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Properties

Label 40.0.13030509980...5936.7
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 $341000$ (GRH)
Class group $[5, 10, 6820]$ (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, 480, 0, 17560, 0, 240976, 0, 1662386, 0, 6857500, 0, 18713229, 0, 35964944, 0, 50713585, 0, 53927016, 0, 44043506, 0, 27947920, 0, 13860054, 0, 5375528, 0, 1622694, 0, 376960, 0, 66044, 0, 8436, 0, 741, 0, 40, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 + 40*x^38 + 741*x^36 + 8436*x^34 + 66044*x^32 + 376960*x^30 + 1622694*x^28 + 5375528*x^26 + 13860054*x^24 + 27947920*x^22 + 44043506*x^20 + 53927016*x^18 + 50713585*x^16 + 35964944*x^14 + 18713229*x^12 + 6857500*x^10 + 1662386*x^8 + 240976*x^6 + 17560*x^4 + 480*x^2 + 1)
 
gp: K = bnfinit(x^40 + 40*x^38 + 741*x^36 + 8436*x^34 + 66044*x^32 + 376960*x^30 + 1622694*x^28 + 5375528*x^26 + 13860054*x^24 + 27947920*x^22 + 44043506*x^20 + 53927016*x^18 + 50713585*x^16 + 35964944*x^14 + 18713229*x^12 + 6857500*x^10 + 1662386*x^8 + 240976*x^6 + 17560*x^4 + 480*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{40} + 40 x^{38} + 741 x^{36} + 8436 x^{34} + 66044 x^{32} + 376960 x^{30} + 1622694 x^{28} + 5375528 x^{26} + 13860054 x^{24} + 27947920 x^{22} + 44043506 x^{20} + 53927016 x^{18} + 50713585 x^{16} + 35964944 x^{14} + 18713229 x^{12} + 6857500 x^{10} + 1662386 x^{8} + 240976 x^{6} + 17560 x^{4} + 480 x^{2} + 1 \)

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$$\chi_{264}(1,·)$, $\chi_{264}(131,·)$, $\chi_{264}(5,·)$, $\chi_{264}(7,·)$, $\chi_{264}(151,·)$, $\chi_{264}(13,·)$, $\chi_{264}(17,·)$, $\chi_{264}(23,·)$, $\chi_{264}(25,·)$, $\chi_{264}(47,·)$, $\chi_{264}(161,·)$, $\chi_{264}(35,·)$, $\chi_{264}(163,·)$, $\chi_{264}(41,·)$, $\chi_{264}(175,·)$, $\chi_{264}(49,·)$, $\chi_{264}(53,·)$, $\chi_{264}(61,·)$, $\chi_{264}(191,·)$, $\chi_{264}(65,·)$, $\chi_{264}(67,·)$, $\chi_{264}(71,·)$, $\chi_{264}(119,·)$, $\chi_{264}(205,·)$, $\chi_{264}(79,·)$, $\chi_{264}(83,·)$, $\chi_{264}(85,·)$, $\chi_{264}(91,·)$, $\chi_{264}(221,·)$, $\chi_{264}(97,·)$, $\chi_{264}(227,·)$, $\chi_{264}(233,·)$, $\chi_{264}(107,·)$, $\chi_{264}(109,·)$, $\chi_{264}(115,·)$, $\chi_{264}(235,·)$, $\chi_{264}(245,·)$, $\chi_{264}(169,·)$, $\chi_{264}(125,·)$, $\chi_{264}(127,·)$$\rbrace$
This is a CM field.

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

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

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

Class group and class number

$C_{5}\times C_{10}\times C_{6820}$, which has order $341000$ (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:  \( -1 \) (order $2$)
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:  \( 45016485591237.79 \) (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

\(\Q(\sqrt{-22}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{-66}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-6}, \sqrt{-22})\), \(\Q(\sqrt{-2}, \sqrt{11})\), \(\Q(\sqrt{3}, \sqrt{-22})\), \(\Q(\sqrt{-2}, \sqrt{33})\), \(\Q(\sqrt{3}, \sqrt{11})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\sqrt{-6}, \sqrt{11})\), \(\Q(\zeta_{11})^+\), 8.0.77720518656.4, 10.0.77265229938688.1, \(\Q(\zeta_{33})^+\), 10.0.1706859170463744.1, 10.0.7024111812608.1, \(\Q(\zeta_{44})^+\), 10.0.18775450875101184.1, 10.10.53339349076992.1, 20.0.352517555563337816067682238201856.2, 20.0.6113193735657808322804901216256.3, 20.0.360977976896857923653306611918700544.1, 20.0.352517555563337816067682238201856.7, \(\Q(\zeta_{132})^+\), 20.0.2983289065263288625233938941476864.1, 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 ${\href{/LocalNumberField/5.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{4}$ 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$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$