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Properties

Label 40.0.11302165783...0000.5
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 $30008$ (GRH)
Class group $[22, 1364]$ (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![1048576, 0, -262144, 0, -196608, 0, 114688, 0, 20480, 0, -33792, 0, 3328, 0, 7616, 0, -2736, 0, -1220, 0, 989, 0, -305, 0, -171, 0, 119, 0, 13, 0, -33, 0, 5, 0, 7, 0, -3, 0, -1, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - x^38 - 3*x^36 + 7*x^34 + 5*x^32 - 33*x^30 + 13*x^28 + 119*x^26 - 171*x^24 - 305*x^22 + 989*x^20 - 1220*x^18 - 2736*x^16 + 7616*x^14 + 3328*x^12 - 33792*x^10 + 20480*x^8 + 114688*x^6 - 196608*x^4 - 262144*x^2 + 1048576)
 
gp: K = bnfinit(x^40 - x^38 - 3*x^36 + 7*x^34 + 5*x^32 - 33*x^30 + 13*x^28 + 119*x^26 - 171*x^24 - 305*x^22 + 989*x^20 - 1220*x^18 - 2736*x^16 + 7616*x^14 + 3328*x^12 - 33792*x^10 + 20480*x^8 + 114688*x^6 - 196608*x^4 - 262144*x^2 + 1048576, 1)
 

Normalized defining polynomial

\( x^{40} - x^{38} - 3 x^{36} + 7 x^{34} + 5 x^{32} - 33 x^{30} + 13 x^{28} + 119 x^{26} - 171 x^{24} - 305 x^{22} + 989 x^{20} - 1220 x^{18} - 2736 x^{16} + 7616 x^{14} + 3328 x^{12} - 33792 x^{10} + 20480 x^{8} + 114688 x^{6} - 196608 x^{4} - 262144 x^{2} + 1048576 \)

