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Properties

Label 40.0.12426862698...4736.2
Degree $40$
Signature $[0, 20]$
Discriminant $2^{60}\cdot 3^{20}\cdot 11^{36}$
Root discriminant $42.40$
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![1048576, 0, 524288, 0, 0, 0, -131072, 0, -65536, 0, 0, 0, 16384, 0, 8192, 0, 0, 0, -2048, 0, -1024, 0, -512, 0, 0, 0, 128, 0, 64, 0, 0, 0, -16, 0, -8, 0, 0, 0, 2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 + 2*x^38 - 8*x^34 - 16*x^32 + 64*x^28 + 128*x^26 - 512*x^22 - 1024*x^20 - 2048*x^18 + 8192*x^14 + 16384*x^12 - 65536*x^8 - 131072*x^6 + 524288*x^2 + 1048576)
 
gp: K = bnfinit(x^40 + 2*x^38 - 8*x^34 - 16*x^32 + 64*x^28 + 128*x^26 - 512*x^22 - 1024*x^20 - 2048*x^18 + 8192*x^14 + 16384*x^12 - 65536*x^8 - 131072*x^6 + 524288*x^2 + 1048576, 1)
 

Normalized defining polynomial

\( x^{40} + 2 x^{38} - 8 x^{34} - 16 x^{32} + 64 x^{28} + 128 x^{26} - 512 x^{22} - 1024 x^{20} - 2048 x^{18} + 8192 x^{14} + 16384 x^{12} - 65536 x^{8} - 131072 x^{6} + 524288 x^{2} + 1048576 \)

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:  \(124268626980350964435787609267597531669991537741004775708201844736=2^{60}\cdot 3^{20}\cdot 11^{36}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.40$
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$$\chi_{264}(1,·)$, $\chi_{264}(131,·)$, $\chi_{264}(257,·)$, $\chi_{264}(137,·)$, $\chi_{264}(139,·)$, $\chi_{264}(115,·)$, $\chi_{264}(17,·)$, $\chi_{264}(19,·)$, $\chi_{264}(89,·)$, $\chi_{264}(25,·)$, $\chi_{264}(155,·)$, $\chi_{264}(161,·)$, $\chi_{264}(35,·)$, $\chi_{264}(113,·)$, $\chi_{264}(41,·)$, $\chi_{264}(193,·)$, $\chi_{264}(43,·)$, $\chi_{264}(49,·)$, $\chi_{264}(179,·)$, $\chi_{264}(185,·)$, $\chi_{264}(59,·)$, $\chi_{264}(65,·)$, $\chi_{264}(83,·)$, $\chi_{264}(67,·)$, $\chi_{264}(73,·)$, $\chi_{264}(203,·)$, $\chi_{264}(163,·)$, $\chi_{264}(217,·)$, $\chi_{264}(91,·)$, $\chi_{264}(97,·)$, $\chi_{264}(227,·)$, $\chi_{264}(145,·)$, $\chi_{264}(259,·)$, $\chi_{264}(233,·)$, $\chi_{264}(107,·)$, $\chi_{264}(241,·)$, $\chi_{264}(211,·)$, $\chi_{264}(235,·)$, $\chi_{264}(169,·)$, $\chi_{264}(251,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{32} a^{11}$, $\frac{1}{64} a^{12}$, $\frac{1}{64} a^{13}$, $\frac{1}{128} a^{14}$, $\frac{1}{128} a^{15}$, $\frac{1}{256} a^{16}$, $\frac{1}{256} a^{17}$, $\frac{1}{512} a^{18}$, $\frac{1}{512} a^{19}$, $\frac{1}{1024} a^{20}$, $\frac{1}{1024} a^{21}$, $\frac{1}{2048} a^{22}$, $\frac{1}{2048} a^{23}$, $\frac{1}{4096} a^{24}$, $\frac{1}{4096} a^{25}$, $\frac{1}{8192} a^{26}$, $\frac{1}{8192} a^{27}$, $\frac{1}{16384} a^{28}$, $\frac{1}{16384} a^{29}$, $\frac{1}{32768} a^{30}$, $\frac{1}{32768} a^{31}$, $\frac{1}{65536} a^{32}$, $\frac{1}{65536} a^{33}$, $\frac{1}{131072} a^{34}$, $\frac{1}{131072} a^{35}$, $\frac{1}{262144} a^{36}$, $\frac{1}{262144} a^{37}$, $\frac{1}{524288} a^{38}$, $\frac{1}{524288} a^{39}$

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:  \( -\frac{1}{131072} a^{34} + \frac{1}{64} a^{12} \) (order $66$)
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

\(\Q(\sqrt{22}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-66}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{6}, \sqrt{22})\), \(\Q(\sqrt{-3}, \sqrt{22})\), \(\Q(\sqrt{-2}, \sqrt{-11})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{6}, \sqrt{-11})\), \(\Q(\sqrt{-3}, \sqrt{-11})\), \(\Q(\sqrt{-2}, \sqrt{33})\), \(\Q(\zeta_{11})^+\), 8.0.4857532416.2, 10.10.77265229938688.1, 10.10.1706859170463744.1, \(\Q(\zeta_{33})^+\), 10.0.52089208083.1, 10.0.18775450875101184.1, 10.0.7024111812608.1, \(\Q(\zeta_{11})\), 20.20.352517555563337816067682238201856.2, 20.0.352517555563337816067682238201856.5, 20.0.5969915757478328440239161344.5, 20.0.2913368227796180298080018497536.2, 20.0.352517555563337816067682238201856.9, \(\Q(\zeta_{33})\), 20.0.352517555563337816067682238201856.7

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
$3$3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
11Data not computed