\frac{1}{5550625683} a^{29} + \frac{62}{19683} a^{24} - \frac{1436}{19683} a^{19} + \frac{9767}{19683} a^{14} + \frac{6805}{19683} a^{9} + \frac{6471298}{22842081} a^{4}$, $\frac{1}{16651877049} a^{30} - \frac{31}{16651877049} a^{25} + \frac{6340}{59049} a^{20} + \frac{24590}{59049} a^{15} + \frac{1}{59049} a^{10} - \frac{14725}{68526243} a^{5} + \frac{718}{282001}$, $\frac{1}{49955631147} a^{31} - \frac{31}{49955631147} a^{26} + \frac{6340}{177147} a^{21} + \frac{83639}{177147} a^{16} - \frac{59048}{177147} a^{11} + \frac{68511518}{205578729} a^{6} - \frac{93761}{282001} a$, $\frac{1}{149866893441} a^{32} - \frac{31}{149866893441} a^{27} + \frac{6340}{531441} a^{22} + \frac{260786}{531441} a^{17} - \frac{59048}{531441} a^{12} + \frac{274090247}{616736187} a^{7} + \frac{188240}{846003} a^{2}$, $\frac{1}{449600680323} a^{33} - \frac{31}{449600680323} a^{28} + \frac{6340}{1594323} a^{23} + \frac{260786}{1594323} a^{18} - \frac{59048}{1594323} a^{13} + \frac{274090247}{1850208561} a^{8} + \frac{1034243}{2538009} a^{3}$, $\frac{1}{1348802040969} a^{34} - \frac{31}{1348802040969} a^{29} + \frac{6340}{4782969} a^{24} + \frac{1855109}{4782969} a^{19} - \frac{1653371}{4782969} a^{14} - \frac{1576118314}{5550625683} a^{9} - \frac{1503766}{7614027} a^{4}$, $\frac{1}{4046406122907} a^{35} - \frac{31}{4046406122907} a^{30} + \frac{718}{4046406122907} a^{25} - \frac{6529849}{14348907} a^{20} + \frac{1}{14348907} a^{15} - \frac{14725}{16651877049} a^{10} + \frac{718}{68526243} a^{5} - \frac{31}{282001}$, $\frac{1}{12139218368721} a^{36} - \frac{31}{12139218368721} a^{31} + \frac{718}{12139218368721} a^{26} - \frac{6529849}{43046721} a^{21} + \frac{14348908}{43046721} a^{16} - \frac{16651891774}{49955631147} a^{11} + \frac{68526961}{205578729} a^{6} - \frac{282032}{846003} a$, $\frac{1}{36417655106163} a^{37} - \frac{31}{36417655106163} a^{32} + \frac{718}{36417655106163} a^{27} - \frac{6529849}{129140163} a^{22} + \frac{14348908}{129140163} a^{17} - \frac{66607522921}{149866893441} a^{12} - \frac{137051768}{616736187} a^{7} - \frac{282032}{2538009} a^{2}$, $\frac{1}{109252965318489} a^{38} - \frac{31}{109252965318489} a^{33} + \frac{718}{109252965318489} a^{28} - \frac{6529849}{387420489} a^{23} - \frac{114791255}{387420489} a^{18} + \frac{83259370520}{449600680323} a^{13} + \frac{479684419}{1850208561} a^{8} - \frac{282032}{7614027} a^{3}$, $\frac{1}{327758895955467} a^{39} - \frac{31}{327758895955467} a^{34} + \frac{718}{327758895955467} a^{29} - \frac{6529849}{1162261467} a^{24} - \frac{502211744}{1162261467} a^{19} + \frac{532860050843}{1348802040969} a^{14} - \frac{1370524142}{5550625683} a^{9} - \frac{7896059}{22842081} a^{4}$

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:  \( \frac{1079}{1348802040969} a^{39} + \frac{500858642}{1348802040969} a^{14} \) (order $50$)
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

\(\Q(\sqrt{-11}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{5}, \sqrt{-11})\), 4.4.15125.1, \(\Q(\zeta_{5})\), 5.5.390625.1, 8.0.228765625.1, 10.0.24574432373046875.1, \(\Q(\zeta_{25})^+\), 10.0.122872161865234375.1, 20.0.15097568161436356604099273681640625.2, 20.20.75487840807181783020496368408203125.1, \(\Q(\zeta_{25})\)

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{10}$ R $20^{2}$ $20^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ $20^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{8}$ $20^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{10}$ $20^{2}$ $20^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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$11.10.5.2$x^{10} + 1331 x^{4} - 14641 x^{2} + 805255$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
11.10.5.2$x^{10} + 1331 x^{4} - 14641 x^{2} + 805255$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
11.10.5.2$x^{10} + 1331 x^{4} - 14641 x^{2} + 805255$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
11.10.5.2$x^{10} + 1331 x^{4} - 14641 x^{2} + 805255$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$