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Properties

Label 32.0.230...944.1
Degree $32$
Signature $[0, 16]$
Discriminant $2.306\times 10^{51}$
Root discriminant \(40.28\)
Ramified primes $2,17$
Class number $72$ (GRH)
Class group [3, 24] (GRH)
Galois group $C_2\times C_{16}$ (as 32T32)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 2*x^30 + 4*x^28 - 8*x^26 + 16*x^24 - 32*x^22 + 64*x^20 - 128*x^18 + 256*x^16 - 512*x^14 + 1024*x^12 - 2048*x^10 + 4096*x^8 - 8192*x^6 + 16384*x^4 - 32768*x^2 + 65536)
 
gp: K = bnfinit(y^32 - 2*y^30 + 4*y^28 - 8*y^26 + 16*y^24 - 32*y^22 + 64*y^20 - 128*y^18 + 256*y^16 - 512*y^14 + 1024*y^12 - 2048*y^10 + 4096*y^8 - 8192*y^6 + 16384*y^4 - 32768*y^2 + 65536, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 2*x^30 + 4*x^28 - 8*x^26 + 16*x^24 - 32*x^22 + 64*x^20 - 128*x^18 + 256*x^16 - 512*x^14 + 1024*x^12 - 2048*x^10 + 4096*x^8 - 8192*x^6 + 16384*x^4 - 32768*x^2 + 65536);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - 2*x^30 + 4*x^28 - 8*x^26 + 16*x^24 - 32*x^22 + 64*x^20 - 128*x^18 + 256*x^16 - 512*x^14 + 1024*x^12 - 2048*x^10 + 4096*x^8 - 8192*x^6 + 16384*x^4 - 32768*x^2 + 65536)
 

\( x^{32} - 2 x^{30} + 4 x^{28} - 8 x^{26} + 16 x^{24} - 32 x^{22} + 64 x^{20} - 128 x^{18} + 256 x^{16} + \cdots + 65536 \) Copy content Toggle raw display

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

Invariants

Degree:  $32$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 16]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(2306255574353269294321282113641705264626385702354944\) \(\medspace = 2^{48}\cdot 17^{30}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(40.28\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2^{3/2}17^{15/16}\approx 40.28012942155011$
Ramified primes:   \(2\), \(17\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $32$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(136=2^{3}\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{136}(1,·)$, $\chi_{136}(3,·)$, $\chi_{136}(129,·)$, $\chi_{136}(9,·)$, $\chi_{136}(11,·)$, $\chi_{136}(19,·)$, $\chi_{136}(25,·)$, $\chi_{136}(27,·)$, $\chi_{136}(33,·)$, $\chi_{136}(35,·)$, $\chi_{136}(41,·)$, $\chi_{136}(43,·)$, $\chi_{136}(49,·)$, $\chi_{136}(57,·)$, $\chi_{136}(59,·)$, $\chi_{136}(65,·)$, $\chi_{136}(67,·)$, $\chi_{136}(73,·)$, $\chi_{136}(75,·)$, $\chi_{136}(81,·)$, $\chi_{136}(83,·)$, $\chi_{136}(89,·)$, $\chi_{136}(91,·)$, $\chi_{136}(97,·)$, $\chi_{136}(99,·)$, $\chi_{136}(131,·)$, $\chi_{136}(105,·)$, $\chi_{136}(107,·)$, $\chi_{136}(113,·)$, $\chi_{136}(115,·)$, $\chi_{136}(121,·)$, $\chi_{136}(123,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{32768}$

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

$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{4}a^{5}$, $\frac{1}{8}a^{6}$, $\frac{1}{8}a^{7}$, $\frac{1}{16}a^{8}$, $\frac{1}{16}a^{9}$, $\frac{1}{32}a^{10}$, $\frac{1}{32}a^{11}$, $\frac{1}{64}a^{12}$, $\frac{1}{64}a^{13}$, $\frac{1}{128}a^{14}$, $\frac{1}{128}a^{15}$, $\frac{1}{256}a^{16}$, $\frac{1}{256}a^{17}$, $\frac{1}{512}a^{18}$, $\frac{1}{512}a^{19}$, $\frac{1}{1024}a^{20}$, $\frac{1}{1024}a^{21}$, $\frac{1}{2048}a^{22}$, $\frac{1}{2048}a^{23}$, $\frac{1}{4096}a^{24}$, $\frac{1}{4096}a^{25}$, $\frac{1}{8192}a^{26}$, $\frac{1}{8192}a^{27}$, $\frac{1}{16384}a^{28}$, $\frac{1}{16384}a^{29}$, $\frac{1}{32768}a^{30}$, $\frac{1}{32768}a^{31}$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

