↑

Properties

Label 40.0.14118416629...4448.1
Degree $40$
Signature $[0, 20]$
Discriminant $2^{155}\cdot 11^{36}$
Root discriminant $126.98$
Ramified primes $2, 11$
Class number Not computed
Class group Not computed
Galois group $C_{40}$ (as 40T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![468512, 0, 18740480, 0, 261429696, 0, 1881544192, 0, 8169678000, 0, 23198669824, 0, 45166558624, 0, 62138831744, 0, 61669393368, 0, 44817432000, 0, 24119005504, 0, 9689664512, 0, 2919917792, 0, 660609664, 0, 111779800, 0, 14004496, 0, 1275010, 0, 81664, 0, 3476, 0, 88, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 + 88*x^38 + 3476*x^36 + 81664*x^34 + 1275010*x^32 + 14004496*x^30 + 111779800*x^28 + 660609664*x^26 + 2919917792*x^24 + 9689664512*x^22 + 24119005504*x^20 + 44817432000*x^18 + 61669393368*x^16 + 62138831744*x^14 + 45166558624*x^12 + 23198669824*x^10 + 8169678000*x^8 + 1881544192*x^6 + 261429696*x^4 + 18740480*x^2 + 468512)
 
gp: K = bnfinit(x^40 + 88*x^38 + 3476*x^36 + 81664*x^34 + 1275010*x^32 + 14004496*x^30 + 111779800*x^28 + 660609664*x^26 + 2919917792*x^24 + 9689664512*x^22 + 24119005504*x^20 + 44817432000*x^18 + 61669393368*x^16 + 62138831744*x^14 + 45166558624*x^12 + 23198669824*x^10 + 8169678000*x^8 + 1881544192*x^6 + 261429696*x^4 + 18740480*x^2 + 468512, 1)
 

Normalized defining polynomial

\( x^{40} + 88 x^{38} + 3476 x^{36} + 81664 x^{34} + 1275010 x^{32} + 14004496 x^{30} + 111779800 x^{28} + 660609664 x^{26} + 2919917792 x^{24} + 9689664512 x^{22} + 24119005504 x^{20} + 44817432000 x^{18} + 61669393368 x^{16} + 62138831744 x^{14} + 45166558624 x^{12} + 23198669824 x^{10} + 8169678000 x^{8} + 1881544192 x^{6} + 261429696 x^{4} + 18740480 x^{2} + 468512 \)

