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Properties

Label 40.0.89000136469...4496.1
Degree $40$
Signature $[0, 20]$
Discriminant $2^{80}\cdot 3^{20}\cdot 11^{32}$
Root discriminant $47.18$
Ramified primes $2, 3, 11$
Class number $125$ (GRH)
Class group $[5, 5, 5]$ (GRH)
Galois group $C_2^2\times C_{10}$ (as 40T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, 0, -155, 0, 0, 0, 23622, 0, 0, 0, -62097, 0, 0, 0, 133864, 0, 0, 0, -66403, 0, 0, 0, 23626, 0, 0, 0, -3794, 0, 0, 0, 441, 0, 0, 0, -25, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - 25*x^36 + 441*x^32 - 3794*x^28 + 23626*x^24 - 66403*x^20 + 133864*x^16 - 62097*x^12 + 23622*x^8 - 155*x^4 + 1)
 
gp: K = bnfinit(x^40 - 25*x^36 + 441*x^32 - 3794*x^28 + 23626*x^24 - 66403*x^20 + 133864*x^16 - 62097*x^12 + 23622*x^8 - 155*x^4 + 1, 1)
 

Normalized defining polynomial

\( x^{40} - 25 x^{36} + 441 x^{32} - 3794 x^{28} + 23626 x^{24} - 66403 x^{20} + 133864 x^{16} - 62097 x^{12} + 23622 x^{8} - 155 x^{4} + 1 \)

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:  \(8900013646919506378267907122148784192909845412083315599549447274496=2^{80}\cdot 3^{20}\cdot 11^{32}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.18$
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:  $2, 3, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(264=2^{3}\cdot 3\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{264}(1,·)$, $\chi_{264}(133,·)$, $\chi_{264}(257,·)$, $\chi_{264}(137,·)$, $\chi_{264}(23,·)$, $\chi_{264}(25,·)$, $\chi_{264}(155,·)$, $\chi_{264}(157,·)$, $\chi_{264}(5,·)$, $\chi_{264}(163,·)$, $\chi_{264}(37,·)$, $\chi_{264}(169,·)$, $\chi_{264}(71,·)$, $\chi_{264}(47,·)$, $\chi_{264}(49,·)$, $\chi_{264}(179,·)$, $\chi_{264}(181,·)$, $\chi_{264}(185,·)$, $\chi_{264}(31,·)$, $\chi_{264}(53,·)$, $\chi_{264}(67,·)$, $\chi_{264}(199,·)$, $\chi_{264}(203,·)$, $\chi_{264}(89,·)$, $\chi_{264}(91,·)$, $\chi_{264}(221,·)$, $\chi_{264}(223,·)$, $\chi_{264}(97,·)$, $\chi_{264}(59,·)$, $\chi_{264}(229,·)$, $\chi_{264}(103,·)$, $\chi_{264}(235,·)$, $\chi_{264}(191,·)$, $\chi_{264}(113,·)$, $\chi_{264}(115,·)$, $\chi_{264}(245,·)$, $\chi_{264}(119,·)$, $\chi_{264}(251,·)$, $\chi_{264}(125,·)$, $\chi_{264}(247,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $\frac{1}{89893} a^{32} + \frac{14059}{89893} a^{28} + \frac{7913}{89893} a^{24} - \frac{39223}{89893} a^{20} - \frac{10045}{89893} a^{16} + \frac{32533}{89893} a^{12} + \frac{42331}{89893} a^{8} + \frac{10410}{89893} a^{4} + \frac{32919}{89893}$, $\frac{1}{89893} a^{33} + \frac{14059}{89893} a^{29} + \frac{7913}{89893} a^{25} - \frac{39223}{89893} a^{21} - \frac{10045}{89893} a^{17} + \frac{32533}{89893} a^{13} + \frac{42331}{89893} a^{9} + \frac{10410}{89893} a^{5} + \frac{32919}{89893} a$, $\frac{1}{89893} a^{34} + \frac{14059}{89893} a^{30} + \frac{7913}{89893} a^{26} - \frac{39223}{89893} a^{22} - \frac{10045}{89893} a^{18} + \frac{32533}{89893} a^{14} + \frac{42331}{89893} a^{10} + \frac{10410}{89893} a^{6} + \frac{32919}{89893} a^{2}$, $\frac{1}{89893} a^{35} + \frac{14059}{89893} a^{31} + \frac{7913}{89893} a^{27} - \frac{39223}{89893} a^{23} - \frac{10045}{89893} a^{19} + \frac{32533}{89893} a^{15} + \frac{42331}{89893} a^{11} + \frac{10410}{89893} a^{7} + \frac{32919}{89893} a^{3}$, $\frac{1}{2273551084786794990343} a^{36} - \frac{4729965627837937}{2273551084786794990343} a^{32} - \frac{150825324919640204024}{2273551084786794990343} a^{28} + \frac{364339556866868606498}{2273551084786794990343} a^{24} - \frac{1120154953666675125164}{2273551084786794990343} a^{20} + \frac{500384154063620494455}{2273551084786794990343} a^{16} + \frac{311678175195741088383}{2273551084786794990343} a^{12} + \frac{752114182503327175171}{2273551084786794990343} a^{8} - \frac{1103175853111986812869}{2273551084786794990343} a^{4} - \frac{1109002502122879428970}{2273551084786794990343}$, $\frac{1}{2273551084786794990343} a^{37} - \frac{4729965627837937}{2273551084786794990343} a^{33} - \frac{150825324919640204024}{2273551084786794990343} a^{29} + \frac{364339556866868606498}{2273551084786794990343} a^{25} - \frac{1120154953666675125164}{2273551084786794990343} a^{21} + \frac{500384154063620494455}{2273551084786794990343} a^{17} + \frac{311678175195741088383}{2273551084786794990343} a^{13} + \frac{752114182503327175171}{2273551084786794990343} a^{9} - \frac{1103175853111986812869}{2273551084786794990343} a^{5} - \frac{1109002502122879428970}{2273551084786794990343} a$, $\frac{1}{2273551084786794990343} a^{38} - \frac{4729965627837937}{2273551084786794990343} a^{34} - \frac{150825324919640204024}{2273551084786794990343} a^{30} + \frac{364339556866868606498}{2273551084786794990343} a^{26} - \frac{1120154953666675125164}{2273551084786794990343} a^{22} + \frac{500384154063620494455}{2273551084786794990343} a^{18} + \frac{311678175195741088383}{2273551084786794990343} a^{14} + \frac{752114182503327175171}{2273551084786794990343} a^{10} - \frac{1103175853111986812869}{2273551084786794990343} a^{6} - \frac{1109002502122879428970}{2273551084786794990343} a^{2}$, $\frac{1}{2273551084786794990343} a^{39} - \frac{4729965627837937}{2273551084786794990343} a^{35} - \frac{150825324919640204024}{2273551084786794990343} a^{31} + \frac{364339556866868606498}{2273551084786794990343} a^{27} - \frac{1120154953666675125164}{2273551084786794990343} a^{23} + \frac{500384154063620494455}{2273551084786794990343} a^{19} + \frac{311678175195741088383}{2273551084786794990343} a^{15} + \frac{752114182503327175171}{2273551084786794990343} a^{11} - \frac{1103175853111986812869}{2273551084786794990343} a^{7} - \frac{1109002502122879428970}{2273551084786794990343} a^{3}$