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$$\chi_{660}(1,·)$, $\chi_{660}(259,·)$, $\chi_{660}(389,·)$, $\chi_{660}(139,·)$, $\chi_{660}(269,·)$, $\chi_{660}(131,·)$, $\chi_{660}(149,·)$, $\chi_{660}(89,·)$, $\chi_{660}(431,·)$, $\chi_{660}(29,·)$, $\chi_{660}(491,·)$, $\chi_{660}(421,·)$, $\chi_{660}(551,·)$, $\chi_{660}(71,·)$, $\chi_{660}(301,·)$, $\chi_{660}(559,·)$, $\chi_{660}(371,·)$, $\chi_{660}(181,·)$, $\chi_{660}(311,·)$, $\chi_{660}(569,·)$, $\chi_{660}(61,·)$, $\chi_{660}(191,·)$, $\chi_{660}(19,·)$, $\chi_{660}(449,·)$, $\chi_{660}(199,·)$, $\chi_{660}(329,·)$, $\chi_{660}(439,·)$, $\chi_{660}(79,·)$, $\chi_{660}(601,·)$, $\chi_{660}(481,·)$, $\chi_{660}(611,·)$, $\chi_{660}(361,·)$, $\chi_{660}(619,·)$, $\chi_{660}(241,·)$, $\chi_{660}(499,·)$, $\chi_{660}(629,·)$, $\chi_{660}(251,·)$, $\chi_{660}(379,·)$, $\chi_{660}(541,·)$, $\chi_{660}(509,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{19} - \frac{1}{2} a^{17} - \frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{3956} a^{22} - \frac{1}{4} a^{20} + \frac{1}{4} a^{18} - \frac{1}{4} a^{16} + \frac{1}{4} a^{14} - \frac{1}{4} a^{12} + \frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{305}{989}$, $\frac{1}{7912} a^{23} - \frac{1}{8} a^{21} - \frac{3}{8} a^{19} - \frac{1}{8} a^{17} - \frac{3}{8} a^{15} - \frac{1}{8} a^{13} - \frac{3}{8} a^{11} - \frac{1}{8} a^{9} - \frac{3}{8} a^{7} - \frac{1}{8} a^{5} - \frac{3}{8} a^{3} + \frac{342}{989} a$, $\frac{1}{15824} a^{24} - \frac{1}{15824} a^{22} + \frac{1}{16} a^{20} + \frac{3}{16} a^{18} - \frac{7}{16} a^{16} - \frac{5}{16} a^{14} + \frac{1}{16} a^{12} + \frac{3}{16} a^{10} - \frac{7}{16} a^{8} - \frac{5}{16} a^{6} + \frac{1}{16} a^{4} - \frac{305}{3956} a^{2} - \frac{171}{989}$, $\frac{1}{31648} a^{25} - \frac{1}{31648} a^{23} + \frac{1}{32} a^{21} - \frac{13}{32} a^{19} + \frac{9}{32} a^{17} + \frac{11}{32} a^{15} - \frac{15}{32} a^{13} + \frac{3}{32} a^{11} - \frac{7}{32} a^{9} - \frac{5}{32} a^{7} + \frac{1}{32} a^{5} - \frac{305}{7912} a^{3} - \frac{171}{1978} a$, $\frac{1}{63296} a^{26} - \frac{1}{63296} a^{24} - \frac{3}{63296} a^{22} - \frac{13}{64} a^{20} + \frac{9}{64} a^{18} - \frac{21}{64} a^{16} - \frac{15}{64} a^{14} - \frac{29}{64} a^{12} + \frac{25}{64} a^{10} + \frac{27}{64} a^{8} + \frac{1}{64} a^{6} - \frac{305}{15824} a^{4} - \frac{171}{3956} a^{2} + \frac{119}{989}$, $\frac{1}{126592} a^{27} - \frac{1}{126592} a^{25} - \frac{3}{126592} a^{23} - \frac{13}{128} a^{21} + \frac{9}{128} a^{19} + \frac{43}{128} a^{17} + \frac{49}{128} a^{15} + \frac{35}{128} a^{13} + \frac{25}{128} a^{11} - \frac{37}{128} a^{9} - \frac{63}{128} a^{7} + \frac{15519}{31648} a^{5} + \frac{3785}{7912} a^{3} - \frac{435}{989} a$, $\frac{1}{253184} a^{28} - \frac{1}{253184} a^{26} - \frac{3}{253184} a^{24} + \frac{7}{253184} a^{22} + \frac{73}{256} a^{20} - \frac{21}{256} a^{18} - \frac{15}{256} a^{16} + \frac{99}{256} a^{14} - \frac{39}{256} a^{12} - \frac{101}{256} a^{10} + \frac{1}{256} a^{8} - \frac{305}{63296} a^{6} - \frac{171}{15824} a^{4} + \frac{119}{3956} a^{2} + \frac{13}{989}$, $\frac{1}{506368} a^{29} - \frac{1}{506368} a^{27} - \frac{3}{506368} a^{25} + \frac{7}{506368} a^{23} + \frac{73}{512} a^{21} + \frac{235}{512} a^{19} - \frac{15}{512} a^{17} + \frac{99}{512} a^{15} - \frac{39}{512} a^{13} + \frac{155}{512} a^{11} + \frac{1}{512} a^{9} - \frac{305}{126592} a^{7} - \frac{171}{31648} a^{5} + \frac{119}{7912} a^{3} + \frac{13}{1978} a$, $\frac{1}{1012736} a^{30} - \frac{1}{1012736} a^{28} - \frac{3}{1012736} a^{26} + \frac{7}{1012736} a^{24} + \frac{5}{1012736} a^{22} + \frac{235}{1024} a^{20} + \frac{497}{1024} a^{18} - \frac{413}{1024} a^{16} + \frac{473}{1024} a^{14} + \frac{155}{1024} a^{12} + \frac{1}{1024} a^{10} - \frac{305}{253184} a^{8} - \frac{171}{63296} a^{6} + \frac{119}{15824} a^{4} + \frac{13}{3956} a^{2} - \frac{33}{989}$, $\frac{1}{2025472} a^{31} - \frac{1}{2025472} a^{29} - \frac{3}{2025472} a^{27} + \frac{7}{2025472} a^{25} + \frac{5}{2025472} a^{23} + \frac{235}{2048} a^{21} - \frac{527}{2048} a^{19} - \frac{413}{2048} a^{17} + \frac{473}{2048} a^{15} - \frac{869}{2048} a^{13} - \frac{1023}{2048} a^{11} + \frac{252879}{506368} a^{9} + \frac