↑

Properties

Label 40.0.67586678490...5625.1
Degree $40$
Signature $[0, 20]$
Discriminant $5^{70}\cdot 7^{20}$
Root discriminant $44.23$
Ramified primes $5, 7$
Class number Not computed
Class group Not computed
Galois group $C_2\times C_{20}$ (as 40T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1048576, 0, 0, 0, 0, -360448, 0, 0, 0, 0, 91136, 0, 0, 0, 0, -20064, 0, 0, 0, 0, 4049, 0, 0, 0, 0, -627, 0, 0, 0, 0, 89, 0, 0, 0, 0, -11, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - 11*x^35 + 89*x^30 - 627*x^25 + 4049*x^20 - 20064*x^15 + 91136*x^10 - 360448*x^5 + 1048576)
 
gp: K = bnfinit(x^40 - 11*x^35 + 89*x^30 - 627*x^25 + 4049*x^20 - 20064*x^15 + 91136*x^10 - 360448*x^5 + 1048576, 1)
 

Normalized defining polynomial

\( x^{40} - 11 x^{35} + 89 x^{30} - 627 x^{25} + 4049 x^{20} - 20064 x^{15} + 91136 x^{10} - 360448 x^{5} + 1048576 \)

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:  \(675866784901662726441095609131171073613586486317217350006103515625=5^{70}\cdot 7^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.23$
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:  $5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(175=5^{2}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{175}(1,·)$, $\chi_{175}(132,·)$, $\chi_{175}(134,·)$, $\chi_{175}(8,·)$, $\chi_{175}(139,·)$, $\chi_{175}(13,·)$, $\chi_{175}(146,·)$, $\chi_{175}(148,·)$, $\chi_{175}(22,·)$, $\chi_{175}(153,·)$, $\chi_{175}(27,·)$, $\chi_{175}(29,·)$, $\chi_{175}(34,·)$, $\chi_{175}(36,·)$, $\chi_{175}(6,·)$, $\chi_{175}(167,·)$, $\chi_{175}(41,·)$, $\chi_{175}(43,·)$, $\chi_{175}(174,·)$, $\chi_{175}(48,·)$, $\chi_{175}(57,·)$, $\chi_{175}(62,·)$, $\chi_{175}(64,·)$, $\chi_{175}(69,·)$, $\chi_{175}(71,·)$, $\chi_{175}(76,·)$, $\chi_{175}(162,·)$, $\chi_{175}(78,·)$, $\chi_{175}(141,·)$, $\chi_{175}(83,·)$, $\chi_{175}(92,·)$, $\chi_{175}(97,·)$, $\chi_{175}(99,·)$, $\chi_{175}(104,·)$, $\chi_{175}(106,·)$, $\chi_{175}(111,·)$, $\chi_{175}(113,·)$, $\chi_{175}(118,·)$, $\chi_{175}(169,·)$, $\chi_{175}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{16} - \frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{4} a^{22} + \frac{1}{4} a^{17} + \frac{1}{4} a^{12} + \frac{1}{4} a^{7} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{23} - \frac{3}{8} a^{18} + \frac{1}{8} a^{13} - \frac{3}{8} a^{8} + \frac{1}{8} a^{3}$, $\frac{1}{16} a^{24} + \frac{5}{16} a^{19} - \frac{7}{16} a^{14} - \frac{3}{16} a^{9} + \frac{1}{16} a^{4}$, $\frac{1}{129568} a^{25} + \frac{5}{32} a^{20} + \frac{9}{32} a^{15} - \frac{3}{32} a^{10} + \frac{1}{32} a^{5} - \frac{627}{4049}$, $\frac{1}{259136} a^{26} + \frac{5}{64} a^{21} - \frac{23}{64} a^{16} + \frac{29}{64} a^{11} - \frac{31}{64} a^{6} + \frac{1711}{4049} a$, $\frac{1}{518272} a^{27} + \frac{5}{128} a^{22} + \frac{41}{128} a^{17} + \frac{29}{128} a^{12} + \frac{33}{128} a^{7} + \frac{1711}{8098} a^{2}$, $\frac{1}{1036544} a^{28} + \frac{5}{256} a^{23} - \frac{87}{256} a^{18} + \frac{29}{256} a^{13} - \frac{95}{256} a^{8} + \frac{1711}{16196} a^{3}$, $\frac{1}{2073088} a^{29} + \frac{5}{512} a^{24} - \frac{87}{512