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Properties

Label 40.0.13030509980...5936.3
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 $12100$ (GRH)
Class group $[110, 110]$ (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, 0, 0, 820, 0, 0, 0, 41990, 0, 0, 0, 222625, 0, 0, 0, 379157, 0, 0, 0, 259698, 0, 0, 0, 85046, 0, 0, 0, 14374, 0, 0, 0, 1280, 0, 0, 0, 57, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 + 57*x^36 + 1280*x^32 + 14374*x^28 + 85046*x^24 + 259698*x^20 + 379157*x^16 + 222625*x^12 + 41990*x^8 + 820*x^4 + 1)
 
gp: K = bnfinit(x^40 + 57*x^36 + 1280*x^32 + 14374*x^28 + 85046*x^24 + 259698*x^20 + 379157*x^16 + 222625*x^12 + 41990*x^8 + 820*x^4 + 1, 1)
 

Normalized defining polynomial

\( x^{40} + 57 x^{36} + 1280 x^{32} + 14374 x^{28} + 85046 x^{24} + 259698 x^{20} + 379157 x^{16} + 222625 x^{12} + 41990 x^{8} + 820 x^{4} + 1 \)

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:  \(130305099804548492884220428175380349368393046678311823693003457545895936=2^{80}\cdot 3^{20}\cdot 11^{36}\)
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$$\chi_{264}(1,·)$, $\chi_{264}(259,·)$, $\chi_{264}(5,·)$, $\chi_{264}(263,·)$, $\chi_{264}(139,·)$, $\chi_{264}(13,·)$, $\chi_{264}(17,·)$, $\chi_{264}(19,·)$, $\chi_{264}(25,·)$, $\chi_{264}(155,·)$, $\chi_{264}(31,·)$, $\chi_{264}(161,·)$, $\chi_{264}(167,·)$, $\chi_{264}(41,·)$, $\chi_{264}(95,·)$, $\chi_{264}(43,·)$, $\chi_{264}(49,·)$, $\chi_{264}(179,·)$, $\chi_{264}(53,·)$, $\chi_{264}(59,·)$, $\chi_{264}(61,·)$, $\chi_{264}(65,·)$, $\chi_{264}(199,·)$, $\chi_{264}(203,·)$, $\chi_{264}(205,·)$, $\chi_{264}(211,·)$, $\chi_{264}(85,·)$, $\chi_{264}(215,·)$, $\chi_{264}(221,·)$, $\chi_{264}(223,·)$, $\chi_{264}(97,·)$, $\chi_{264}(103,·)$, $\chi_{264}(233,·)$, $\chi_{264}(109,·)$, $\chi_{264}(239,·)$, $\chi_{264}(245,·)$, $\chi_{264}(169,·)$, $\chi_{264}(251,·)$, $\chi_{264}(125,·)$, $\chi_{264}(247,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $\frac{1}{14118201587196095474227} a^{36} - \frac{2651947530383184348186}{14118201587196095474227} a^{32} - \frac{6340003673962139395778}{14118201587196095474227} a^{28} - \frac{3400349059480953373554}{14118201587196095474227} a^{24} + \frac{2902670951706543829076}{14118201587196095474227} a^{20} + \frac{6369967583850831293907}{14118201587196095474227} a^{16} - \frac{6125273777511648692738}{14118201587196095474227} a^{12} + \frac{4476873473504989455572}{14118201587196095474227} a^{8} + \frac{2582026534985203037790}{14118201587196095474227} a^{4} - \frac{2015164774139463173484}{14118201587196095474227}$, $\frac{1}{14118201587196095474227} a^{37} - \frac{2651947530383184348186}{14118201587196095474227} a^{33} - \frac{6340003673962139395778}{14118201587196095474227} a^{29} - \frac{3400349059480953373554}{14118201587196095474227} a^{25} + \frac{2902670951706543829076}{14118201587196095474227} a^{21} + \frac{6369967583850831293907}{14118201587196095474227} a^{17} - \frac{6125273777511648692738}{14118201587196095474227} a^{13} + \frac{4476873473504989455572}{14118201587196095474227} a^{9} + \frac{2582026534985203037790}{14118201587196095474227} a^{5} - \frac{2015164774139463173484}{14118201587196095474227} a$, $\frac{1}{14118201587196095474227} a^{38} - \frac{2651947530383184348186}{14118201587196095474227} a^{34} - \frac{6340003673962139395778}{14118201587196095474227} a^{30} - \frac{3400349059480953373554}{14118201587196095474227} a^{26} + \frac{2902670951706543829076}{14118201587196095474227} a^{22} + \frac{6369967583850831293907}{14118201587196095474227} a^{18} - \frac{6125273777511648692738}{14118201587196095474227} a^{14} + \frac{4476873473504989455572}{14118201587196095474227} a^{10} + \frac{2582026534985203037790}{14118201587196095474227} a^{6} - \frac{2015164774139463173484}{14118201587196095474227} a^{2}$, $\frac{1}{14118201587196095474227} a^{39} - \frac{2651947530383184348186}{14118201587196095474227} a^{35} - \frac{6340003673962139395778}{14118201587196095474227} a^{31} - \frac{3400349059480953373554}{14118201587196095474227} a^{27} + \frac{2902670951706543829076}{14118201587196095474227} a^{23} + \frac{6369967583850831293907}{14118201587196095474227} a^{19} - \frac{6125273777511648692738}{14118201587196095474227} a^{15} + \frac{4476873473504989455572}{14118201587196095474227} a^{11} + \frac{2582026534985203037790}{14118201587196095474227} a^{7} - \frac{2015164774139463173484}{14118201587196095474227} a^{3}$

