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Properties

Label 40.0.27120475053...7136.1
Degree $40$
Signature $[0, 20]$
Discriminant $2^{40}\cdot 7^{20}\cdot 11^{36}$
Root discriminant $45.80$
Ramified primes $2, 7, 11$
Class number Not computed
Class group Not computed
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, 786432, 0, 327680, 0, 49152, 0, -45056, 0, -46080, 0, -23296, 0, -5952, 0, 1360, 0, 2508, 0, 1541, 0, 627, 0, 85, 0, -93, 0, -91, 0, -45, 0, -11, 0, 3, 0, 5, 0, 3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 + 3*x^38 + 5*x^36 + 3*x^34 - 11*x^32 - 45*x^30 - 91*x^28 - 93*x^26 + 85*x^24 + 627*x^22 + 1541*x^20 + 2508*x^18 + 1360*x^16 - 5952*x^14 - 23296*x^12 - 46080*x^10 - 45056*x^8 + 49152*x^6 + 327680*x^4 + 786432*x^2 + 1048576)
 
gp: K = bnfinit(x^40 + 3*x^38 + 5*x^36 + 3*x^34 - 11*x^32 - 45*x^30 - 91*x^28 - 93*x^26 + 85*x^24 + 627*x^22 + 1541*x^20 + 2508*x^18 + 1360*x^16 - 5952*x^14 - 23296*x^12 - 46080*x^10 - 45056*x^8 + 49152*x^6 + 327680*x^4 + 786432*x^2 + 1048576, 1)
 

Normalized defining polynomial

\( x^{40} + 3 x^{38} + 5 x^{36} + 3 x^{34} - 11 x^{32} - 45 x^{30} - 91 x^{28} - 93 x^{26} + 85 x^{24} + 627 x^{22} + 1541 x^{20} + 2508 x^{18} + 1360 x^{16} - 5952 x^{14} - 23296 x^{12} - 46080 x^{10} - 45056 x^{8} + 49152 x^{6} + 327680 x^{4} + 786432 x^{2} + 1048576 \)