} a^{19} - \frac{227}{512} a^{14} + \frac{161}{512} a^{9} - \frac{14485}{32392} a^{4}$, $\frac{1}{4146176} a^{30} - \frac{11}{4146176} a^{25} - \frac{503}{1024} a^{20} + \frac{253}{1024} a^{15} + \frac{1}{1024} a^{10} - \frac{627}{129568} a^{5} + \frac{89}{4049}$, $\frac{1}{8292352} a^{31} - \frac{11}{8292352} a^{26} - \frac{503}{2048} a^{21} + \frac{253}{2048} a^{16} - \frac{1023}{2048} a^{11} + \frac{128941}{259136} a^{6} - \frac{1980}{4049} a$, $\frac{1}{16584704} a^{32} - \frac{11}{16584704} a^{27} - \frac{503}{4096} a^{22} - \frac{1795}{4096} a^{17} - \frac{1023}{4096} a^{12} - \frac{130195}{518272} a^{7} - \frac{990}{4049} a^{2}$, $\frac{1}{33169408} a^{33} - \frac{11}{33169408} a^{28} - \frac{503}{8192} a^{23} - \frac{1795}{8192} a^{18} + \frac{3073}{8192} a^{13} - \frac{130195}{1036544} a^{8} + \frac{3059}{8098} a^{3}$, $\frac{1}{66338816} a^{34} - \frac{11}{66338816} a^{29} - \frac{503}{16384} a^{24} - \frac{1795}{16384} a^{19} + \frac{3073}{16384} a^{14} - \frac{130195}{2073088} a^{9} - \frac{5039}{16196} a^{4}$, $\frac{1}{132677632} a^{35} - \frac{11}{132677632} a^{30} + \frac{89}{132677632} a^{25} + \frac{7421}{32768} a^{20} + \frac{1}{32768} a^{15} - \frac{627}{4146176} a^{10} + \frac{89}{129568} a^{5} - \frac{11}{4049}$, $\frac{1}{265355264} a^{36} - \frac{11}{265355264} a^{31} + \frac{89}{265355264} a^{26} + \frac{7421}{65536} a^{21} - \frac{32767}{65536} a^{16} + \frac{4145549}{8292352} a^{11} - \frac{129479}{259136} a^{6} + \frac{2019}{4049} a$, $\frac{1}{530710528} a^{37} - \frac{11}{530710528} a^{32} + \frac{89}{530710528} a^{27} + \frac{7421}{131072} a^{22} + \frac{32769}{131072} a^{17} + \frac{4145549}{16584704} a^{12} + \frac{129657}{518272} a^{7} + \frac{2019}{8098} a^{2}$, $\frac{1}{1061421056} a^{38} - \frac{11}{1061421056} a^{33} + \frac{89}{1061421056} a^{28} + \frac{7421}{262144} a^{23} + \frac{32769}{262144} a^{18} - \frac{12439155}{33169408} a^{13} + \frac{129657}{1036544} a^{8} - \frac{6079}{16196} a^{3}$, $\frac{1}{2122842112} a^{39} - \frac{11}{2122842112} a^{34} + \frac{89}{2122842112} a^{29} + \frac{7421}{524288} a^{24} - \frac{229375}{524288} a^{19} - \frac{12439155}{66338816} a^{14} + \frac{129657}{2073088} a^{9} + \frac{10117}{32392} a^{4}$

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:  \( \frac{1}{33169408} a^{38} - \frac{364229}{33169408} a^{13} \) (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

\(\Q(\sqrt{-35}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{5}, \sqrt{-7})\), \(\Q(\zeta_{5})\), 4.4.6125.1, 5.5.390625.1, 8.0.37515625.1, 10.0.12822723388671875.1, \(\Q(\zeta_{25})^+\), 10.0.2564544677734375.1, 20.0.164422235102392733097076416015625.1, \(\Q(\zeta_{25})\), 20.20.822111175511963665485382080078125.1

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 $20^{2}$ $20^{2}$ R R ${\href{/LocalNumberField/11.5.0.1}{5} }^{8}$ $20^{2}$ $20^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ $20^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ $20^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{10}$ $20^{2}$ $20^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$