{63125}{126592} a^{7} - \frac{15705}{31648} a^{5} - \frac{3943}{7912} a^{3} + \frac{478}{989} a$, $\frac{1}{4050944} a^{32} - \frac{1}{4050944} a^{30} - \frac{3}{4050944} a^{28} + \frac{7}{4050944} a^{26} + \frac{5}{4050944} a^{24} - \frac{33}{4050944} a^{22} + \frac{497}{4096} a^{20} - \frac{1437}{4096} a^{18} - \frac{551}{4096} a^{16} - \frac{1893}{4096} a^{14} + \frac{1}{4096} a^{12} - \frac{305}{1012736} a^{10} - \frac{171}{253184} a^{8} + \frac{119}{63296} a^{6} + \frac{13}{15824} a^{4} - \frac{33}{3956} a^{2} + \frac{5}{989}$, $\frac{1}{8101888} a^{33} - \frac{1}{8101888} a^{31} - \frac{3}{8101888} a^{29} + \frac{7}{8101888} a^{27} + \frac{5}{8101888} a^{25} - \frac{33}{8101888} a^{23} + \frac{497}{8192} a^{21} + \frac{2659}{8192} a^{19} + \frac{3545}{8192} a^{17} + \frac{2203}{8192} a^{15} + \frac{1}{8192} a^{13} - \frac{305}{2025472} a^{11} - \frac{171}{506368} a^{9} + \frac{119}{126592} a^{7} + \frac{13}{31648} a^{5} - \frac{33}{7912} a^{3} + \frac{5}{1978} a$, $\frac{1}{16203776} a^{34} - \frac{1}{16203776} a^{32} - \frac{3}{16203776} a^{30} + \frac{7}{16203776} a^{28} + \frac{5}{16203776} a^{26} - \frac{33}{16203776} a^{24} + \frac{13}{16203776} a^{22} - \frac{5533}{16384} a^{20} + \frac{3545}{16384} a^{18} + \frac{2203}{16384} a^{16} + \frac{1}{16384} a^{14} - \frac{305}{4050944} a^{12} - \frac{171}{1012736} a^{10} + \frac{119}{253184} a^{8} + \frac{13}{63296} a^{6} - \frac{33}{15824} a^{4} + \frac{5}{3956} a^{2} + \frac{7}{989}$, $\frac{1}{32407552} a^{35} - \frac{1}{32407552} a^{33} - \frac{3}{32407552} a^{31} + \frac{7}{32407552} a^{29} + \frac{5}{32407552} a^{27} - \frac{33}{32407552} a^{25} + \frac{13}{32407552} a^{23} - \frac{5533}{32768} a^{21} + \frac{3545}{32768} a^{19} - \frac{14181}{32768} a^{17} + \frac{1}{32768} a^{15} - \frac{305}{8101888} a^{13} - \frac{171}{2025472} a^{11} + \frac{119}{506368} a^{9} + \frac{13}{126592} a^{7} - \frac{33}{31648} a^{5} + \frac{5}{7912} a^{3} + \frac{7}{1978} a$, $\frac{1}{64815104} a^{36} - \frac{1}{64815104} a^{34} - \frac{3}{64815104} a^{32} + \frac{7}{64815104} a^{30} + \frac{5}{64815104} a^{28} - \frac{33}{64815104} a^{26} + \frac{13}{64815104} a^{24} + \frac{119}{64815104} a^{22} - \frac{29223}{65536} a^{20} - \frac{14181}{65536} a^{18} + \frac{1}{65536} a^{16} - \frac{305}{16203776} a^{14} - \frac{171}{4050944} a^{12} + \frac{119}{1012736} a^{10} + \frac{13}{253184} a^{8} - \frac{33}{63296} a^{6} + \frac{5}{15824} a^{4} + \frac{7}{3956} a^{2} - \frac{3}{989}$, $\frac{1}{129630208} a^{37} - \frac{1}{129630208} a^{35} - \frac{3}{129630208} a^{33} + \frac{7}{129630208} a^{31} + \frac{5}{129630208} a^{29} - \frac{33}{129630208} a^{27} + \frac{13}{129630208} a^{25} + \frac{119}{129630208} a^{23} - \frac{29223}{131072} a^{21} + \frac{51355}{131072} a^{19} - \frac{65535}{131072} a^{17} + \frac{16203471}{32407552} a^{15} + \frac{4050773}{8101888} a^{13} - \frac{1012617}{2025472} a^{11} - \frac{253171}{506368} a^{9} + \frac{63263}{126592} a^{7} - \frac{15819}{31648} a^{5} - \frac{3949}{7912} a^{3} + \frac{493}{989} a$, $\frac{1}{259260416} a^{38} - \frac{1}{259260416} a^{36} - \frac{3}{259260416} a^{34} + \frac{7}{259260416} a^{32} + \frac{5}{259260416} a^{30} - \frac{33}{259260416} a^{28} + \frac{13}{259260416} a^{26} + \frac{119}{259260416} a^{24} - \frac{171}{259260416} a^{22} + \frac{116891}{262144} a^{20} + \frac{1}{262144} a^{18} - \frac{305}{64815104} a^{16} - \frac{171}{16203776} a^{14} + \frac{119}{4050944} a^{12} + \frac{13}{1012736} a^{10} - \frac{33}{253184} a^{8} + \frac{5}{63296} a^{6} + \frac{7}{15824} a^{4} - \frac{3}{3956} a^{2} - \frac{1}{989}$, $\frac{1}{518520832} a^{39} - \frac{1}{518520832} a^{37} - \frac{3}{518520832} a^{35} + \frac{7}{518520832} a^{33} + \frac{5}{518520832} a^{31} - \frac{33}{518520832} a^{29} + \frac{13}{518520832} a^{27} + \frac{119}{518520832} a^{25} - \frac{171}{518520832} a^{23} + \frac{116891}{524288} a^{21} - \frac{262143}{524288} a^{19} + \frac{64814799}{129630208} a^{17} + \frac{16203605}{32407552} a^{15} - \frac{4050825}{8101888} a^{13} - \frac{1012723}{2025472} a^{11} + \frac{253151}{506368} a^{9} - \frac{63291}{126592} a^{7} - \frac{15817}{31648} a^{5} + \frac{3953}{7912} a^{3} + \frac{494}{989} a$

