↑

Properties

Label 40.0.32473210254...0000.2
Degree $40$
Signature $[0, 20]$
Discriminant $2^{40}\cdot 3^{20}\cdot 5^{70}$
Root discriminant $57.91$
Ramified primes $2, 3, 5$
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![25, 0, 625, 0, 11250, 0, 82625, 0, 409750, 0, 1285650, 0, 2915250, 0, 4782125, 0, 5985150, 0, 5771300, 0, 4412520, 0, 2703375, 0, 1346750, 0, 546975, 0, 181750, 0, 49005, 0, 10625, 0, 1800, 0, 230, 0, 20, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 + 20*x^38 + 230*x^36 + 1800*x^34 + 10625*x^32 + 49005*x^30 + 181750*x^28 + 546975*x^26 + 1346750*x^24 + 2703375*x^22 + 4412520*x^20 + 5771300*x^18 + 5985150*x^16 + 4782125*x^14 + 2915250*x^12 + 1285650*x^10 + 409750*x^8 + 82625*x^6 + 11250*x^4 + 625*x^2 + 25)
 
gp: K = bnfinit(x^40 + 20*x^38 + 230*x^36 + 1800*x^34 + 10625*x^32 + 49005*x^30 + 181750*x^28 + 546975*x^26 + 1346750*x^24 + 2703375*x^22 + 4412520*x^20 + 5771300*x^18 + 5985150*x^16 + 4782125*x^14 + 2915250*x^12 + 1285650*x^10 + 409750*x^8 + 82625*x^6 + 11250*x^4 + 625*x^2 + 25, 1)
 

Normalized defining polynomial

\( x^{40} + 20 x^{38} + 230 x^{36} + 1800 x^{34} + 10625 x^{32} + 49005 x^{30} + 181750 x^{28} + 546975 x^{26} + 1346750 x^{24} + 2703375 x^{22} + 4412520 x^{20} + 5771300 x^{18} + 5985150 x^{16} + 4782125 x^{14} + 2915250 x^{12} + 1285650 x^{10} + 409750 x^{8} + 82625 x^{6} + 11250 x^{4} + 625 x^{2} + 25 \)