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:  \(1411841662908675517629776705295515492024702234241930698046194396081616318012166504448=2^{155}\cdot 11^{36}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $126.98$
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:  $2, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(352=2^{5}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{352}(1,·)$, $\chi_{352}(261,·)$, $\chi_{352}(257,·)$, $\chi_{352}(9,·)$, $\chi_{352}(13,·)$, $\chi_{352}(109,·)$, $\chi_{352}(273,·)$, $\chi_{352}(149,·)$, $\chi_{352}(89,·)$, $\chi_{352}(25,·)$, $\chi_{352}(265,·)$, $\chi_{352}(29,·)$, $\chi_{352}(325,·)$, $\chi_{352}(289,·)$, $\chi_{352}(293,·)$, $\chi_{352}(49,·)$, $\chi_{352}(169,·)$, $\chi_{352}(173,·)$, $\chi_{352}(285,·)$, $\chi_{352}(177,·)$, $\chi_{352}(137,·)$, $\chi_{352}(313,·)$, $\chi_{352}(61,·)$, $\chi_{352}(197,·)$, $\chi_{352}(97,·)$, $\chi_{352}(201,·)$, $\chi_{352}(205,·)$, $\chi_{352}(81,·)$, $\chi_{352}(85,·)$, $\chi_{352}(185,·)$, $\chi_{352}(345,·)$, $\chi_{352}(349,·)$, $\chi_{352}(225,·)$, $\chi_{352}(101,·)$, $\chi_{352}(237,·)$, $\chi_{352}(189,·)$, $\chi_{352}(113,·)$, $\chi_{352}(117,·)$, $\chi_{352}(21,·)$, $\chi_{352}(277,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{22} a^{10}$, $\frac{1}{22} a^{11}$, $\frac{1}{22} a^{12}$, $\frac{1}{22} a^{13}$, $\frac{1}{22} a^{14}$, $\frac{1}{22} a^{15}$, $\frac{1}{44} a^{16}$, $\frac{1}{44} a^{17}$, $\frac{1}{44} a^{18}$, $\frac{1}{44} a^{19}$, $\frac{1}{484} a^{20}$, $\frac{1}{484} a^{21}$, $\frac{1}{484} a^{22}$, $\frac{1}{484} a^{23}$, $\frac{1}{968} a^{24}$, $\frac{1}{968} a^{25}$, $\frac{1}{968} a^{26}$, $\frac{1}{968} a^{27}$, $\frac{1}{968} a^{28}$, $\frac{1}{968} a^{29}$, $\frac{1}{10648} a^{30}$, $\frac{1}{10648} a^{31}$, $\frac{1}{21296} a^{32}$, $\frac{1}{21296} a^{33}$, $\frac{1}{21296} a^{34}$, $\frac{1}{21296} a^{35}$, $\frac{1}{12628528} a^{36} - \frac{185}{12628528} a^{34} - \frac{5}{12628528} a^{32} - \frac{30}{789283} a^{30} + \frac{223}{574024} a^{28} + \frac{35}{71753} a^{26} - \frac{63}{143506} a^{24} + \frac{31}{287012} a^{22} + \frac{24}{71753} a^{20} - \frac{3}{26092} a^{18} + \frac{2}{593} a^{16} + \frac{111}{13046} a^{14} - \frac{197}{13046} a^{12} - \frac{9}{593} a^{10} - \frac{3}{593} a^{8} - \frac{157}{593} a^{6} + \frac{188}{593} a^{4} - \frac{278}{593} a^{2} + \frac{233}{593}$, $\frac{1}{12628528} a^{37} - \frac{185}{12628528} a^{35} - \frac{5}{12628528} a^{33} - \frac{30}{789283} a^{31} + \frac{223}{574024} a^{29} + \frac{35}{71753} a^{27} - \frac{63}{143506} a^{25} + \frac{31}{287012} a^{23} + \frac{24}{71753} a^{21} - \frac{3}{26092} a^{19} + \frac{2}{593} a^{17} + \frac{111}{13046} a^{15} - \frac{197}{13046} a^{13} - \frac{9}{593} a^{11} - \frac{3}{593} a^{9} - \frac{157}{593} a^{7} + \frac{188}{593} a^{5} - \frac{278}{593} a^{3} + \frac{233}{593} a$, $\frac{1}{791511224274688215168014909669168} a^{38} - \frac{2157876358078093559102377}{395755612137344107584007454834584} a^{36} - \frac{2829139382746872149809601697}{791511224274688215168014909669168} a^{34} - \frac{33404360086911144142345885}{791511224274688215168014909669168} a^{32} + \frac{1521157875179169395190080615}{35977782921576737053091586803144} a^{30} - \frac{9213011251690050797674695743}{17988891460788368526545793401572} a^{28} + \frac{524880950838172400662676653}{3270707538325157913917416982104} a^{26} + \frac{14607947977648550842552911195}{35977782921576737053091586803144} a^{24} + \frac{52203028330993083928174391}{148668524469325359723518953732} a^{22} + \frac{8309365215424094927240598603}{8994445730394184263272896700786} a^{20} + \frac{499266564399139750445238209}{74334262234662679861759476866} a^{18} + \frac{5070514597439565857943839233}{1635353769162578956958708491052} a^{16} - \frac{3963732011874850120007530932}{408838442290644739239677122763} a^{14} - \frac{6901921729114742833595390811}{408838442290644739239677122763} a^{12} - \frac{12011777515535292632236960609}{817676884581289478479354245526} a^{10} - \frac{2694916966650719643325665393}{37167131117331339930879738433} a^{8} + \frac{2072499877393610020876743631}{37167131117331339930879738433} a^{6} - \frac{8461369723998809660507220594}{37167131117331339930879738433} a^{4} - \frac{4939112971512775719492213548}{37167131117331339930879738433} a^{2} - \frac{6686746252477004080811094082}{37167131117331339930879738433}$, $\frac{1}{791511224274688215168014909669168} a^{39} - \frac{2157876358078093559102377}{395755612137344107584007454834584} a^{37} - \frac{2829139382746872149809601697}{791511224274688215168014909669168} a^{35} - \frac{33404360086911144142345885}{791511224274688215168014909669168} a^{33} + \frac{1521157875179169395190080615}{35977782921576737053091586803144} a^{31} - \frac{9213011251690050797674695743}{17988891460788368526545793401572} a^{29} + \frac{524880950838172400662676653}{3270707538325157913917416982104} a^{27} + \frac{14607947977648550842552911195}{35977782921576737053091586803144} a^{25} + \frac{52203028330993083928174391}{148668524469325359723518953732} a^{23} + \frac{8309365215424094927240598603}{8994445730394184263272896700786} a^{21} + \frac{499266564399139750445238209}{74334262234662679861759476866} a^{19} + \frac{5070514597439565857943839233}{1635353769162578956958708491052} a^{17} - \frac{3963732011874850120007530932}{408838442290644739239677122763} a^{15} - \frac{6901921729114742833595390811}{408838442290644739239677122763} a^{13} - \frac{12011777515535292632236960609}{817676884581289478479354245526} a^{11} - \frac{2694916966650719643325665393}{37167131117331339930879738433} a^{9} + \frac{2072499877393610020876743631}{37167131117331339930879738433} a^{7} - \frac{8461369723998809660507220594}{37167131117331339930879738433} a^{5} - \frac{4939112971512775719492213548}{37167131117331339930879738433} a^{3} - \frac{6686746252477004080811094082}{37167131117331339930879738433} a$

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:  \( -1 \) (order $2$)
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

$C_{40}$ (as 40T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 40
The 40 conjugacy class representatives for $C_{40}$
Character table for $C_{40}$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), \(\Q(\zeta_{11})^+\), 8.0.31441308090368.8, 10.10.7024111812608.1, 20.20.1655513490330868290261743826894848.1

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 $40$ $40$ $20^{2}$ R $40$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{8}$ $40$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{10}$ $40$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{8}$ $40$ $20^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{5}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{4}$ $40$ $40$

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
11Data not computed