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Properties

Label 40.0.13030509980...5936.9
Degree $40$
Signature $[0, 20]$
Discriminant $2^{80}\cdot 3^{20}\cdot 11^{36}$
Root discriminant $59.96$
Ramified primes $2, 3, 11$
Class number Not computed
Class group Not computed
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![3486784401, 0, 0, 0, -387420489, 0, 0, 0, 43046721, 0, 0, 0, -4782969, 0, 0, 0, 531441, 0, 0, 0, -59049, 0, 0, 0, 6561, 0, 0, 0, -729, 0, 0, 0, 81, 0, 0, 0, -9, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - 9*x^36 + 81*x^32 - 729*x^28 + 6561*x^24 - 59049*x^20 + 531441*x^16 - 4782969*x^12 + 43046721*x^8 - 387420489*x^4 + 3486784401)
 
gp: K = bnfinit(x^40 - 9*x^36 + 81*x^32 - 729*x^28 + 6561*x^24 - 59049*x^20 + 531441*x^16 - 4782969*x^12 + 43046721*x^8 - 387420489*x^4 + 3486784401, 1)
 

Normalized defining polynomial

\( x^{40} - 9 x^{36} + 81 x^{32} - 729 x^{28} + 6561 x^{24} - 59049 x^{20} + 531441 x^{16} - 4782969 x^{12} + 43046721 x^{8} - 387420489 x^{4} + 3486784401 \)

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:  \(130305099804548492884220428175380349368393046678311823693003457545895936=2^{80}\cdot 3^{20}\cdot 11^{36}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.96$
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$$\chi_{264}(1,·)$, $\chi_{264}(131,·)$, $\chi_{264}(5,·)$, $\chi_{264}(7,·)$, $\chi_{264}(145,·)$, $\chi_{264}(149,·)$, $\chi_{264}(151,·)$, $\chi_{264}(25,·)$, $\chi_{264}(155,·)$, $\chi_{264}(29,·)$, $\chi_{264}(31,·)$, $\chi_{264}(35,·)$, $\chi_{264}(169,·)$, $\chi_{264}(173,·)$, $\chi_{264}(175,·)$, $\chi_{264}(49,·)$, $\chi_{264}(179,·)$, $\chi_{264}(53,·)$, $\chi_{264}(59,·)$, $\chi_{264}(193,·)$, $\chi_{264}(197,·)$, $\chi_{264}(199,·)$, $\chi_{264}(73,·)$, $\chi_{264}(203,·)$, $\chi_{264}(79,·)$, $\chi_{264}(83,·)$, $\chi_{264}(217,·)$, $\chi_{264}(221,·)$, $\chi_{264}(223,·)$, $\chi_{264}(97,·)$, $\chi_{264}(227,·)$, $\chi_{264}(101,·)$, $\chi_{264}(103,·)$, $\chi_{264}(107,·)$, $\chi_{264}(241,·)$, $\chi_{264}(245,·)$, $\chi_{264}(247,·)$, $\chi_{264}(251,·)$, $\chi_{264}(125,·)$, $\chi_{264}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{9} a^{4}$, $\frac{1}{9} a^{5}$, $\frac{1}{27} a^{6}$, $\frac{1}{27} a^{7}$, $\frac{1}{81} a^{8}$, $\frac{1}{81} a^{9}$, $\frac{1}{243} a^{10}$, $\frac{1}{243} a^{11}$, $\frac{1}{729} a^{12}$, $\frac{1}{729} a^{13}$, $\frac{1}{2187} a^{14}$, $\frac{1}{2187} a^{15}$, $\frac{1}{6561} a^{16}$, $\frac{1}{6561} a^{17}$, $\frac{1}{19683} a^{18}$, $\frac{1}{19683} a^{19}$, $\frac{1}{59049} a^{20}$, $\frac{1}{59049} a^{21}$, $\frac{1}{177147} a^{22}$, $\frac{1}{177147} a^{23}$, $\frac{1}{531441} a^{24}$, $\frac{1}{531441} a^{25}$, $\frac{1}{1594323} a^{26}$, $\frac{1}{1594323} a^{27}$, $\frac{1}{4782969} a^{28}$, $\frac{1}{4782969} a^{29}$, $\frac{1}{14348907} a^{30}$, $\frac{1}{14348907} a^{31}$, $\frac{1}{43046721} a^{32}$, $\frac{1}{43046721} a^{33}$, $\frac{1}{129140163} a^{34}$, $\frac{1}{129140163} a^{35}$, $\frac{1}{387420489} a^{36}$, $\frac{1}{387420489} a^{37}$, $\frac{1}{1162261467} a^{38}$, $\frac{1}{1162261467} a^{39}$

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:  \( \frac{1}{27} a^{6} \) (order $44$)
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

$C_2^2\times C_{10}$ (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{-66}) \), \(\Q(\sqrt{66}) \), \(\Q(i, \sqrt{6})\), \(\Q(i, \sqrt{11})\), \(\Q(i, \sqrt{66})\), \(\Q(\sqrt{6}, \sqrt{-11})\), \(\Q(\sqrt{6}, \sqrt{11})\), \(\Q(\sqrt{-6}, \sqrt{-11})\), \(\Q(\sqrt{-6}, \sqrt{11})\), \(\Q(\zeta_{11})^+\), 8.0.77720518656.3, 10.0.219503494144.1, 10.10.1706859170463744.1, 10.0.1706859170463744.1, \(\Q(\zeta_{11})\), \(\Q(\zeta_{44})^+\), 10.0.18775450875101184.1, 10.10.18775450875101184.1, 20.0.2983289065263288625233938941476864.4, \(\Q(\zeta_{44})\), 20.0.360977976896857923653306611918700544.10, 20.0.352517555563337816067682238201856.9, 20.20.360977976896857923653306611918700544.2, 20.0.352517555563337816067682238201856.8, 20.0.360977976896857923653306611918700544.3

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 ${\href{/LocalNumberField/5.5.0.1}{5} }^{8}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{8}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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
11Data not computed