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Properties

Label 32.32.448...808.1
Degree $32$
Signature $[32, 0]$
Discriminant $4.490\times 10^{78}$
Root discriminant \(287.00\)
Ramified primes $2,3,7$
Class number not computed
Class group not computed
Galois group $C_{32}$ (as 32T33)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 672*x^30 + 204624*x^28 - 37340352*x^26 + 4550855400*x^24 - 390766783680*x^22 + 24302688046560*x^20 - 1108202574923136*x^18 + 37090154929458708*x^16 - 903064641760733760*x^14 + 15688695730952383776*x^12 - 188264348771428605312*x^10 + 1482581746575000266832*x^8 - 7058769773166802889856*x^6 + 17646924432917007224640*x^4 - 17439313557235630669056*x^2 + 2861137380483970656642)
 
gp: K = bnfinit(y^32 - 672*y^30 + 204624*y^28 - 37340352*y^26 + 4550855400*y^24 - 390766783680*y^22 + 24302688046560*y^20 - 1108202574923136*y^18 + 37090154929458708*y^16 - 903064641760733760*y^14 + 15688695730952383776*y^12 - 188264348771428605312*y^10 + 1482581746575000266832*y^8 - 7058769773166802889856*y^6 + 17646924432917007224640*y^4 - 17439313557235630669056*y^2 + 2861137380483970656642, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 672*x^30 + 204624*x^28 - 37340352*x^26 + 4550855400*x^24 - 390766783680*x^22 + 24302688046560*x^20 - 1108202574923136*x^18 + 37090154929458708*x^16 - 903064641760733760*x^14 + 15688695730952383776*x^12 - 188264348771428605312*x^10 + 1482581746575000266832*x^8 - 7058769773166802889856*x^6 + 17646924432917007224640*x^4 - 17439313557235630669056*x^2 + 2861137380483970656642);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - 672*x^30 + 204624*x^28 - 37340352*x^26 + 4550855400*x^24 - 390766783680*x^22 + 24302688046560*x^20 - 1108202574923136*x^18 + 37090154929458708*x^16 - 903064641760733760*x^14 + 15688695730952383776*x^12 - 188264348771428605312*x^10 + 1482581746575000266832*x^8 - 7058769773166802889856*x^6 + 17646924432917007224640*x^4 - 17439313557235630669056*x^2 + 2861137380483970656642)
 

\( x^{32} - 672 x^{30} + 204624 x^{28} - 37340352 x^{26} + 4550855400 x^{24} - 390766783680 x^{22} + \cdots + 28\!\cdots\!42 \) 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:  $[32, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(4489912604053908534055314729400632754872833954383027744299245049859706934263808\) \(\medspace = 2^{191}\cdot 3^{16}\cdot 7^{16}\) 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:  \(287.00\)
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:  $2^{191/32}3^{1/2}7^{1/2}\approx 287.00035503909834$
Ramified primes:   \(2\), \(3\), \(7\) 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:  \(\Q(\sqrt{2}) \)
$\card{ \Gal(K/\Q) }$:  $32$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2688=2^{7}\cdot 3\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{2688}(1,·)$, $\chi_{2688}(2309,·)$, $\chi_{2688}(2185,·)$, $\chi_{2688}(1805,·)$, $\chi_{2688}(1681,·)$, $\chi_{2688}(1301,·)$, $\chi_{2688}(1177,·)$, $\chi_{2688}(797,·)$, $\chi_{2688}(673,·)$, $\chi_{2688}(293,·)$, $\chi_{2688}(169,·)$, $\chi_{2688}(2477,·)$, $\chi_{2688}(2353,·)$, $\chi_{2688}(1973,·)$, $\chi_{2688}(1849,·)$, $\chi_{2688}(1469,·)$, $\chi_{2688}(1345,·)$, $\chi_{2688}(965,·)$, $\chi_{2688}(841,·)$, $\chi_{2688}(461,·)$, $\chi_{2688}(337,·)$, $\chi_{2688}(2645,·)$, $\chi_{2688}(2521,·)$, $\chi_{2688}(2141,·)$, $\chi_{2688}(2017,·)$, $\chi_{2688}(1637,·)$, $\chi_{2688}(1513,·)$, $\chi_{2688}(1133,·)$, $\chi_{2688}(1009,·)$, $\chi_{2688}(629,·)$, $\chi_{2688}(505,·)$, $\chi_{2688}(125,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $\frac{1}{21}a^{2}$, $\frac{1}{21}a^{3}$, $\frac{1}{441}a^{4}$, $\frac{1}{441}a^{5}$, $\frac{1}{9261}a^{6}$, $\frac{1}{9261}a^{7}$, $\frac{1}{194481}a^{8}$, $\frac{1}{194481}a^{9}$, $\frac{1}{4084101}a^{10}$, $\frac{1}{4084101}a^{11}$, $\frac{1}{85766121}a^{12}$, $\frac{1}{85766121}a^{13}$, $\frac{1}{1801088541}a^{14}$, $\frac{1}{1801088541}a^{15}$, $\frac{1}{37822859361}a^{16}$, $\frac{1}{37822859361}a^{17}$, $\frac{1}{794280046581}a^{18}$, $\frac{1}{794280046581}a^{19}$, $\frac{1}{16679880978201}a^{20}$, $\frac{1}{16679880978201}a^{21}$, $\frac{1}{350277500542221}a^{22}$, $\frac{1}{350277500542221}a^{23}$, $\frac{1}{73\!\cdots\!41}a^{24}$, $\frac{1}{73\!\cdots\!41}a^{25}$, $\frac{1}{15\!\cdots\!61}a^{26}$, $\frac{1}{15\!\cdots\!61}a^{27}$, $\frac{1}{32\!\cdots\!81}a^{28}$, $\frac{1}{32\!\cdots\!81}a^{29}$, $\frac{1}{68\!\cdots\!01}a^{30}$, $\frac{1}{68\!\cdots\!01}a^{31}$ 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