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:  \(32473210254684090614318847656250000000000000000000000000000000000000000=2^{40}\cdot 3^{20}\cdot 5^{70}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.91$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(300=2^{2}\cdot 3\cdot 5^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{300}(1,·)$, $\chi_{300}(7,·)$, $\chi_{300}(269,·)$, $\chi_{300}(143,·)$, $\chi_{300}(149,·)$, $\chi_{300}(23,·)$, $\chi_{300}(281,·)$, $\chi_{300}(283,·)$, $\chi_{300}(29,·)$, $\chi_{300}(287,·)$, $\chi_{300}(161,·)$, $\chi_{300}(163,·)$, $\chi_{300}(167,·)$, $\chi_{300}(41,·)$, $\chi_{300}(43,·)$, $\chi_{300}(47,·)$, $\chi_{300}(49,·)$, $\chi_{300}(181,·)$, $\chi_{300}(187,·)$, $\chi_{300}(61,·)$, $\chi_{300}(67,·)$, $\chi_{300}(289,·)$, $\chi_{300}(203,·)$, $\chi_{300}(209,·)$, $\chi_{300}(83,·)$, $\chi_{300}(89,·)$, $\chi_{300}(221,·)$, $\chi_{300}(223,·)$, $\chi_{300}(263,·)$, $\chi_{300}(227,·)$, $\chi_{300}(101,·)$, $\chi_{300}(103,·)$, $\chi_{300}(107,·)$, $\chi_{300}(109,·)$, $\chi_{300}(229,·)$, $\chi_{300}(241,·)$, $\chi_{300}(169,·)$, $\chi_{300}(121,·)$, $\chi_{300}(127,·)$, $\chi_{300}(247,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{5} a^{20}$, $\frac{1}{5} a^{21}$, $\frac{1}{5} a^{22}$, $\frac{1}{5} a^{23}$, $\frac{1}{5} a^{24}$, $\frac{1}{5} a^{25}$, $\frac{1}{5} a^{26}$, $\frac{1}{5} a^{27}$, $\frac{1}{5} a^{28}$, $\frac{1}{5} a^{29}$, $\frac{1}{5} a^{30}$, $\frac{1}{5} a^{31}$, $\frac{1}{5} a^{32}$, $\frac{1}{5} a^{33}$, $\frac{1}{5} a^{34}$, $\frac{1}{5} a^{35}$, $\frac{1}{159215} a^{36} + \frac{8053}{159215} a^{34} + \frac{2127}{159215} a^{32} + \frac{7682}{159215} a^{30} + \frac{2753}{159215} a^{28} + \frac{4261}{159215} a^{26} + \frac{2812}{159215} a^{24} - \frac{1039}{159215} a^{22} + \frac{2399}{31843} a^{20} - \frac{5561}{31843} a^{18} - \frac{1878}{31843} a^{16} - \frac{474}{31843} a^{14} + \frac{3072}{31843} a^{12} + \frac{3375}{31843} a^{10} - \frac{8693}{31843} a^{8} + \frac{1671}{31843} a^{6} + \frac{10754}{31843} a^{4} + \frac{695}{31843} a^{2} - \frac{15756}{31843}$, $\frac{1}{159215} a^{37} + \frac{8053}{159215} a^{35} + \frac{2127}{159215} a^{33} + \frac{7682}{159215} a^{31} + \frac{2753}{159215} a^{29} + \frac{4261}{159215} a^{27} + \frac{2812}{159215} a^{25} - \frac{1039}{159215} a^{23} + \frac{2399}{31843} a^{21} - \frac{5561}{31843} a^{19} - \frac{1878}{31843} a^{17} - \frac{474}{31843} a^{15} + \frac{3072}{31843} a^{13} + \frac{3375}{31843} a^{11} - \frac{8693}{31843} a^{9} + \frac{1671}{31843} a^{7} + \frac{10754}{31843} a^{5} + \frac{695}{31843} a^{3} - \frac{15756}{31843} a$, $\frac{1}{10876794422132693343948145156820465} a^{38} + \frac{14355522572032803866175124596}{10876794422132693343948145156820465} a^{36} + \frac{485846662657128933365659544638007}{10876794422132693343948145156820465} a^{34} - \frac{300875521503833909750788197173597}{10876794422132693343948145156820465} a^{32} + \frac{865345983103069741592158676659313}{10876794422132693343948145156820465} a^{30} - \frac{884478005145794918317061643038661}{10876794422132693343948145156820465} a^{28} + \frac{1058154049078152100580338820567644}{10876794422132693343948145156820465} a^{26} - \frac{153881528056580792036365666764644}{1553827774590384763421163593831495} a^{24} - \frac{131686810222769340819645764031292}{10876794422132693343948145156820465} a^{22} + \frac{66254246094453883806450084787442}{1553827774590384763421163593831495} a^{20} + \frac{486026532708855576287110615432147}{2175358884426538668789629031364093} a^{18} + \frac{79002672534628199647942534400362}{2175358884426538668789629031364093} a^{16} + \frac{285273036151113713803538402910557}{2175358884426538668789629031364093} a^{14} - \frac{496331422646368633908738106239974}{2175358884426538668789629031364093} a^{12} + \frac{875248195021723048288807824797370}{2175358884426538668789629031364093} a^{10} + \frac{113211757827163550668905558206330}{310765554918076952684232718766299} a^{8} - \frac{540475433033592763116404868399206}{2175358884426538668789629031364093} a^{6} + \frac{426690240287441484021000649092671}{2175358884426538668789629031364093} a^{4} - \frac{857867852469423921386809185858628}{2175358884426538668789629031364093} a^{2} - \frac{637916619361443036488681813770952}{2175358884426538668789629031364093}$, $\frac{1}{10876794422132693343948145156820465} a^{39} + \frac{14355522572032803866175124596}{10876794422132693343948145156820465} a^{37} + \frac{485846662657128933365659544638007}{10876794422132693343948145156820465} a^{35} - \frac{300875521503833909750788197173597}{10876794422132693343948145156820465} a^{33} + \frac{865345983103069741592158676659313}{10876794422132693343948145156820465} a^{31} - \frac{884478005145794918317061643038661}{10876794422132693343948145156820465} a^{29} + \frac{1058154049078152100580338820567644}{10876794422132693343948145156820465} a^{27} - \frac{153881528056580792036365666764644}{1553827774590384763421163593831495} a^{25} - \frac{131686810222769340819645764031292}{10876794422132693343948145156820465} a^{23} + \frac{66254246094453883806450084787442}{1553827774590384763421163593831495} a^{21} + \frac{486026532708855576287110615432147}{2175358884426538668789629031364093} a^{19} + \frac{79002672534628199647942534400362}{2175358884426538668789629031364093} a^{17} + \frac{285273036151113713803538402910557}{2175358884426538668789629031364093} a^{15} - \frac{496331422646368633908738106239974}{2175358884426538668789629031364093} a^{13} + \frac{875248195021723048288807824797370}{2175358884426538668789629031364093} a^{11} + \frac{113211757827163550668905558206330}{310765554918076952684232718766299} a^{9} - \frac{540475433033592763116404868399206}{2175358884426538668789629031364093} a^{7} + \frac{426690240287441484021000649092671}{2175358884426538668789629031364093} a^{5} - \frac{857867852469423921386809185858628}{2175358884426538668789629031364093} a^{3} - \frac{637916619361443036488681813770952}{2175358884426538668789629031364093} a$