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

Class group and class number

$C_{110}\times C_{110}$, which has order $12100$ (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:  \( \frac{398772571518731004694}{14118201587196095474227} a^{38} + \frac{22731418606447681518718}{14118201587196095474227} a^{34} + \frac{510507728716834005253668}{14118201587196095474227} a^{30} + \frac{5733729459516139143490252}{14118201587196095474227} a^{26} + \frac{33933956932744042548205033}{14118201587196095474227} a^{22} + \frac{103678878142004921547988119}{14118201587196095474227} a^{18} + \frac{151561773478616221575224766}{14118201587196095474227} a^{14} + \frac{89318133696368223029564203}{14118201587196095474227} a^{10} + \frac{17080580897272744456909535}{14118201587196095474227} a^{6} + \frac{412784981084028852042315}{14118201587196095474227} a^{2} \) (order $4$)
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:  \( 1535102994471753.0 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-33}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{22}) \), \(\Q(\sqrt{-22}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{6}) \), \(\Q(i, \sqrt{33})\), \(\Q(i, \sqrt{22})\), \(\Q(i, \sqrt{6})\), \(\Q(\sqrt{-6}, \sqrt{22})\), \(\Q(\sqrt{6}, \sqrt{-22})\), \(\Q(\sqrt{6}, \sqrt{22})\), \(\Q(\sqrt{-6}, \sqrt{-22})\), \(\Q(\zeta_{11})^+\), 8.0.77720518656.1, 10.0.219503494144.1, 10.0.586732839846912.1, \(\Q(\zeta_{33})^+\), 10.10.77265229938688.1, 10.0.77265229938688.1, 10.0.1706859170463744.1, 10.10.1706859170463744.1, 20.0.344255425354822086003595935744.1, 20.0.6113193735657808322804901216256.4, 20.0.2983289065263288625233938941476864.4, 20.0.360977976896857923653306611918700544.4, 20.0.360977976896857923653306611918700544.5, 20.20.352517555563337816067682238201856.2, 20.0.352517555563337816067682238201856.2

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 ${\href{/LocalNumberField/13.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{8}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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
11Data not computed