↑

Properties

Label 40.0.11302165783...0000.2
Degree $40$
Signature $[0, 20]$
Discriminant $2^{40}\cdot 3^{20}\cdot 5^{20}\cdot 11^{36}$
Root discriminant $67.04$
Ramified primes $2, 3, 5, 11$
Class number Not computed
Class group Not computed
Galois group $C_2^2\times C_{10}$ (as 40T7)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 360, 0, 22030, 0, 457722, 0, 4509483, 0, 25054116, 0, 87735123, 0, 207509745, 0, 346146943, 0, 418309584, 0, 372028031, 0, 245357970, 0, 120157564, 0, 43529874, 0, 11569924, 0, 2227290, 0, 304524, 0, 28656, 0, 1757, 0, 63, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 + 63*x^38 + 1757*x^36 + 28656*x^34 + 304524*x^32 + 2227290*x^30 + 11569924*x^28 + 43529874*x^26 + 120157564*x^24 + 245357970*x^22 + 372028031*x^20 + 418309584*x^18 + 346146943*x^16 + 207509745*x^14 + 87735123*x^12 + 25054116*x^10 + 4509483*x^8 + 457722*x^6 + 22030*x^4 + 360*x^2 + 1)
 
gp: K = bnfinit(x^40 + 63*x^38 + 1757*x^36 + 28656*x^34 + 304524*x^32 + 2227290*x^30 + 11569924*x^28 + 43529874*x^26 + 120157564*x^24 + 245357970*x^22 + 372028031*x^20 + 418309584*x^18 + 346146943*x^16 + 207509745*x^14 + 87735123*x^12 + 25054116*x^10 + 4509483*x^8 + 457722*x^6 + 22030*x^4 + 360*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{40} + 63 x^{38} + 1757 x^{36} + 28656 x^{34} + 304524 x^{32} + 2227290 x^{30} + 11569924 x^{28} + 43529874 x^{26} + 120157564 x^{24} + 245357970 x^{22} + 372028031 x^{20} + 418309584 x^{18} + 346146943 x^{16} + 207509745 x^{14} + 87735123 x^{12} + 25054116 x^{10} + 4509483 x^{8} + 457722 x^{6} + 22030 x^{4} + 360 x^{2} + 1 \)

