Properties

Label 32.32.2088443876...4768.1
Degree $32$
Signature $[32, 0]$
Discriminant $2^{191}\cdot 13^{16}$
Root discriminant $225.81$
Ramified primes $2, 13$
Class number Not computed
Class group Not computed
Galois group $C_{32}$ (as 32T33)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1330833218366359682, 0, -13103588611607233792, 0, 21419327538204132160, 0, -13840180870839593088, 0, 4695775652606290512, 0, -963236031303854464, 0, 129666388829365024, 0, -12056891607126080, 0, 799928385472788, 0, -38608911516032, 0, 1367724598240, 0, -35525314240, 0, 668327400, 0, -8858304, 0, 78416, 0, -416, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 416*x^30 + 78416*x^28 - 8858304*x^26 + 668327400*x^24 - 35525314240*x^22 + 1367724598240*x^20 - 38608911516032*x^18 + 799928385472788*x^16 - 12056891607126080*x^14 + 129666388829365024*x^12 - 963236031303854464*x^10 + 4695775652606290512*x^8 - 13840180870839593088*x^6 + 21419327538204132160*x^4 - 13103588611607233792*x^2 + 1330833218366359682)
 
gp: K = bnfinit(x^32 - 416*x^30 + 78416*x^28 - 8858304*x^26 + 668327400*x^24 - 35525314240*x^22 + 1367724598240*x^20 - 38608911516032*x^18 + 799928385472788*x^16 - 12056891607126080*x^14 + 129666388829365024*x^12 - 963236031303854464*x^10 + 4695775652606290512*x^8 - 13840180870839593088*x^6 + 21419327538204132160*x^4 - 13103588611607233792*x^2 + 1330833218366359682, 1)
 

Normalized defining polynomial

\( x^{32} - 416 x^{30} + 78416 x^{28} - 8858304 x^{26} + 668327400 x^{24} - 35525314240 x^{22} + 1367724598240 x^{20} - 38608911516032 x^{18} + 799928385472788 x^{16} - 12056891607126080 x^{14} + 129666388829365024 x^{12} - 963236031303854464 x^{10} + 4695775652606290512 x^{8} - 13840180870839593088 x^{6} + 21419327538204132160 x^{4} - 13103588611607233792 x^{2} + 1330833218366359682 \)

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:  \(2088443876129429457733048543333054873029337200425307489036314630977400864768=2^{191}\cdot 13^{16}\)
magma: Discriminant(K);
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $225.81$
magma: Abs(Discriminant(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(K));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1664=2^{7}\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{1664}(1,·)$, $\chi_{1664}(389,·)$, $\chi_{1664}(521,·)$, $\chi_{1664}(909,·)$, $\chi_{1664}(1041,·)$, $\chi_{1664}(1429,·)$, $\chi_{1664}(1561,·)$, $\chi_{1664}(285,·)$, $\chi_{1664}(417,·)$, $\chi_{1664}(805,·)$, $\chi_{1664}(937,·)$, $\chi_{1664}(1325,·)$, $\chi_{1664}(1457,·)$, $\chi_{1664}(181,·)$, $\chi_{1664}(313,·)$, $\chi_{1664}(701,·)$, $\chi_{1664}(833,·)$, $\chi_{1664}(1221,·)$, $\chi_{1664}(1353,·)$, $\chi_{1664}(77,·)$, $\chi_{1664}(209,·)$, $\chi_{1664}(597,·)$, $\chi_{1664}(729,·)$, $\chi_{1664}(1117,·)$, $\chi_{1664}(1249,·)$, $\chi_{1664}(1637,·)$, $\chi_{1664}(105,·)$, $\chi_{1664}(493,·)$, $\chi_{1664}(625,·)$, $\chi_{1664}(1013,·)$, $\chi_{1664}(1145,·)$, $\chi_{1664}(1533,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $\frac{1}{13} a^{2}$, $\frac{1}{13} a^{3}$, $\frac{1}{169} a^{4}$, $\frac{1}{169} a^{5}$, $\frac{1}{2197} a^{6}$, $\frac{1}{2197} a^{7}$, $\frac{1}{28561} a^{8}$, $\frac{1}{28561} a^{9}$, $\frac{1}{371293} a^{10}$, $\frac{1}{371293} a^{11}$, $\frac{1}{4826809} a^{12}$, $\frac{1}{4826809} a^{13}$, $\frac{1}{62748517} a^{14}$, $\frac{1}{62748517} a^{15}$, $\frac{1}{815730721} a^{16}$, $\frac{1}{815730721} a^{17}$, $\frac{1}{10604499373} a^{18}$, $\frac{1}{10604499373} a^{19}$, $\frac{1}{137858491849} a^{20}$, $\frac{1}{137858491849} a^{21}$, $\frac{1}{1792160394037} a^{22}$, $\frac{1}{1792160394037} a^{23}$, $\frac{1}{23298085122481} a^{24}$, $\frac{1}{23298085122481} a^{25}$, $\frac{1}{302875106592253} a^{26}$, $\frac{1}{302875106592253} a^{27}$, $\frac{1}{3937376385699289} a^{28}$, $\frac{1}{3937376385699289} a^{29}$, $\frac{1}{51185893014090757} a^{30}$, $\frac{1}{51185893014090757} a^{31}$

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:  $31$
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:  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

$C_{32}$ (as 32T33):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 32
The 32 conjugacy class representatives for $C_{32}$
Character table for $C_{32}$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), \(\Q(\zeta_{32})^+\), \(\Q(\zeta_{64})^+\)

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 $32$ $32$ $16^{2}$ $32$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{4}$ $32$ $16^{2}$ $32$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ $32$ $16^{2}$ $32$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{4}$ $32$ $32$

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