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Properties

Label 40.0.13030509980...5936.2
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 $5500$ (GRH)
Class group $[5, 10, 110]$ (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, -4, 0, 15, 0, -56, 0, 209, 0, -780, 0, 2911, 0, -10864, 0, 40545, 0, -151316, 0, 564719, 0, -151316, 0, 40545, 0, -10864, 0, 2911, 0, -780, 0, 209, 0, -56, 0, 15, 0, -4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - 4*x^38 + 15*x^36 - 56*x^34 + 209*x^32 - 780*x^30 + 2911*x^28 - 10864*x^26 + 40545*x^24 - 151316*x^22 + 564719*x^20 - 151316*x^18 + 40545*x^16 - 10864*x^14 + 2911*x^12 - 780*x^10 + 209*x^8 - 56*x^6 + 15*x^4 - 4*x^2 + 1)
 
gp: K = bnfinit(x^40 - 4*x^38 + 15*x^36 - 56*x^34 + 209*x^32 - 780*x^30 + 2911*x^28 - 10864*x^26 + 40545*x^24 - 151316*x^22 + 564719*x^20 - 151316*x^18 + 40545*x^16 - 10864*x^14 + 2911*x^12 - 780*x^10 + 209*x^8 - 56*x^6 + 15*x^4 - 4*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{40} - 4 x^{38} + 15 x^{36} - 56 x^{34} + 209 x^{32} - 780 x^{30} + 2911 x^{28} - 10864 x^{26} + 40545 x^{24} - 151316 x^{22} + 564719 x^{20} - 151316 x^{18} + 40545 x^{16} - 10864 x^{14} + 2911 x^{12} - 780 x^{10} + 209 x^{8} - 56 x^{6} + 15 x^{4} - 4 x^{2} + 1 \)

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:  \(130305099804548492884220428175380349368393046678311823693003457545895936=2^{80}\cdot 3^{20}\cdot 11^{36}\)
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$$\chi_{264}(1,·)$, $\chi_{264}(259,·)$, $\chi_{264}(5,·)$, $\chi_{264}(263,·)$, $\chi_{264}(139,·)$, $\chi_{264}(145,·)$, $\chi_{264}(19,·)$, $\chi_{264}(149,·)$, $\chi_{264}(23,·)$, $\chi_{264}(25,·)$, $\chi_{264}(29,·)$, $\chi_{264}(163,·)$, $\chi_{264}(167,·)$, $\chi_{264}(169,·)$, $\chi_{264}(43,·)$, $\chi_{264}(173,·)$, $\chi_{264}(47,·)$, $\chi_{264}(49,·)$, $\chi_{264}(53,·)$, $\chi_{264}(191,·)$, $\chi_{264}(193,·)$, $\chi_{264}(67,·)$, $\chi_{264}(197,·)$, $\chi_{264}(71,·)$, $\chi_{264}(73,·)$, $\chi_{264}(211,·)$, $\chi_{264}(215,·)$, $\chi_{264}(217,·)$, $\chi_{264}(91,·)$, $\chi_{264}(221,·)$, $\chi_{264}(95,·)$, $\chi_{264}(97,·)$, $\chi_{264}(101,·)$, $\chi_{264}(235,·)$, $\chi_{264}(239,·)$, $\chi_{264}(241,·)$, $\chi_{264}(115,·)$, $\chi_{264}(245,·)$, $\chi_{264}(119,·)$, $\chi_{264}(125,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{564719} a^{22} - \frac{151316}{564719}$, $\frac{1}{564719} a^{23} - \frac{151316}{564719} a$, $\frac{1}{564719} a^{24} - \frac{151316}{564719} a^{2}$, $\frac{1}{564719} a^{25} - \frac{151316}{564719} a^{3}$, $\frac{1}{564719} a^{26} - \frac{151316}{564719} a^{4}$, $\frac{1}{564719} a^{27} - \frac{151316}{564719} a^{5}$, $\frac{1}{564719} a^{28} - \frac{151316}{564719} a^{6}$, $\frac{1}{564719} a^{29} - \frac{151316}{564719} a^{7}$, $\frac{1}{564719} a^{30} - \frac{151316}{564719} a^{8}$, $\frac{1}{564719} a^{31} - \frac{151316}{564719} a^{9}$, $\frac{1}{564719} a^{32} - \frac{151316}{564719} a^{10}$, $\frac{1}{564719} a^{33} - \frac{151316}{564719} a^{11}$, $\frac{1}{564719} a^{34} - \frac{151316}{564719} a^{12}$, $\frac{1}{564719} a^{35} - \frac{151316}{564719} a^{13}$, $\frac{1}{564719} a^{36} - \frac{151316}{564719} a^{14}$, $\frac{1}{564719} a^{37} - \frac{151316}{564719} a^{15}$, $\frac{1}{564719} a^{38} - \frac{151316}{564719} a^{16}$, $\frac{1}{564719} a^{39} - \frac{151316}{564719} a^{17}$

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

Class group and class number

$C_{5}\times C_{10}\times C_{110}$, which has order $5500$ (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:  \( \frac{209}{564719} a^{32} + \frac{408855776}{564719} a^{10} \) (order $22$)
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:  \( 24039412784024376 \) (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

\(\Q(\sqrt{22}) \), \(\Q(\sqrt{-33}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{66}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-6}, \sqrt{22})\), \(\Q(\sqrt{3}, \sqrt{22})\), \(\Q(\sqrt{-2}, \sqrt{-11})\), \(\Q(\sqrt{-2}, \sqrt{-33})\), \(\Q(\sqrt{3}, \sqrt{-11})\), \(\Q(\sqrt{-6}, \sqrt{-11})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\zeta_{11})^+\), 8.0.77720518656.2, 10.10.77265229938688.1, 10.0.586732839846912.1, 10.0.1706859170463744.1, 10.10.18775450875101184.1, 10.10.53339349076992.1, 10.0.7024111812608.1, \(\Q(\zeta_{11})\), 20.0.360977976896857923653306611918700544.4, 20.20.360977976896857923653306611918700544.1, 20.0.5969915757478328440239161344.5, 20.0.360977976896857923653306611918700544.6, 20.0.344255425354822086003595935744.3, 20.0.352517555563337816067682238201856.8, 20.0.2983289065263288625233938941476864.1

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 ${\href{/LocalNumberField/13.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{8}$

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$3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$