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:  \(11302165783522556415463223790320401501047994110286233600000000000000000000=2^{40}\cdot 3^{20}\cdot 5^{20}\cdot 11^{36}\)
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$$\chi_{660}(1,·)$, $\chi_{660}(131,·)$, $\chi_{660}(631,·)$, $\chi_{660}(529,·)$, $\chi_{660}(659,·)$, $\chi_{660}(149,·)$, $\chi_{660}(281,·)$, $\chi_{660}(559,·)$, $\chi_{660}(29,·)$, $\chi_{660}(31,·)$, $\chi_{660}(289,·)$, $\chi_{660}(619,·)$, $\chi_{660}(421,·)$, $\chi_{660}(41,·)$, $\chi_{660}(299,·)$, $\chi_{660}(301,·)$, $\chi_{660}(431,·)$, $\chi_{660}(49,·)$, $\chi_{660}(371,·)$, $\chi_{660}(181,·)$, $\chi_{660}(569,·)$, $\chi_{660}(161,·)$, $\chi_{660}(329,·)$, $\chi_{660}(331,·)$, $\chi_{660}(461,·)$, $\chi_{660}(91,·)$, $\chi_{660}(199,·)$, $\chi_{660}(479,·)$, $\chi_{660}(611,·)$, $\chi_{660}(101,·)$, $\chi_{660}(359,·)$, $\chi_{660}(361,·)$, $\chi_{660}(491,·)$, $\chi_{660}(239,·)$, $\chi_{660}(229,·)$, $\chi_{660}(499,·)$, $\chi_{660}(629,·)$, $\chi_{660}(169,·)$, $\chi_{660}(379,·)$, $\chi_{660}(511,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $\frac{1}{26069} a^{36} + \frac{11948}{26069} a^{34} - \frac{35}{26069} a^{32} - \frac{6503}{26069} a^{30} - \frac{2911}{26069} a^{28} + \frac{7989}{26069} a^{26} + \frac{2896}{26069} a^{24} + \frac{3116}{26069} a^{22} + \frac{4241}{26069} a^{20} - \frac{8462}{26069} a^{18} + \frac{12360}{26069} a^{16} - \frac{5432}{26069} a^{14} - \frac{12196}{26069} a^{12} + \frac{5798}{26069} a^{10} - \frac{9590}{26069} a^{8} + \frac{6994}{26069} a^{6} - \frac{8184}{26069} a^{4} - \frac{6568}{26069} a^{2} - \frac{5606}{26069}$, $\frac{1}{26069} a^{37} + \frac{11948}{26069} a^{35} - \frac{35}{26069} a^{33} - \frac{6503}{26069} a^{31} - \frac{2911}{26069} a^{29} + \frac{7989}{26069} a^{27} + \frac{2896}{26069} a^{25} + \frac{3116}{26069} a^{23} + \frac{4241}{26069} a^{21} - \frac{8462}{26069} a^{19} + \frac{12360}{26069} a^{17} - \frac{5432}{26069} a^{15} - \frac{12196}{26069} a^{13} + \frac{5798}{26069} a^{11} - \frac{9590}{26069} a^{9} + \frac{6994}{26069} a^{7} - \frac{8184}{26069} a^{5} - \frac{6568}{26069} a^{3} - \frac{5606}{26069} a$, $\frac{1}{3352707566614500670696620129140499407} a^{38} - \frac{59978996665568388524850165177905}{3352707566614500670696620129140499407} a^{36} + \frac{1512864769609980147742633781683014018}{3352707566614500670696620129140499407} a^{34} + \frac{1163132690816370739340338748523055105}{3352707566614500670696620129140499407} a^{32} + \frac{1446202989280347352780149272521872682}{3352707566614500670696620129140499407} a^{30} + \frac{774569351830498538920438975100159853}{3352707566614500670696620129140499407} a^{28} - \frac{8004358470421150719461345168768936}{16847776716655782264807136327339193} a^{26} + \frac{152513628075479605641444537111799934}{3352707566614500670696620129140499407} a^{24} - \frac{30017718046238061375628344866769407}{3352707566614500670696620129140499407} a^{22} - \frac{764412130455071002101275076850427050}{3352707566614500670696620129140499407} a^{20} + \frac{808970023271771640072362057745280035}{3352707566614500670696620129140499407} a^{18} - \frac{1409438931657309291387381933716330467}{3352707566614500670696620129140499407} a^{16} + \frac{436224623535146109720443635630316086}{3352707566614500670696620129140499407} a^{14} + \frac{852814672505301561483030658156840141}{3352707566614500670696620129140499407} a^{12} + \frac{673681996912936261237650592683166111}{3352707566614500670696620129140499407} a^{10} - \frac{709526457373791935347587091956106183}{3352707566614500670696620129140499407} a^{8} + \frac{930830649901782369310296093419753232}{3352707566614500670696620129140499407} a^{6} + \frac{1036746290074442474519707789314264150}{3352707566614500670696620129140499407} a^{4} - \frac{1403153261942343471406926078963338278}{3352707566614500670696620129140499407} a^{2} - \frac{1115834755201645245578378770401917148}{3352707566614500670696620129140499407}$, $\frac{1}{3352707566614500670696620129140499407} a^{39} - \frac{59978996665568388524850165177905}{3352707566614500670696620129140499407} a^{37} + \frac{1512864769609980147742633781683014018}{3352707566614500670696620129140499407} a^{35} + \frac{1163132690816370739340338748523055105}{3352707566614500670696620129140499407} a^{33} + \frac{1446202989280347352780149272521872682}{3352707566614500670696620129140499407} a^{31} + \frac{774569351830498538920438975100159853}{3352707566614500670696620129140499407} a^{29} - \frac{8004358470421150719461345168768936}{16847776716655782264807136327339193} a^{27} + \frac{152513628075479605641444537111799934}{3352707566614500670696620129140499407} a^{25} - \frac{30017718046238061375628344866769407}{3352707566614500670696620129140499407} a^{23} - \frac{764412130455071002101275076850427050}{3352707566614500670696620129140499407} a^{21} + \frac{808970023271771640072362057745280035}{3352707566614500670696620129140499407} a^{19} - \frac{1409438931657309291387381933716330467}{3352707566614500670696620129140499407} a^{17} + \frac{436224623535146109720443635630316086}{3352707566614500670696620129140499407} a^{15} + \frac{852814672505301561483030658156840141}{3352707566614500670696620129140499407} a^{13} + \frac{673681996912936261237650592683166111}{3352707566614500670696620129140499407} a^{11} - \frac{709526457373791935347587091956106183}{3352707566614500670696620129140499407} a^{9} + \frac{930830649901782369310296093419753232}{3352707566614500670696620129140499407} a^{7} + \frac{1036746290074442474519707789314264150}{3352707566614500670696620129140499407} a^{5} - \frac{1403153261942343471406926078963338278}{3352707566614500670696620129140499407} a^{3} - \frac{1115834755201645245578378770401917148}{3352707566614500670696620129140499407} a$

