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Properties

Label 32.32.4867972999...5376.1
Degree $32$
Signature $[32, 0]$
Discriminant $2^{128}\cdot 3^{16}\cdot 7^{16}$
Root discriminant $73.32$
Ramified primes $2, 3, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2\times C_8$ (as 32T37)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![37249, 0, -4710400, 0, 79473664, 0, -364686336, 0, 820815552, 0, -1099478272, 0, 962963040, 0, -582109152, 0, 251040319, 0, -78757888, 0, 18135040, 0, -3061760, 0, 374400, 0, -32256, 0, 1856, 0, -64, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 64*x^30 + 1856*x^28 - 32256*x^26 + 374400*x^24 - 3061760*x^22 + 18135040*x^20 - 78757888*x^18 + 251040319*x^16 - 582109152*x^14 + 962963040*x^12 - 1099478272*x^10 + 820815552*x^8 - 364686336*x^6 + 79473664*x^4 - 4710400*x^2 + 37249)
 
gp: K = bnfinit(x^32 - 64*x^30 + 1856*x^28 - 32256*x^26 + 374400*x^24 - 3061760*x^22 + 18135040*x^20 - 78757888*x^18 + 251040319*x^16 - 582109152*x^14 + 962963040*x^12 - 1099478272*x^10 + 820815552*x^8 - 364686336*x^6 + 79473664*x^4 - 4710400*x^2 + 37249, 1)
 

Normalized defining polynomial

\( x^{32} - 64 x^{30} + 1856 x^{28} - 32256 x^{26} + 374400 x^{24} - 3061760 x^{22} + 18135040 x^{20} - 78757888 x^{18} + 251040319 x^{16} - 582109152 x^{14} + 962963040 x^{12} - 1099478272 x^{10} + 820815552 x^{8} - 364686336 x^{6} + 79473664 x^{4} - 4710400 x^{2} + 37249 \)

