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Properties

Label 40.40.1130216578...0000.1
Degree $40$
Signature $[40, 0]$
Discriminant $2^{40}\cdot 3^{20}\cdot 5^{20}\cdot 11^{36}$
Root discriminant $67.04$
Ramified primes $2, 3, 5, 11$
Class number $1$ (GRH)
Class group Trivial (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, -120, 0, 4330, 0, -72774, 0, 681395, 0, -3949296, 0, 15148783, 0, -40198503, 0, 76047611, 0, -104672040, 0, 106185927, 0, -79942830, 0, 44732012, 0, -18536622, 0, 5639500, 0, -1241646, 0, 193556, 0, -20652, 0, 1425, 0, -57, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - 57*x^38 + 1425*x^36 - 20652*x^34 + 193556*x^32 - 1241646*x^30 + 5639500*x^28 - 18536622*x^26 + 44732012*x^24 - 79942830*x^22 + 106185927*x^20 - 104672040*x^18 + 76047611*x^16 - 40198503*x^14 + 15148783*x^12 - 3949296*x^10 + 681395*x^8 - 72774*x^6 + 4330*x^4 - 120*x^2 + 1)
 
gp: K = bnfinit(x^40 - 57*x^38 + 1425*x^36 - 20652*x^34 + 193556*x^32 - 1241646*x^30 + 5639500*x^28 - 18536622*x^26 + 44732012*x^24 - 79942830*x^22 + 106185927*x^20 - 104672040*x^18 + 76047611*x^16 - 40198503*x^14 + 15148783*x^12 - 3949296*x^10 + 681395*x^8 - 72774*x^6 + 4330*x^4 - 120*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{40} - 57 x^{38} + 1425 x^{36} - 20652 x^{34} + 193556 x^{32} - 1241646 x^{30} + 5639500 x^{28} - 18536622 x^{26} + 44732012 x^{24} - 79942830 x^{22} + 106185927 x^{20} - 104672040 x^{18} + 76047611 x^{16} - 40198503 x^{14} + 15148783 x^{12} - 3949296 x^{10} + 681395 x^{8} - 72774 x^{6} + 4330 x^{4} - 120 x^{2} + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $40$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[40, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(11302165783522556415463223790320401501047994110286233600000000000000000000=2^{40}\cdot 3^{20}\cdot 5^{20}\cdot 11^{36}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.04$
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:  $2, 3, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(660=2^{2}\cdot 3\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{660}(1,·)$, $\chi_{660}(259,·)$, $\chi_{660}(391,·)$, $\chi_{660}(139,·)$, $\chi_{660}(271,·)$, $\chi_{660}(529,·)$, $\chi_{660}(19,·)$, $\chi_{660}(149,·)$, $\chi_{660}(151,·)$, $\chi_{660}(281,·)$, $\chi_{660}(29,·)$, $\chi_{660}(161,·)$, $\chi_{660}(419,·)$, $\chi_{660}(421,·)$, $\chi_{660}(551,·)$, $\chi_{660}(169,·)$, $\chi_{660}(71,·)$, $\chi_{660}(301,·)$, $\chi_{660}(49,·)$, $\chi_{660}(179,·)$, $\chi_{660}(181,·)$, $\chi_{660}(311,·)$, $\chi_{660}(569,·)$, $\chi_{660}(59,·)$, $\chi_{660}(191,·)$, $\chi_{660}(289,·)$, $\chi_{660}(329,·)$, $\chi_{660}(439,·)$, $\chi_{660}(461,·)$, $\chi_{660}(79,·)$, $\chi_{660}(211,·)$, $\chi_{660}(251,·)$, $\chi_{660}(599,·)$, $\chi_{660}(101,·)$, $\chi_{660}(571,·)$, $\chi_{660}(229,·)$, $\chi_{660}(361,·)$, $\chi_{660}(629,·)$, $\chi_{660}(41,·)$, $\chi_{660}(119,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $\frac{1}{43} a^{36} - \frac{1}{43} a^{34} + \frac{8}{43} a^{32} - \frac{9}{43} a^{30} + \frac{16}{43} a^{28} + \frac{9}{43} a^{26} + \frac{20}{43} a^{24} - \frac{2}{43} a^{22} - \frac{12}{43} a^{20} + \frac{4}{43} a^{18} + \frac{8}{43} a^{16} + \frac{14}{43} a^{14} + \frac{5}{43} a^{12} + \frac{21}{43} a^{10} + \frac{16}{43} a^{8} + \frac{3}{43} a^{6} - \frac{5}{43} a^{4} + \frac{5}{43} a^{2} + \frac{20}{43}$, $\frac{1}{43} a^{37} - \frac{1}{43} a^{35} + \frac{8}{43} a^{33} - \frac{9}{43} a^{31} + \frac{16}{43} a^{29} + \frac{9}{43} a^{27} + \frac{20}{43} a^{25} - \frac{2}{43} a^{23} - \frac{12}{43} a^{21} + \frac{4}{43} a^{19} + \frac{8}{43} a^{17} + \frac{14}{43} a^{15} + \frac{5}{43} a^{13} + \frac{21}{43} a^{11} + \frac{16}{43} a^{9} + \frac{3}{43} a^{7} - \frac{5}{43} a^{5} + \frac{5}{43} a^{3} + \frac{20}{43} a$, $\frac{1}{11779850768470767736091164007281} a^{38} - \frac{104149634653874014884603201447}{11779850768470767736091164007281} a^{36} - \frac{5377651676694521217843332536714}{11779850768470767736091164007281} a^{34} + \frac{4611175375244917765629740518204}{11779850768470767736091164007281} a^{32} - \frac{3670876016133604166478561815893}{11779850768470767736091164007281} a^{30} + \frac{5095946805339043377761481926821}{11779850768470767736091164007281} a^{28} + \frac{1942700540901303335114061561559}{11779850768470767736091164007281} a^{26} - \frac{3329489896198655860787595735663}{11779850768470767736091164007281} a^{24} + \frac{5277780155769692499873597121384}{11779850768470767736091164007281} a^{22} - \frac{2157508012168933224462445102967}{11779850768470767736091164007281} a^{20} + \frac{1462773598614393377459364044762}{11779850768470767736091164007281} a^{18} - \frac{135981509753506197371808370755}{273950017871413203164910790867} a^{16} - \frac{714006898902038560520997388914}{11779850768470767736091164007281} a^{14} + \frac{696138276232508111825750606962}{11779850768470767736091164007281} a^{12} - \frac{3847446416094753666655601035014}{11779850768470767736091164007281} a^{10} + \frac{50464044798993822172186615558}{512167424716120336351789739447} a^{8} - \frac{774428790760829460683038757344}{11779850768470767736091164007281} a^{6} + \frac{2063152560074271105115518306491}{11779850768470767736091164007281} a^{4} - \frac{2577253213598548182807296352060}{11779850768470767736091164007281} a^{2} + \frac{569466228461565869621141555609}{11779850768470767736091164007281}$, $\frac{1}{11779850768470767736091164007281} a^{39} - \frac{104149634653874014884603201447}{11779850768470767736091164007281} a^{37} - \frac{5377651676694521217843332536714}{11779850768470767736091164007281} a^{35} + \frac{4611175375244917765629740518204}{11779850768470767736091164007281} a^{33} - \frac{3670876016133604166478561815893}{11779850768470767736091164007281} a^{31} + \frac{5095946805339043377761481926821}{11779850768470767736091164007281} a^{29} + \frac{1942700540901303335114061561559}{11779850768470767736091164007281} a^{27} - \frac{3329489896198655860787595735663}{11779850768470767736091164007281} a^{25} + \frac{5277780155769692499873597121384}{11779850768470767736091164007281} a^{23} - \frac{2157508012168933224462445102967}{11779850768470767736091164007281} a^{21} + \frac{1462773598614393377459364044762}{11779850768470767736091164007281} a^{19} - \frac{135981509753506197371808370755}{273950017871413203164910790867} a^{17} - \frac{714006898902038560520997388914}{11779850768470767736091164007281} a^{15} + \frac{696138276232508111825750606962}{11779850768470767736091164007281} a^{13} - \frac{3847446416094753666655601035014}{11779850768470767736091164007281} a^{11} + \frac{50464044798993822172186615558}{512167424716120336351789739447} a^{9} - \frac{774428790760829460683038757344}{11779850768470767736091164007281} a^{7} + \frac{2063152560074271105115518306491}{11779850768470767736091164007281} a^{5} - \frac{2577253213598548182807296352060}{11779850768470767736091164007281} a^{3} + \frac{569466228461565869621141555609}{11779850768470767736091164007281} a$