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:  \(2712047505327472012144594086234943329183650768987920496114727387136=2^{40}\cdot 7^{20}\cdot 11^{36}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.80$
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:  $2, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(308=2^{2}\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{308}(1,·)$, $\chi_{308}(265,·)$, $\chi_{308}(139,·)$, $\chi_{308}(13,·)$, $\chi_{308}(15,·)$, $\chi_{308}(279,·)$, $\chi_{308}(153,·)$, $\chi_{308}(27,·)$, $\chi_{308}(29,·)$, $\chi_{308}(69,·)$, $\chi_{308}(155,·)$, $\chi_{308}(293,·)$, $\chi_{308}(295,·)$, $\chi_{308}(41,·)$, $\chi_{308}(43,·)$, $\chi_{308}(307,·)$, $\chi_{308}(181,·)$, $\chi_{308}(183,·)$, $\chi_{308}(111,·)$, $\chi_{308}(57,·)$, $\chi_{308}(195,·)$, $\chi_{308}(197,·)$, $\chi_{308}(71,·)$, $\chi_{308}(141,·)$, $\chi_{308}(83,·)$, $\chi_{308}(85,·)$, $\chi_{308}(267,·)$, $\chi_{308}(223,·)$, $\chi_{308}(225,·)$, $\chi_{308}(97,·)$, $\chi_{308}(167,·)$, $\chi_{308}(237,·)$, $\chi_{308}(239,·)$, $\chi_{308}(113,·)$, $\chi_{308}(211,·)$, $\chi_{308}(169,·)$, $\chi_{308}(251,·)$, $\chi_{308}(281,·)$, $\chi_{308}(125,·)$, $\chi_{308}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{19} - \frac{1}{2} a^{17} - \frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{6164} a^{22} - \frac{1}{4} a^{20} + \frac{1}{4} a^{18} - \frac{1}{4} a^{16} + \frac{1}{4} a^{14} - \frac{1}{4} a^{12} + \frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} + \frac{627}{1541}$, $\frac{1}{12328} a^{23} - \frac{1}{8} a^{21} + \frac{1}{8} a^{19} - \frac{1}{8} a^{17} + \frac{1}{8} a^{15} - \frac{1}{8} a^{13} + \frac{1}{8} a^{11} - \frac{1}{8} a^{9} + \frac{1}{8} a^{7} - \frac{1}{8} a^{5} + \frac{1}{8} a^{3} + \frac{627}{3082} a$, $\frac{1}{24656} a^{24} - \frac{1}{24656} a^{22} + \frac{5}{16} a^{20} + \frac{3}{16} a^{18} + \frac{5}{16} a^{16} + \frac{3}{16} a^{14} + \frac{5}{16} a^{12} + \frac{3}{16} a^{10} + \frac{5}{16} a^{8} + \frac{3}{16} a^{6} + \frac{5}{16} a^{4} - \frac{457}{3082} a^{2} - \frac{542}{1541}$, $\frac{1}{49312} a^{25} - \frac{1}{49312} a^{23} + \frac{5}{32} a^{21} + \frac{3}{32} a^{19} - \frac{11}{32} a^{17} - \frac{13}{32} a^{15} + \frac{5}{32} a^{13} + \frac{3}{32} a^{11} - \frac{11}{32} a^{9} - \frac{13}{32} a^{7} + \frac{5}{32} a^{5} - \frac{457}{6164} a^{3} - \frac{271}{1541} a$, $\frac{1}{98624} a^{26} - \frac{1}{98624} a^{24} - \frac{7}{98624} a^{22} - \frac{29}{64} a^{20} + \frac{21}{64} a^{18} - \frac{13}{64} a^{16} + \frac{5}{64} a^{14} + \frac{3}{64} a^{12} - \frac{11}{64} a^{10} + \frac{19}{64} a^{8} - \frac{27}{64} a^{6} - \frac{457}{12328} a^{4} - \frac{271}{3082} a^{2} - \frac{178}{1541}$, $\frac{1}{197248} a^{27} - \frac{1}{197248} a^{25} - \frac{7}{197248} a^{23} - \frac{29}{128} a^{21} + \frac{21}{128} a^{19} + \frac{51}{128} a^{17} - \frac{59}{128} a^{15} + \frac{3}{128} a^{13} - \frac{11}{128} a^{11} - \frac{45}{128} a^{9} + \frac{37}{128} a^{7} - \frac{457}{24656} a^{5} - \frac{271}{6164} a^{3} - \frac{89}{1541} a$, $\frac{1}{394496} a^{28} - \frac{1}{394496} a^{26} - \frac{7}{394496} a^{24} - \frac{17}{394496} a^{22} - \frac{107}{256} a^{20} + \frac{51}{256} a^{18} + \frac{69}{256} a^{16} + \frac{3}{256} a^{14} - \frac{11}{256} a^{12} - \frac{45}{256} a^{10} - \frac{91}{256} a^{8} - \frac{457}{49312} a^{6} - \frac{271}{12328} a^{4} - \frac{89}{3082} a^{2} + \frac{2}{1541}$, $\frac{1}{788992} a^{29} - \frac{1}{788992} a^{27} - \frac{7}{788992} a^{25} - \frac{17}{788992} a^{23} - \frac{107}{512} a^{21} - \frac{205}{512} a^{19} - \frac{187}{512} a^{17} - \frac{253}{512} a^{15} - \frac{11}{512} a^{13} - \frac{45}{512} a^{11} - \frac{91}{512} a^{9} - \frac{457}{98624} a^{7} - \frac{271}{24656} a^{5} - \frac{89}{6164} a^{3} + \frac{1}{1541} a$, $\frac{1}{1577984} a^{30} - \frac{1}{1577984} a^{28} - \frac{7}{1577984} a^{26} - \frac{17}{1577984} a^{24} - \frac{1}{68608} a^{22} + \frac{307}{1024} a^{20} + \frac{325}{1024} a^{18} - \frac{253}{1024} a^{16} - \frac{11}{1024} a^{14} - \frac{45}{1024} a^{12} - \frac{91}{1024} a^{10} - \frac{457}{197248} a^{8} - \frac{271}{49312} a^{6} - \frac{89}{12328} a^{4} + \frac{1}{3082} a^{2} + \frac{2}{67}$, $\frac{1}{3155968} a^{31} - \frac{1}{3155968} a^{29} - \frac{7}{3155968} a^{27} - \frac{17}{3155968} a^{25} - \frac{1}{137216} a^{23} + \frac{307}{2048} a^{21} - \frac{699}{2048} a^{19} + \frac{771}{2048} a^{17} + \frac{1013}{2048} a^{15} - \frac{45}{2048} a^{13} - \frac{91}{2048} a^{11} - \frac{457}{394496} a^{9} - \frac{271}{98624} a^{7} - \frac{89}{24656} a^{5} + \frac{1}{6164} a^{3} + \frac{1}{67} a$, $\frac{1}{6311936} a^{32} - \frac{1}{6311936} a^{30} - \frac{7}{6311936} a^{28} - \frac{17}{6311936} a^{26} - \frac{1}{274432} a^{24} - \frac{1}{6311936} a^{22} - \frac{699}{4096} a^{20} + \frac{771}{4096} a^{18} + \frac{1013}{4096} a^{16} - \frac{45}{4096} a^{14} - \frac{91}{4096} a^{12} - \frac{457}{788992} a^{10} - \frac{271}{197248} a^{8} - \frac{89}{49312} a^{6} + \frac{1}{12328} a^{4} + \frac{1}{134} a^{2} + \frac{34}{1541}$, $\frac{1}{12623872} a^{33} - \frac{1}{12623872} a^{31} - \frac{7}{12623872} a^{29} - \frac{17}{12623872} a^{27} - \frac{1}{548864} a^{25} - \frac{1}{12623872} a^{23} - \frac{699}{8192} a^{21} - \frac{3325}{8192} a^{19} + \frac{1013}{8192} a^{17} - \frac{45}{8192} a^{15} + \frac{4005}{8192} a^{13} + \frac{788535}{1577984} a^{11} + \frac{196977}{394496} a^{9} + \frac{49223}{98624} a^{7} - \frac{12327}{24656} a^{5} - \frac{133}{268} a^{3} - \frac{1507}{3082} a$, $\frac{1}{25247744} a^{34} - \frac{1}{25247744} a^{32} - \frac{7}{25247744} a^{30} - \frac{17}{25247744} a^{28} - \frac{1}{1097728} a^{26} - \frac{1}{25247744} a^{24} + \frac{89}{25247744} a^{22} - \frac{7421}{16384} a^{20} - \frac{3083}{16384} a^{18} + \frac{4051}{16384} a^{16} + \frac{8101}{16384} a^{14} - \frac{457}{3155968} a^{12} - \frac{271}{788992} a^{10} - \frac{89}{197248} a^{8} + \frac{1}{49312} a^{6} + \frac{1}{536} a^{4} + \frac{17}{3082} a^{2} + \frac{14}{1541}$, $\frac{1}{50495488} a^{35} - \frac{1}{50495488} a^{33} - \frac{7}{50495488} a^{31} - \frac{17}{50495488} a^{29} - \frac{1}{2195456} a^{27} - \frac{1}{50495488} a^{25} + \frac{89}{50495488} a^{23} - \frac{7421}{32768} a^{21} - \frac{3083}{32768} a^{19} - \frac{12333}{32768} a^{17} + \frac{8101}{32768} a^{15} - \frac{457}{6311936} a^{13} - \frac{271}{1577984} a^{11} - \frac{89}{394496} a^{9} + \frac{1}{98624} a^{7} + \frac{1}{1072} a^{5} + \frac{17}{6164} a^{3} + \frac{7}{1541} a$, $\frac{1}{100990976} a^{36} - \frac{1}{100990976} a^{34} - \frac{7}{100990976} a^{32} - \frac{17}{100990976} a^{30} - \frac{1}{4390912} a^{28} - \frac{1}{100990976} a^{26} + \frac{89}{100990976} a^{24} + \frac{271}{100990976} a^{22} + \frac{29685}{65536} a^{20} - \frac{12333}{65536} a^{18} - \frac{24667}{65536} a^{16} - \frac{457}{12623872} a^{14} - \frac{271}{3155968} a^{12} - \frac{89}{788992} a^{10} + \frac{1}{197248} a^{8} + \frac{1}{2144} a^{6} + \frac{17}{12328} a^{4} + \frac{7}{3082} a^{2} + \frac{2}{1541}$, $\frac{1}{201981952} a^{37} - \frac{1}{201981952} a^{35} - \frac{7}{201981952} a^{33} - \frac{17}{201981952} a^{31} - \frac{1}{8781824} a^{29} - \frac{1}{201981952} a^{27} + \frac{89}{201981952} a^{25} + \frac{271}{201981952} a^{23} + \frac{29685}{131072} a^{21} - \frac{12333}{131072} a^{19} - \frac{24667}{131072} a^{17} - \frac{457}{25247744} a^{15} - \frac{271}{6311936} a^{13} - \frac{89}{1577984} a^{11} + \frac{1}{394496} a^{9} + \frac{1}{4288} a^{7} + \frac{17}{24656} a^{5} + \frac{7}{6164} a^{3} + \frac{1}{1541} a$, $\frac{1}{403963904} a^{38} - \frac{1}{403963904} a^{36} - \frac{7}{403963904} a^{34} - \frac{17}{403963904} a^{32} - \frac{1}{17563648} a^{30} - \frac{1}{403963904} a^{28} + \frac{89}{403963904} a^{26} + \frac{271}{403963904} a^{24} + \frac{457}{403963904} a^{22} + \frac{118739}{262144} a^{20} - \frac{24667}{262144} a^{18} - \frac{457}{50495488} a^{16} - \frac{271}{12623872} a^{14} - \frac{89}{3155968} a^{12} + \frac{1}{788992} a^{10} + \frac{1}{8576} a^{8} + \frac{17}{49312} a^{6} + \frac{7}{12328} a^{4} + \frac{1}{3082} a^{2} - \frac{2}{1541}$, $\frac{1}{807927808} a^{39} - \frac{1}{807927808} a^{37} - \frac{7}{807927808} a^{35} - \frac{17}{807927808} a^{33} - \frac{1}{35127296} a^{31} - \frac{1}{807927808} a^{29} + \frac{89}{807927808} a^{27} + \frac{271}{807927808} a^{25} + \frac{457}{807927808} a^{23} + \frac{118739}{524288} a^{21} + \frac{237477}{524288} a^{19} - \frac{457}{100990976} a^{17} - \frac{271}{25247744} a^{15} - \frac{89}{6311936} a^{13} + \frac{1}{1577984} a^{11} + \frac{1}{17152} a^{9} + \frac{17}{98624} a^{7} + \frac{7}{24656} a^{5} + \frac{1}{6164} a^{3} - \frac{1}{1541} a$