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:  \( \frac{34771395730757978124566515565983}{10876794422132693343948145156820465} a^{38} + \frac{692283897168111053725510776099378}{10876794422132693343948145156820465} a^{36} + \frac{1587052547053860559766113339320657}{2175358884426538668789629031364093} a^{34} + \frac{12375918588351769666644002884613743}{2175358884426538668789629031364093} a^{32} + \frac{363948310199062342368789764166523708}{10876794422132693343948145156820465} a^{30} + \frac{334362934312097178180994489480061962}{2175358884426538668789629031364093} a^{28} + \frac{1234580133632388750749612083207465799}{2175358884426538668789629031364093} a^{26} + \frac{3696145712529795454860576250672591245}{2175358884426538668789629031364093} a^{24} + \frac{9045892737790323647042167510486708285}{2175358884426538668789629031364093} a^{22} + \frac{90123004371087405788337777342915478407}{10876794422132693343948145156820465} a^{20} + \frac{29158312147958218840435273548763128420}{2175358884426538668789629031364093} a^{18} + \frac{37698071349679749583427711324594752322}{2175358884426538668789629031364093} a^{16} + \frac{38528234138397276629821371218314527235}{2175358884426538668789629031364093} a^{14} + \frac{30166374776853215627973237685016843565}{2175358884426538668789629031364093} a^{12} + \frac{17933471660986256714922124117907747132}{2175358884426538668789629031364093} a^{10} + \frac{7607606223328870595462112891904849700}{2175358884426538668789629031364093} a^{8} + \frac{2324457873859495166602237549634081505}{2175358884426538668789629031364093} a^{6} + \frac{427937480517537119948273759350450550}{2175358884426538668789629031364093} a^{4} + \frac{59899867790215038918631270886861075}{2175358884426538668789629031364093} a^{2} + \frac{471995768929960333928148596728894}{310765554918076952684232718766299} \) (order $6$)
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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\zeta_{20})^+\), 4.0.18000.1, 5.5.390625.1, 8.0.324000000.2, 10.0.37078857421875.1, \(\Q(\zeta_{25})^+\), 10.0.185394287109375.1, 20.0.34371041692793369293212890625.1, \(\Q(\zeta_{100})^+\), 20.0.180203247070312500000000000000000000.1

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{10}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{4}$ $20^{2}$ $20^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{8}$ $20^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ $20^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{10}$ $20^{2}$ $20^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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
2Data not computed
3Data not computed
5Data not computed