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Properties

Label 40.40.9643068526...4528.1
Degree $40$
Signature $[40, 0]$
Discriminant $2^{155}\cdot 11^{32}$
Root discriminant $99.91$
Ramified primes $2, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{40}$ (as 40T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![32, 0, -3840, 0, 121280, 0, -1828864, 0, 15867056, 0, -86580736, 0, 312117152, 0, -765197952, 0, 1299305304, 0, -1548248640, 0, 1309625664, 0, -794852864, 0, 349153248, 0, -111555328, 0, 25920664, 0, -4350192, 0, 518914, 0, -42688, 0, 2292, 0, -72, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - 72*x^38 + 2292*x^36 - 42688*x^34 + 518914*x^32 - 4350192*x^30 + 25920664*x^28 - 111555328*x^26 + 349153248*x^24 - 794852864*x^22 + 1309625664*x^20 - 1548248640*x^18 + 1299305304*x^16 - 765197952*x^14 + 312117152*x^12 - 86580736*x^10 + 15867056*x^8 - 1828864*x^6 + 121280*x^4 - 3840*x^2 + 32)
 
gp: K = bnfinit(x^40 - 72*x^38 + 2292*x^36 - 42688*x^34 + 518914*x^32 - 4350192*x^30 + 25920664*x^28 - 111555328*x^26 + 349153248*x^24 - 794852864*x^22 + 1309625664*x^20 - 1548248640*x^18 + 1299305304*x^16 - 765197952*x^14 + 312117152*x^12 - 86580736*x^10 + 15867056*x^8 - 1828864*x^6 + 121280*x^4 - 3840*x^2 + 32, 1)
 