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:  \( -\frac{1}{49312} a^{27} + \frac{8279}{49312} a^{5} \) (order $44$)
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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{77}) \), \(\Q(\sqrt{-77}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{11}) \), \(\Q(i, \sqrt{77})\), \(\Q(i, \sqrt{7})\), \(\Q(i, \sqrt{11})\), \(\Q(\sqrt{-7}, \sqrt{-11})\), \(\Q(\sqrt{7}, \sqrt{11})\), \(\Q(\sqrt{-7}, \sqrt{11})\), \(\Q(\sqrt{7}, \sqrt{-11})\), \(\Q(\zeta_{11})^+\), 8.0.8999178496.1, 10.0.219503494144.1, 10.10.39630026842637.1, 10.0.40581147486860288.1, 10.0.3602729712967.1, 10.10.3689195226078208.1, \(\Q(\zeta_{11})\), \(\Q(\zeta_{44})^+\), 20.0.1646829531350307068613612031442944.1, 20.0.13610161416118240236476132491264.1, \(\Q(\zeta_{44})\), 20.0.1570539027548129147161113769.2, 20.20.1646829531350307068613612031442944.1, 20.0.1646829531350307068613612031442944.3, 20.0.1646829531350307068613612031442944.4

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{4}$ R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{8}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{8}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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
7Data not computed
11Data not computed