$C_{3}\times C_{24}$, which has order $72$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $15$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -\frac{1}{4096} a^{24} \)  (order $34$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{1}{4}a^{4}+1$, $\frac{1}{256}a^{16}+\frac{1}{16}a^{8}+1$, $\frac{1}{64}a^{12}+1$, $\frac{1}{2}a^{2}-1$, $\frac{1}{16}a^{8}+1$, $\frac{1}{32768}a^{30}-\frac{1}{16384}a^{28}-\frac{1}{1024}a^{20}-\frac{1}{64}a^{12}-\frac{1}{4}a^{4}$, $\frac{1}{1024}a^{20}-\frac{1}{32}a^{10}$, $\frac{1}{8}a^{6}-\frac{1}{4}a^{5}+\frac{1}{4}a^{4}$, $\frac{1}{32768}a^{30}-\frac{1}{16384}a^{28}+\frac{1}{8192}a^{26}-\frac{1}{4096}a^{24}+\frac{1}{2048}a^{22}+\frac{1}{1024}a^{21}-\frac{1}{1024}a^{20}+\frac{1}{512}a^{18}-\frac{1}{256}a^{16}+\frac{1}{128}a^{14}-\frac{1}{64}a^{12}+\frac{1}{16}a^{10}-\frac{1}{16}a^{8}+\frac{1}{8}a^{6}-\frac{1}{4}a^{4}+\frac{1}{2}a^{2}-1$, $\frac{1}{512}a^{18}+\frac{1}{16}a^{9}+1$, $\frac{1}{64}a^{12}+\frac{1}{8}a^{7}+\frac{1}{2}a^{2}$, $\frac{1}{16384}a^{28}+\frac{1}{128}a^{15}+\frac{1}{2}a^{2}$, $\frac{1}{32768}a^{30}-\frac{1}{32}a^{10}+\frac{1}{2}a^{3}$, $\frac{1}{1024}a^{20}+\frac{1}{256}a^{17}+\frac{1}{128}a^{14}$, $\frac{1}{16384}a^{29}+\frac{1}{128}a^{14}-\frac{1}{32}a^{10}$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 598124168304.8442 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 598124168304.8442 \cdot 72}{34\cdot\sqrt{2306255574353269294321282113641705264626385702354944}}\cr\approx \mathstrut & 0.155621324660132 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^32 - 2*x^30 + 4*x^28 - 8*x^26 + 16*x^24 - 32*x^22 + 64*x^20 - 128*x^18 + 256*x^16 - 512*x^14 + 1024*x^12 - 2048*x^10 + 4096*x^8 - 8192*x^6 + 16384*x^4 - 32768*x^2 + 65536)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^32 - 2*x^30 + 4*x^28 - 8*x^26 + 16*x^24 - 32*x^22 + 64*x^20 - 128*x^18 + 256*x^16 - 512*x^14 + 1024*x^12 - 2048*x^10 + 4096*x^8 - 8192*x^6 + 16384*x^4 - 32768*x^2 + 65536, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^32 - 2*x^30 + 4*x^28 - 8*x^26 + 16*x^24 - 32*x^22 + 64*x^20 - 128*x^18 + 256*x^16 - 512*x^14 + 1024*x^12 - 2048*x^10 + 4096*x^8 - 8192*x^6 + 16384*x^4 - 32768*x^2 + 65536);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - 2*x^30 + 4*x^28 - 8*x^26 + 16*x^24 - 32*x^22 + 64*x^20 - 128*x^18 + 256*x^16 - 512*x^14 + 1024*x^12 - 2048*x^10 + 4096*x^8 - 8192*x^6 + 16384*x^4 - 32768*x^2 + 65536);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$C_2\times C_{16}$ (as 32T32):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
An abelian group of order 32
The 32 conjugacy class representatives for $C_2\times C_{16}$
Character table for $C_2\times C_{16}$ is not computed

Intermediate fields

\(\Q(\sqrt{-2}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-34}) \), \(\Q(\sqrt{-2}, \sqrt{17})\), 4.4.4913.1, 4.0.314432.2, 8.0.98867482624.1, \(\Q(\zeta_{17})^+\), 8.0.1680747204608.1, 16.0.2824911165797606216433664.1, \(\Q(\zeta_{17})\), 16.16.48023489818559305679372288.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R $16^{2}$ $16^{2}$ $16^{2}$ $16^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{8}$ R ${\href{/padicField/19.8.0.1}{8} }^{4}$ $16^{2}$ $16^{2}$ $16^{2}$ $16^{2}$ $16^{2}$ ${\href{/padicField/43.8.0.1}{8} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{8}$ ${\href{/padicField/53.8.0.1}{8} }^{4}$ ${\href{/padicField/59.8.0.1}{8} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display Deg $16$$2$$8$$24$
Deg $16$$2$$8$$24$
\(17\) Copy content Toggle raw display 17.16.15.5$x^{16} + 17$$16$$1$$15$$C_{16}$$[\ ]_{16}$
17.16.15.5$x^{16} + 17$$16$$1$$15$$C_{16}$$[\ ]_{16}$