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

Class group and class number

Trivial group, which has order $1$ (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:  $39$
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:  \( 1149329825881126000000000 \) (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

\(\Q(\sqrt{33}) \), \(\Q(\sqrt{165}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{55}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{5}, \sqrt{33})\), \(\Q(\sqrt{15}, \sqrt{33})\), \(\Q(\sqrt{3}, \sqrt{11})\), \(\Q(\sqrt{11}, \sqrt{15})\), \(\Q(\sqrt{3}, \sqrt{55})\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\sqrt{5}, \sqrt{11})\), \(\Q(\zeta_{11})^+\), 8.8.189747360000.1, \(\Q(\zeta_{33})^+\), 10.10.1790566527853125.1, 10.10.669871503125.1, 10.10.166685465865600000.1, 10.10.7545432611200000.1, \(\Q(\zeta_{44})^+\), 10.10.53339349076992.1, 20.20.3206128490667995866421572265625.1, 20.20.3361869388230684433628866560000000000.3, \(\Q(\zeta_{132})^+\), 20.20.3361869388230684433628866560000000000.2, 20.20.3361869388230684433628866560000000000.1, 20.20.27784044530832102757263360000000000.1, 20.20.56933553290160450365440000000000.1

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 R ${\href{/LocalNumberField/7.10.0.1}{10} }^{4}$ 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
5Data 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}$