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Properties

Label 32.0.12676506002...0000.2
Degree $32$
Signature $[0, 16]$
Discriminant $2^{124}\cdot 5^{24}$
Root discriminant $49.06$
Ramified primes $2, 5$
Class number $1845$ (GRH)
Class group $[3, 615]$ (GRH)
Galois group $C_4\times C_8$ (as 32T43)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, 0, 128, 0, 864, 0, 5696, 0, 37432, 0, 50240, 0, 49952, 0, 44096, 0, 35492, 0, 17664, 0, 7440, 0, 2800, 0, 910, 0, 208, 0, 44, 0, 8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 8*x^30 + 44*x^28 + 208*x^26 + 910*x^24 + 2800*x^22 + 7440*x^20 + 17664*x^18 + 35492*x^16 + 44096*x^14 + 49952*x^12 + 50240*x^10 + 37432*x^8 + 5696*x^6 + 864*x^4 + 128*x^2 + 16)
 
gp: K = bnfinit(x^32 + 8*x^30 + 44*x^28 + 208*x^26 + 910*x^24 + 2800*x^22 + 7440*x^20 + 17664*x^18 + 35492*x^16 + 44096*x^14 + 49952*x^12 + 50240*x^10 + 37432*x^8 + 5696*x^6 + 864*x^4 + 128*x^2 + 16, 1)
 

Normalized defining polynomial

\( x^{32} + 8 x^{30} + 44 x^{28} + 208 x^{26} + 910 x^{24} + 2800 x^{22} + 7440 x^{20} + 17664 x^{18} + 35492 x^{16} + 44096 x^{14} + 49952 x^{12} + 50240 x^{10} + 37432 x^{8} + 5696 x^{6} + 864 x^{4} + 128 x^{2} + 16 \)