not computed

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:  $31$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) 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:  not computed
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:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^32 - 672*x^30 + 204624*x^28 - 37340352*x^26 + 4550855400*x^24 - 390766783680*x^22 + 24302688046560*x^20 - 1108202574923136*x^18 + 37090154929458708*x^16 - 903064641760733760*x^14 + 15688695730952383776*x^12 - 188264348771428605312*x^10 + 1482581746575000266832*x^8 - 7058769773166802889856*x^6 + 17646924432917007224640*x^4 - 17439313557235630669056*x^2 + 2861137380483970656642)
 
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 - 672*x^30 + 204624*x^28 - 37340352*x^26 + 4550855400*x^24 - 390766783680*x^22 + 24302688046560*x^20 - 1108202574923136*x^18 + 37090154929458708*x^16 - 903064641760733760*x^14 + 15688695730952383776*x^12 - 188264348771428605312*x^10 + 1482581746575000266832*x^8 - 7058769773166802889856*x^6 + 17646924432917007224640*x^4 - 17439313557235630669056*x^2 + 2861137380483970656642, 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 - 672*x^30 + 204624*x^28 - 37340352*x^26 + 4550855400*x^24 - 390766783680*x^22 + 24302688046560*x^20 - 1108202574923136*x^18 + 37090154929458708*x^16 - 903064641760733760*x^14 + 15688695730952383776*x^12 - 188264348771428605312*x^10 + 1482581746575000266832*x^8 - 7058769773166802889856*x^6 + 17646924432917007224640*x^4 - 17439313557235630669056*x^2 + 2861137380483970656642);
 
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 - 672*x^30 + 204624*x^28 - 37340352*x^26 + 4550855400*x^24 - 390766783680*x^22 + 24302688046560*x^20 - 1108202574923136*x^18 + 37090154929458708*x^16 - 903064641760733760*x^14 + 15688695730952383776*x^12 - 188264348771428605312*x^10 + 1482581746575000266832*x^8 - 7058769773166802889856*x^6 + 17646924432917007224640*x^4 - 17439313557235630669056*x^2 + 2861137380483970656642);
 
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

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

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)
 
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.

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

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

# 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 $32$$32$$1$$191$
\(3\) Copy content Toggle raw display Deg $32$$2$$16$$16$
\(7\) Copy content Toggle raw display 7.16.8.2$x^{16} - 67228 x^{6} + 705894 x^{4} - 1647086 x^{2} + 17294403$$2$$8$$8$$C_{16}$$[\ ]_{2}^{8}$
7.16.8.2$x^{16} - 67228 x^{6} + 705894 x^{4} - 1647086 x^{2} + 17294403$$2$$8$$8$$C_{16}$$[\ ]_{2}^{8}$