Normalized defining polynomial

\( x^{40} - 72 x^{38} + 2292 x^{36} - 42688 x^{34} + 518914 x^{32} - 4350192 x^{30} + 25920664 x^{28} - 111555328 x^{26} + 349153248 x^{24} - 794852864 x^{22} + 1309625664 x^{20} - 1548248640 x^{18} + 1299305304 x^{16} - 765197952 x^{14} + 312117152 x^{12} - 86580736 x^{10} + 15867056 x^{8} - 1828864 x^{6} + 121280 x^{4} - 3840 x^{2} + 32 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $40$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[40, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(96430685261162182749113906515642066253992366248338958954046471967872161601814528=2^{155}\cdot 11^{32}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $99.91$
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$$\chi_{352}(1,·)$, $\chi_{352}(5,·)$, $\chi_{352}(257,·)$, $\chi_{352}(9,·)$, $\chi_{352}(269,·)$, $\chi_{352}(45,·)$, $\chi_{352}(273,·)$, $\chi_{352}(89,·)$, $\chi_{352}(25,·)$, $\chi_{352}(265,·)$, $\chi_{352}(157,·)$, $\chi_{352}(133,·)$, $\chi_{352}(289,·)$, $\chi_{352}(37,·)$, $\chi_{352}(49,·)$, $\chi_{352}(169,·)$, $\chi_{352}(301,·)$, $\chi_{352}(93,·)$, $\chi_{352}(177,·)$, $\chi_{352}(309,·)$, $\chi_{352}(137,·)$, $\chi_{352}(313,·)$, $\chi_{352}(317,·)$, $\chi_{352}(53,·)$, $\chi_{352}(181,·)$, $\chi_{352}(69,·)$, $\chi_{352}(97,·)$, $\chi_{352}(201,·)$, $\chi_{352}(333,·)$, $\chi_{352}(141,·)$, $\chi_{352}(81,·)$, $\chi_{352}(213,·)$, $\chi_{352}(185,·)$, $\chi_{352}(345,·)$, $\chi_{352}(221,·)$, $\chi_{352}(225,·)$, $\chi_{352}(229,·)$, $\chi_{352}(113,·)$, $\chi_{352}(245,·)$, $\chi_{352}(125,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{2} a^{12}$, $\frac{1}{2} a^{13}$, $\frac{1}{2} a^{14}$, $\frac{1}{2} a^{15}$, $\frac{1}{4} a^{16}$, $\frac{1}{4} a^{17}$, $\frac{1}{4} a^{18}$, $\frac{1}{4} a^{19}$, $\frac{1}{4} a^{20}$, $\frac{1}{4} a^{21}$, $\frac{1}{4} a^{22}$, $\frac{1}{4} a^{23}$, $\frac{1}{8} a^{24}$, $\frac{1}{8} a^{25}$, $\frac{1}{8} a^{26}$, $\frac{1}{8} a^{27}$, $\frac{1}{8} a^{28}$, $\frac{1}{8} a^{29}$, $\frac{1}{8} a^{30}$, $\frac{1}{8} a^{31}$, $\frac{1}{16} a^{32}$, $\frac{1}{16} a^{33}$, $\frac{1}{16} a^{34}$, $\frac{1}{16} a^{35}$, $\frac{1}{100336} a^{36} - \frac{1795}{100336} a^{34} + \frac{2579}{100336} a^{32} + \frac{305}{6271} a^{30} + \frac{2693}{50168} a^{28} - \frac{25}{25084} a^{26} + \frac{631}{50168} a^{24} + \frac{351}{12542} a^{22} - \frac{293}{25084} a^{20} - \frac{1035}{12542} a^{18} + \frac{1041}{25084} a^{16} - \frac{465}{12542} a^{14} + \frac{383}{12542} a^{12} + \frac{1327}{6271} a^{10} - \frac{2681}{12542} a^{8} - \frac{3051}{6271} a^{6} + \frac{1499}{6271} a^{4} + \frac{2209}{6271} a^{2} + \frac{2088}{6271}$, $\frac{1}{100336} a^{37} - \frac{1795}{100336} a^{35} + \frac{2579}{100336} a^{33} + \frac{305}{6271} a^{31} + \frac{2693}{50168} a^{29} - \frac{25}{25084} a^{27} + \frac{631}{50168} a^{25} + \frac{351}{12542} a^{23} - \frac{293}{25084} a^{21} - \frac{1035}{12542} a^{19} + \frac{1041}{25084} a^{17} - \frac{465}{12542} a^{15} + \frac{383}{12542} a^{13} + \frac{1327}{6271} a^{11} - \frac{2681}{12542} a^{9} - \frac{3051}{6271} a^{7} + \frac{1499}{6271} a^{5} + \frac{2209}{6271} a^{3} + \frac{2088}{6271} a$, $\frac{1}{50277416338424686093777005462144418517776} a^{38} + \frac{64841939243783995601803937325143123}{25138708169212343046888502731072209258888} a^{36} + \frac{179865112797785542516501751504162436853}{6284677042303085761722125682768052314722} a^{34} - \frac{606663352054164231707861068630606228533}{50277416338424686093777005462144418517776} a^{32} + \frac{1337789427633593458131914258060193148995}{25138708169212343046888502731072209258888} a^{30} - \frac{312003109023529472263213426510897665817}{6284677042303085761722125682768052314722} a^{28} + \frac{276305956565823144435674642509982864023}{25138708169212343046888502731072209258888} a^{26} - \frac{34629264057161305932238204117558506583}{6284677042303085761722125682768052314722} a^{24} - \frac{1077448491385276236500578669027796067575}{12569354084606171523444251365536104629444} a^{22} - \frac{787193084522227703810987886912977206927}{12569354084606171523444251365536104629444} a^{20} + \frac{246696994416685941658239577589291980133}{12569354084606171523444251365536104629444} a^{18} + \frac{519461728574506300858527415592226822411}{12569354084606171523444251365536104629444} a^{16} - \frac{166195963772033674541165883693570159737}{3142338521151542880861062841384026157361} a^{14} - \frac{497459244361003441529332978806172702615}{6284677042303085761722125682768052314722} a^{12} - \frac{416064899811288606300132351105451403162}{3142338521151542880861062841384026157361} a^{10} + \frac{635729164679928178838317738376904668653}{3142338521151542880861062841384026157361} a^{8} - \frac{886648884067543375413804118958530872588}{3142338521151542880861062841384026157361} a^{6} + \frac{1214456868241144568525247728176946636258}{3142338521151542880861062841384026157361} a^{4} + \frac{784713030257190449083064562976278778147}{3142338521151542880861062841384026157361} a^{2} - \frac{99636909455647602176679607998294349347}{3142338521151542880861062841384026157361}$, $\frac{1}{50277416338424686093777005462144418517776} a^{39} + \frac{64841939243783995601803937325143123}{25138708169212343046888502731072209258888} a^{37} + \frac{179865112797785542516501751504162436853}{6284677042303085761722125682768052314722} a^{35} - \frac{606663352054164231707861068630606228533}{50277416338424686093777005462144418517776} a^{33} + \frac{1337789427633593458131914258060193148995}{25138708169212343046888502731072209258888} a^{31} - \frac{312003109023529472263213426510897665817}{6284677042303085761722125682768052314722} a^{29} + \frac{276305956565823144435674642509982864023}{25138708169212343046888502731072209258888} a^{27} - \frac{34629264057161305932238204117558506583}{6284677042303085761722125682768052314722} a^{25} - \frac{1077448491385276236500578669027796067575}{12569354084606171523444251365536104629444} a^{23} - \frac{787193084522227703810987886912977206927}{12569354084606171523444251365536104629444} a^{21} + \frac{246696994416685941658239577589291980133}{12569354084606171523444251365536104629444} a^{19} + \frac{519461728574506300858527415592226822411}{12569354084606171523444251365536104629444} a^{17} - \frac{166195963772033674541165883693570159737}{3142338521151542880861062841384026157361} a^{15} - \frac{497459244361003441529332978806172702615}{6284677042303085761722125682768052314722} a^{13} - \frac{416064899811288606300132351105451403162}{3142338521151542880861062841384026157361} a^{11} + \frac{635729164679928178838317738376904668653}{3142338521151542880861062841384026157361} a^{9} - \frac{886648884067543375413804118958530872588}{3142338521151542880861062841384026157361} a^{7} + \frac{1214456868241144568525247728176946636258}{3142338521151542880861062841384026157361} a^{5} + \frac{784713030257190449083064562976278778147}{3142338521151542880861062841384026157361} a^{3} - \frac{99636909455647602176679607998294349347}{3142338521151542880861062841384026157361} a$

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

Class group and class number

Trivial group, which has order $1$ (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:  $39$
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:  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:  \( 1748425349013661700000000000 \) (assuming GRH)
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

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{32})^+\), 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 $40$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{4}$ $40$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{10}$ $40$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{8}$ $40$ $20^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{5}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{4}$ $40$ $40$

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