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

Class group and class number

$C_{5}\times C_{5}\times C_{5}$, which has order $125$ (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:  $19$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{53581368526244142240}{2273551084786794990343} a^{37} + \frac{1339224586826730664500}{2273551084786794990343} a^{33} - \frac{23621564515385087475643}{2273551084786794990343} a^{29} + \frac{203149221171675960743220}{2273551084786794990343} a^{25} - \frac{1264704590067397349415492}{2273551084786794990343} a^{21} + \frac{3550359893243219213346364}{2273551084786794990343} a^{17} - \frac{7150303866121008375115512}{2273551084786794990343} a^{13} + \frac{3281262936679115684907084}{2273551084786794990343} a^{9} - \frac{1238571944153848101248881}{2273551084786794990343} a^{5} + \frac{139324111753014574752}{2273551084786794990343} a \) (order $24$)
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:  \( 7121371366090164.0 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_{10}$ (as 40T7):

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 $C_2^2\times C_{10}$
Character table for $C_2^2\times C_{10}$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(i, \sqrt{6})\), \(\Q(\zeta_{8})\), \(\Q(\zeta_{12})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{24})\), 10.0.219503494144.1, 10.0.1706859170463744.1, 10.10.1706859170463744.1, 10.0.7024111812608.1, 10.10.7024111812608.1, 10.10.53339349076992.1, 10.0.52089208083.1, 20.0.2983289065263288625233938941476864.4, 20.0.50522262278163705147147943936.1, 20.0.2845086159957207322343768064.1, 20.0.2983289065263288625233938941476864.1, 20.0.2913368227796180298080018497536.4, 20.0.2913368227796180298080018497536.2, 20.20.2983289065263288625233938941476864.1

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 ${\href{/LocalNumberField/5.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$