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:  \( \frac{420266113026570167555}{925490360222769284989} a^{39} + \frac{26219033725248837567985}{925490360222769284989} a^{37} + \frac{722428088952141305699075}{925490360222769284989} a^{35} + \frac{11606268314634663311728296}{925490360222769284989} a^{33} + \frac{121031305967670493156765267}{925490360222769284989} a^{31} + \frac{864490508064208024979431375}{925490360222769284989} a^{29} + \frac{4359540862926437222178083159}{925490360222769284989} a^{27} + \frac{15810888177680088822917008047}{925490360222769284989} a^{25} + \frac{41733912706752907935116292195}{925490360222769284989} a^{23} + \frac{80776718775489358050650401290}{925490360222769284989} a^{21} + \frac{114999826327308175808042469355}{925490360222769284989} a^{19} + \frac{120177871233900719454958176104}{925490360222769284989} a^{17} + \frac{91403575107408003252037278372}{925490360222769284989} a^{15} + \frac{49746589358729697894688334436}{925490360222769284989} a^{13} + \frac{18830734669818577692117874967}{925490360222769284989} a^{11} + \frac{4737202792732091628545993460}{925490360222769284989} a^{9} + \frac{736435377146514428680961642}{925490360222769284989} a^{7} + \frac{62724503287815445718553406}{925490360222769284989} a^{5} + \frac{2362165047801810349080004}{925490360222769284989} a^{3} + \frac{23186095649371596206408}{925490360222769284989} a \) (order $4$)
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

\(\Q(\sqrt{-165}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-33}) \), \(\Q(\sqrt{165}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{5}, \sqrt{-33})\), \(\Q(i, \sqrt{165})\), \(\Q(\sqrt{-5}, \sqrt{33})\), \(\Q(\sqrt{5}, \sqrt{33})\), \(\Q(i, \sqrt{5})\), \(\Q(\sqrt{-5}, \sqrt{-33})\), \(\Q(i, \sqrt{33})\), \(\Q(\zeta_{11})^+\), 8.0.189747360000.8, 10.0.1833540124521600000.1, 10.10.669871503125.1, 10.0.586732839846912.1, 10.10.1790566527853125.1, 10.0.219503494144.1, \(\Q(\zeta_{33})^+\), 10.0.685948419200000.1, 20.0.3361869388230684433628866560000000000.6, 20.0.3361869388230684433628866560000000000.3, 20.0.3361869388230684433628866560000000000.7, 20.20.3206128490667995866421572265625.1, 20.0.470525233802978928640000000000.1, 20.0.3361869388230684433628866560000000000.1, 20.0.344255425354822086003595935744.1

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 ${\href{/LocalNumberField/13.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{8}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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
11Data not computed