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

Invariants

Degree:  $32$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[32, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(486797299958529611528622959063550881784243829894751113445376=2^{128}\cdot 3^{16}\cdot 7^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73.32$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(672=2^{5}\cdot 3\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{672}(253,·)$, $\chi_{672}(1,·)$, $\chi_{672}(643,·)$, $\chi_{672}(391,·)$, $\chi_{672}(139,·)$, $\chi_{672}(659,·)$, $\chi_{672}(407,·)$, $\chi_{672}(155,·)$, $\chi_{672}(545,·)$, $\chi_{672}(293,·)$, $\chi_{672}(41,·)$, $\chi_{672}(559,·)$, $\chi_{672}(307,·)$, $\chi_{672}(55,·)$, $\chi_{672}(575,·)$, $\chi_{672}(323,·)$, $\chi_{672}(71,·)$, $\chi_{672}(461,·)$, $\chi_{672}(589,·)$, $\chi_{672}(337,·)$, $\chi_{672}(85,·)$, $\chi_{672}(377,·)$, $\chi_{672}(475,·)$, $\chi_{672}(223,·)$, $\chi_{672}(209,·)$, $\chi_{672}(421,·)$, $\chi_{672}(491,·)$, $\chi_{672}(239,·)$, $\chi_{672}(629,·)$, $\chi_{672}(169,·)$, $\chi_{672}(505,·)$, $\chi_{672}(125,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{93} a^{16} - \frac{32}{93} a^{14} + \frac{44}{93} a^{12} - \frac{26}{93} a^{10} - \frac{14}{31} a^{8} - \frac{7}{31} a^{6} + \frac{7}{31} a^{4} - \frac{8}{93} a^{2} - \frac{38}{93}$, $\frac{1}{93} a^{17} - \frac{32}{93} a^{15} + \frac{44}{93} a^{13} - \frac{26}{93} a^{11} - \frac{14}{31} a^{9} - \frac{7}{31} a^{7} + \frac{7}{31} a^{5} - \frac{8}{93} a^{3} - \frac{38}{93} a$, $\frac{1}{93} a^{18} + \frac{43}{93} a^{14} - \frac{13}{93} a^{12} - \frac{37}{93} a^{10} + \frac{10}{31} a^{8} + \frac{13}{93} a^{4} - \frac{5}{31} a^{2} - \frac{7}{93}$, $\frac{1}{93} a^{19} + \frac{43}{93} a^{15} - \frac{13}{93} a^{13} - \frac{37}{93} a^{11} + \frac{10}{31} a^{9} + \frac{13}{93} a^{5} - \frac{5}{31} a^{3} - \frac{7}{93} a$, $\frac{1}{93} a^{20} - \frac{32}{93} a^{14} + \frac{8}{31} a^{12} + \frac{32}{93} a^{10} + \frac{13}{31} a^{8} - \frac{14}{93} a^{6} + \frac{4}{31} a^{4} - \frac{35}{93} a^{2} - \frac{40}{93}$, $\frac{1}{93} a^{21} - \frac{32}{93} a^{15} + \frac{8}{31} a^{13} + \frac{32}{93} a^{11} + \frac{13}{31} a^{9} - \frac{14}{93} a^{7} + \frac{4}{31} a^{5} - \frac{35}{93} a^{3} - \frac{40}{93} a$, $\frac{1}{93} a^{22} + \frac{23}{93} a^{14} + \frac{15}{31} a^{12} + \frac{44}{93} a^{10} + \frac{37}{93} a^{8} - \frac{3}{31} a^{6} - \frac{14}{93} a^{4} - \frac{17}{93} a^{2} - \frac{7}{93}$, $\frac{1}{93} a^{23} + \frac{23}{93} a^{15} + \frac{15}{31} a^{13} + \frac{44}{93} a^{11} + \frac{37}{93} a^{9} - \frac{3}{31} a^{7} - \frac{14}{93} a^{5} - \frac{17}{93} a^{3} - \frac{7}{93} a$, $\frac{1}{65565} a^{24} - \frac{16}{21855} a^{22} + \frac{101}{21855} a^{20} - \frac{35}{13113} a^{18} - \frac{4}{7285} a^{16} + \frac{9943}{21855} a^{14} + \frac{4928}{65565} a^{12} + \frac{4249}{21855} a^{10} - \frac{438}{1457} a^{8} + \frac{6529}{65565} a^{6} - \frac{3182}{21855} a^{4} - \frac{1954}{21855} a^{2} - \frac{9433}{65565}$, $\frac{1}{12654045} a^{25} - \frac{7447}{1406005} a^{23} - \frac{5293}{1406005} a^{21} + \frac{1657}{2530809} a^{19} + \frac{2581}{1406005} a^{17} - \frac{2086022}{4218015} a^{15} + \frac{5146493}{12654045} a^{13} + \frac{761654}{4218015} a^{11} + \frac{276926}{843603} a^{9} - \frac{2092961}{12654045} a^{7} - \frac{994177}{4218015} a^{5} + \frac{2034556}{4218015} a^{3} + \frac{5302037}{12654045} a$, $\frac{1}{12654045} a^{26} - \frac{52}{12654045} a^{24} + \frac{221}{843603} a^{22} + \frac{26813}{12654045} a^{20} + \frac{4894}{12654045} a^{18} + \frac{13046}{4218015} a^{16} + \frac{2805017}{12654045} a^{14} - \frac{806926}{2530809} a^{12} - \frac{971321}{4218015} a^{10} + \frac{5358769}{12654045} a^{8} - \frac{320482}{12654045} a^{6} - \frac{1708486}{4218015} a^{4} - \frac{23966}{81639} a^{2} - \frac{17561}{65565}$, $\frac{1}{12654045} a^{27} + \frac{6201}{1406005} a^{23} - \frac{1141}{12654045} a^{21} + \frac{9173}{4218015} a^{19} + \frac{7487}{4218015} a^{17} + \frac{1142086}{2530809} a^{15} + \frac{915467}{4218015} a^{13} + \frac{1534297}{4218015} a^{11} + \frac{6106219}{12654045} a^{9} + \frac{760927}{4218015} a^{7} + \frac{83139}{281201} a^{5} - \frac{196024}{408195} a^{3} - \frac{387456}{1406005} a$, $\frac{1}{12654045} a^{28} + \frac{32}{12654045} a^{24} - \frac{9029}{2530809} a^{22} - \frac{284}{4218015} a^{20} - \frac{13244}{12654045} a^{18} - \frac{37303}{12654045} a^{16} + \frac{1878151}{4218015} a^{14} + \frac{155063}{2530809} a^{12} + \frac{204404}{2530809} a^{10} - \frac{1611043}{4218015} a^{8} - \frac{4344673}{12654045} a^{6} - \frac{4146937}{12654045} a^{4} + \frac{19279}{281201} a^{2} + \frac{30217}{65565}$, $\frac{1}{12654045} a^{29} + \frac{58616}{12654045} a^{23} + \frac{8939}{4218015} a^{21} - \frac{2078}{4218015} a^{19} + \frac{35759}{12654045} a^{17} + \frac{174097}{843603} a^{15} - \frac{2071042}{4218015} a^{13} - \frac{662639}{12654045} a^{11} + \frac{658559}{1406005} a^{9} - \frac{418849}{1406005} a^{7} - \frac{790289}{2530809} a^{5} - \frac{1364962}{4218015} a^{3} + \frac{1311614}{4218015} a$, $\frac{1}{12654045} a^{30} - \frac{56}{12654045} a^{24} - \frac{1588}{1406005} a^{22} + \frac{2711}{843603} a^{20} - \frac{12527}{4218015} a^{18} - \frac{4292}{1406005} a^{16} - \frac{228241}{1406005} a^{14} - \frac{485795}{2530809} a^{12} - \frac{1996456}{4218015} a^{10} - \frac{55044}{1406005} a^{8} - \frac{4678283}{12654045} a^{6} + \frac{668144}{1406005} a^{4} + \frac{417969}{1406005} a^{2} - \frac{308}{65565}$, $\frac{1}{12654045} a^{31} + \frac{2816}{843603} a^{23} - \frac{13924}{4218015} a^{21} + \frac{18184}{12654045} a^{19} + \frac{4179}{1406005} a^{17} + \frac{202132}{843603} a^{15} + \frac{276367}{1406005} a^{13} + \frac{562348}{4218015} a^{11} - \frac{635464}{1406005} a^{9} - \frac{715633}{4218015} a^{7} + \frac{51523}{843603} a^{5} + \frac{1257283}{4218015} a^{3} + \frac{1049318}{12654045} a$