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

Class group and class number

$C_{22}\times C_{1364}$, which has order $30008$ (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:  \( -\frac{33}{4050944} a^{34} + \frac{27403}{4050944} a^{12} \) (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:  \( 152923482011716860 \) (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

\(\Q(\sqrt{165}) \), \(\Q(\sqrt{-33}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{55}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-5}, \sqrt{-33})\), \(\Q(\sqrt{3}, \sqrt{55})\), \(\Q(\sqrt{-11}, \sqrt{-15})\), \(\Q(\sqrt{3}, \sqrt{-11})\), \(\Q(\sqrt{-15}, \sqrt{-33})\), \(\Q(\sqrt{3}, \sqrt{-5})\), \(\Q(\sqrt{-5}, \sqrt{-11})\), \(\Q(\zeta_{11})^+\), 8.0.189747360000.1, 10.10.1790566527853125.1, 10.0.586732839846912.1, 10.0.685948419200000.1, 10.10.53339349076992.1, 10.10.7545432611200000.1, \(\Q(\zeta_{11})\), 10.0.162778775259375.1, 20.0.3361869388230684433628866560000000000.1, 20.20.3361869388230684433628866560000000000.1, 20.0.3206128490667995866421572265625.3, 20.0.344255425354822086003595935744.3, 20.0.3361869388230684433628866560000000000.2, 20.0.27784044530832102757263360000000000.1, 20.0.56933553290160450365440000000000.3

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 R ${\href{/LocalNumberField/7.10.0.1}{10} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{40}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{8}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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
$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}$
5Data not computed
11Data not computed