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

Invariants

Degree:  $32$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 16]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1267650600228229401496703205376000000000000000000000000=2^{124}\cdot 5^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.06$
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:  $2, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(160=2^{5}\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{160}(1,·)$, $\chi_{160}(133,·)$, $\chi_{160}(129,·)$, $\chi_{160}(9,·)$, $\chi_{160}(13,·)$, $\chi_{160}(17,·)$, $\chi_{160}(21,·)$, $\chi_{160}(153,·)$, $\chi_{160}(157,·)$, $\chi_{160}(33,·)$, $\chi_{160}(37,·)$, $\chi_{160}(41,·)$, $\chi_{160}(29,·)$, $\chi_{160}(49,·)$, $\chi_{160}(53,·)$, $\chi_{160}(137,·)$, $\chi_{160}(57,·)$, $\chi_{160}(61,·)$, $\chi_{160}(69,·)$, $\chi_{160}(73,·)$, $\chi_{160}(77,·)$, $\chi_{160}(141,·)$, $\chi_{160}(81,·)$, $\chi_{160}(89,·)$, $\chi_{160}(93,·)$, $\chi_{160}(97,·)$, $\chi_{160}(101,·)$, $\chi_{160}(109,·)$, $\chi_{160}(113,·)$, $\chi_{160}(117,·)$, $\chi_{160}(121,·)$, $\chi_{160}(149,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{2} a^{12}$, $\frac{1}{2} a^{13}$, $\frac{1}{2} a^{14}$, $\frac{1}{2} a^{15}$, $\frac{1}{4} a^{16}$, $\frac{1}{4} a^{17}$, $\frac{1}{124} a^{18} - \frac{15}{124} a^{16} + \frac{2}{31} a^{14} + \frac{1}{31} a^{12} + \frac{1}{62} a^{10} + \frac{13}{62} a^{8} + \frac{11}{31} a^{6} - \frac{10}{31} a^{4} - \frac{5}{31} a^{2} + \frac{13}{31}$, $\frac{1}{124} a^{19} - \frac{15}{124} a^{17} + \frac{2}{31} a^{15} + \frac{1}{31} a^{13} + \frac{1}{62} a^{11} + \frac{13}{62} a^{9} + \frac{11}{31} a^{7} - \frac{10}{31} a^{5} - \frac{5}{31} a^{3} + \frac{13}{31} a$, $\frac{1}{124} a^{20} - \frac{3}{62} a^{10} + \frac{9}{31}$, $\frac{1}{124} a^{21} - \frac{3}{62} a^{11} + \frac{9}{31} a$, $\frac{1}{124} a^{22} - \frac{3}{62} a^{12} + \frac{9}{31} a^{2}$, $\frac{1}{124} a^{23} - \frac{3}{62} a^{13} + \frac{9}{31} a^{3}$, $\frac{1}{248} a^{24} + \frac{7}{31} a^{14} - \frac{11}{31} a^{4}$, $\frac{1}{248} a^{25} + \frac{7}{31} a^{15} - \frac{11}{31} a^{5}$, $\frac{1}{102362248} a^{26} - \frac{3173}{1651004} a^{24} + \frac{85069}{51181124} a^{22} + \frac{85657}{25590562} a^{20} + \frac{79539}{25590562} a^{18} - \frac{411907}{51181124} a^{16} - \frac{573153}{25590562} a^{14} + \frac{31366}{12795281} a^{12} + \frac{3144819}{25590562} a^{10} - \frac{3841175}{25590562} a^{8} - \frac{5143468}{12795281} a^{6} - \frac{3127171}{12795281} a^{4} - \frac{1006672}{12795281} a^{2} + \frac{4928301}{12795281}$, $\frac{1}{102362248} a^{27} - \frac{3173}{1651004} a^{25} + \frac{85069}{51181124} a^{23} + \frac{85657}{25590562} a^{21} + \frac{79539}{25590562} a^{19} - \frac{411907}{51181124} a^{17} - \frac{573153}{25590562} a^{15} + \frac{31366}{12795281} a^{13} + \frac{3144819}{25590562} a^{11} - \frac{3841175}{25590562} a^{9} - \frac{5143468}{12795281} a^{7} - \frac{3127171}{12795281} a^{5} - \frac{1006672}{12795281} a^{3} + \frac{4928301}{12795281} a$, $\frac{1}{3173229688} a^{28} - \frac{15}{3173229688} a^{26} - \frac{1353545}{3173229688} a^{24} + \frac{2071545}{793307422} a^{22} - \frac{4501075}{1586614844} a^{20} - \frac{1974767}{1586614844} a^{18} + \frac{44767237}{1586614844} a^{16} + \frac{122974903}{793307422} a^{14} + \frac{11094585}{396653711} a^{12} - \frac{181713821}{793307422} a^{10} - \frac{35684806}{396653711} a^{8} - \frac{174213533}{396653711} a^{6} - \frac{90689607}{396653711} a^{4} - \frac{125886043}{396653711} a^{2} + \frac{9949030}{396653711}$, $\frac{1}{3173229688} a^{29} - \frac{15}{3173229688} a^{27} - \frac{1353545}{3173229688} a^{25} + \frac{2071545}{793307422} a^{23} - \frac{4501075}{1586614844} a^{21} - \frac{1974767}{1586614844} a^{19} + \frac{44767237}{1586614844} a^{17} + \frac{122974903}{793307422} a^{15} + \frac{11094585}{396653711} a^{13} - \frac{181713821}{793307422} a^{11} - \frac{35684806}{396653711} a^{9} - \frac{174213533}{396653711} a^{7} - \frac{90689607}{396653711} a^{5} - \frac{125886043}{396653711} a^{3} + \frac{9949030}{396653711} a$, $\frac{1}{3173229688} a^{30} - \frac{57208}{396653711} a^{20} - \frac{54751068}{396653711} a^{10} + \frac{130294233}{396653711}$, $\frac{1}{3173229688} a^{31} - \frac{57208}{396653711} a^{21} - \frac{54751068}{396653711} a^{11} + \frac{130294233}{396653711} a$

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

Class group and class number

$C_{3}\times C_{615}$, which has order $1845$ (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:  $15$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{7546805}{793307422} a^{30} + \frac{120744621}{1586614844} a^{28} + \frac{166029710}{396653711} a^{26} + \frac{784867720}{396653711} a^{24} + \frac{3433796275}{396653711} a^{22} + \frac{10565527000}{396653711} a^{20} + \frac{28075191666}{396653711} a^{18} + \frac{66653381760}{396653711} a^{16} + \frac{133925601530}{396653711} a^{14} + \frac{166391956640}{396653711} a^{12} + \frac{188489001680}{396653711} a^{10} + \frac{378845991847}{793307422} a^{8} + \frac{141246002380}{396653711} a^{6} + \frac{21493300640}{396653711} a^{4} + \frac{3260219760}{396653711} a^{2} + \frac{482995520}{396653711} \) (order $10$)
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:  \( 323273529801.8212 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times C_8$ (as 32T43):

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 $C_4\times C_8$
Character table for $C_4\times C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), \(\Q(\zeta_{5})\), \(\Q(\sqrt{2}, \sqrt{5})\), 4.0.8000.2, 4.4.51200.1, \(\Q(\zeta_{16})^+\), 4.0.256000.2, 4.0.256000.4, 8.0.64000000.2, 8.8.2621440000.1, 8.0.65536000000.1, 8.0.33554432000000.2, 8.0.33554432000000.1, \(\Q(\zeta_{32})^+\), 8.8.1342177280000.1, 16.0.4294967296000000000000.1, 16.0.1125899906842624000000000000.1, 16.16.1801439850948198400000000.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{32}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

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
5Data not computed