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:  $31$
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:  \( 51823553187063560000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_8$ (as 32T37):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 32
The 32 conjugacy class representatives for $C_2^2\times C_8$
Character table for $C_2^2\times C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{14}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{42}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{6}, \sqrt{14})\), \(\Q(\sqrt{2}, \sqrt{7})\), \(\Q(\sqrt{3}, \sqrt{14})\), \(\Q(\sqrt{2}, \sqrt{21})\), \(\Q(\sqrt{3}, \sqrt{7})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{6}, \sqrt{7})\), \(\Q(\zeta_{16})^+\), 4.4.100352.1, 4.4.903168.2, 4.4.18432.1, 8.8.12745506816.1, 8.8.40282095616.1, 8.8.3262849744896.1, 8.8.815712436224.1, 8.8.815712436224.2, \(\Q(\zeta_{48})^+\), 8.8.3262849744896.2, 8.8.417644767346688.4, 8.8.173946175488.1, \(\Q(\zeta_{32})^+\), 8.8.5156108238848.1, 16.16.10646188457767892278050816.1, 16.16.697708606768276588334338277376.3, 16.16.106341808682864896865468416.1, 16.16.174427151692069147083584569344.1, 16.16.174427151692069147083584569344.2, 16.16.697708606768276588334338277376.2, \(\Q(\zeta_{96})^+\)

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 ${\href{/LocalNumberField/5.8.0.1}{8} }^{4}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

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
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$