↑

Properties

Label 32.0.43005107648...1296.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{96}\cdot 13^{24}$
Root discriminant $54.77$
Ramified primes $2, 13$
Class number $375$ (GRH)
Class group $[5, 5, 15]$ (GRH)
Galois group $C_2\times C_4^2$ (as 32T36)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6561, 0, 0, 0, 0, 0, 0, 0, 35572, 0, 0, 0, 0, 0, 0, 0, 117486, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 3*x^24 + 117486*x^16 + 35572*x^8 + 6561)
 
gp: K = bnfinit(x^32 - 3*x^24 + 117486*x^16 + 35572*x^8 + 6561, 1)
 

Normalized defining polynomial

\( x^{32} - 3 x^{24} + 117486 x^{16} + 35572 x^{8} + 6561 \)

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:  \(43005107648088506255732033296018914516707307660871991296=2^{96}\cdot 13^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $54.77$
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:  $2, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(208=2^{4}\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{208}(1,·)$, $\chi_{208}(131,·)$, $\chi_{208}(5,·)$, $\chi_{208}(129,·)$, $\chi_{208}(21,·)$, $\chi_{208}(151,·)$, $\chi_{208}(25,·)$, $\chi_{208}(155,·)$, $\chi_{208}(157,·)$, $\chi_{208}(31,·)$, $\chi_{208}(161,·)$, $\chi_{208}(27,·)$, $\chi_{208}(135,·)$, $\chi_{208}(47,·)$, $\chi_{208}(177,·)$, $\chi_{208}(51,·)$, $\chi_{208}(181,·)$, $\chi_{208}(183,·)$, $\chi_{208}(57,·)$, $\chi_{208}(187,·)$, $\chi_{208}(53,·)$, $\chi_{208}(73,·)$, $\chi_{208}(203,·)$, $\chi_{208}(77,·)$, $\chi_{208}(79,·)$, $\chi_{208}(83,·)$, $\chi_{208}(207,·)$, $\chi_{208}(99,·)$, $\chi_{208}(103,·)$, $\chi_{208}(105,·)$, $\chi_{208}(109,·)$, $\chi_{208}(125,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3} a$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{5}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{6}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{7}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{8}$, $\frac{1}{3} a^{17} - \frac{1}{3} a$, $\frac{1}{9} a^{18} - \frac{1}{9} a^{10} - \frac{2}{9} a^{2}$, $\frac{1}{27} a^{19} + \frac{2}{27} a^{11} - \frac{8}{27} a^{3}$, $\frac{1}{27} a^{20} + \frac{2}{27} a^{12} - \frac{8}{27} a^{4}$, $\frac{1}{27} a^{21} + \frac{2}{27} a^{13} - \frac{8}{27} a^{5}$, $\frac{1}{27} a^{22} + \frac{2}{27} a^{14} - \frac{8}{27} a^{6}$, $\frac{1}{27} a^{23} + \frac{2}{27} a^{15} - \frac{8}{27} a^{7}$, $\frac{1}{24421218093} a^{24} - \frac{2831412994}{24421218093} a^{16} + \frac{12033653563}{24421218093} a^{8} + \frac{169209443}{904489559}$, $\frac{1}{24421218093} a^{25} - \frac{2831412994}{24421218093} a^{17} + \frac{3893247532}{24421218093} a^{9} - \frac{396861230}{2713468677} a$, $\frac{1}{24421218093} a^{26} - \frac{117944317}{24421218093} a^{18} + \frac{1179778855}{24421218093} a^{10} - \frac{2999562808}{8140406031} a^{2}$, $\frac{1}{24421218093} a^{27} - \frac{117944317}{24421218093} a^{19} + \frac{1179778855}{24421218093} a^{11} - \frac{2999562808}{8140406031} a^{3}$, $\frac{1}{73263654279} a^{28} - \frac{113603764}{8140406031} a^{20} - \frac{2923202098}{24421218093} a^{12} - \frac{9903177983}{73263654279} a^{4}$, $\frac{1}{219790962837} a^{29} + \frac{563678267}{73263654279} a^{21} - \frac{371407660}{24421218093} a^{13} - \frac{31610927399}{219790962837} a^{5}$, $\frac{1}{659372888511} a^{30} - \frac{2149790410}{219790962837} a^{22} + \frac{5960019253}{73263654279} a^{14} + \frac{326566937965}{659372888511} a^{6}$, $\frac{1}{1978118665533} a^{31} - \frac{2149790410}{659372888511} a^{23} - \frac{18461198840}{219790962837} a^{15} - \frac{552596913383}{1978118665533} a^{7}$

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

Class group and class number

$C_{5}\times C_{5}\times C_{15}$, which has order $375$ (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:  \( \frac{327797006}{1978118665533} a^{31} - \frac{343926968}{659372888511} a^{23} + \frac{4279075670711}{219790962837} a^{15} + \frac{5965382536868}{1978118665533} a^{7} \) (order $16$)
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:  \( 6740125694616.533 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_4^2$ (as 32T36):

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 $C_2\times C_4^2$
Character table for $C_2\times C_4^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{26}) \), \(\Q(\sqrt{-26}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-13}) \), \(\Q(\zeta_{8})\), \(\Q(\zeta_{16})^+\), 4.0.2048.2, \(\Q(i, \sqrt{26})\), \(\Q(i, \sqrt{13})\), \(\Q(\sqrt{2}, \sqrt{13})\), \(\Q(\sqrt{2}, \sqrt{-13})\), 4.4.346112.1, 4.0.346112.2, \(\Q(\sqrt{-2}, \sqrt{-13})\), \(\Q(\sqrt{-2}, \sqrt{13})\), 4.4.4499456.1, 4.0.4499456.1, 4.4.4499456.2, 4.0.4499456.2, 4.4.35152.1, 4.0.2197.1, 4.4.140608.1, 4.0.140608.2, \(\Q(\zeta_{16})\), 8.0.1871773696.1, 8.0.479174066176.3, 8.8.119793516544.1, 8.0.479174066176.2, 8.0.119793516544.1, 8.0.479174066176.1, 8.0.80980417183744.3, 8.0.80980417183744.4, 8.0.1235663104.1, 8.0.316329754624.2, 8.8.20245104295936.1, 8.0.20245104295936.1, 8.8.316329754624.1, 8.0.19770609664.2, 8.0.80980417183744.2, 8.0.80980417183744.1, 8.0.316329754624.1, 8.0.19770609664.1, 16.0.229607785695641627262976.1, 16.0.6557827967253220516257857536.2, 16.0.100064513660480049381376.1, 16.16.6557827967253220516257857536.1, 16.0.409864247953326282266116096.2, 16.0.409864247953326282266116096.1, 16.0.6557827967253220516257857536.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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
13Data not computed